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Geometry in a Frechet Context:A Projective Limit Approach

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London Mathematical Society Lecture Notes Series 428 Cambridge University Press 2016

Geometry in a Frechet Context:

A Projective Limit Approach

by

C.T.J. DodsonUniversity of Manchester,

Manchester, UK

George GalanisHellenic Naval Academy,

Piraeus, Greece

Efstathios VassiliouNational and Kapodistrian University of Athens,

Athens, Greece

Contents

Preface ix

1 Banach manifolds and bundles 1

1.1 Banach manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Ordinary derivatives in Banach spaces . . . . . . . . . 1

1.1.2 Smooth structures . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Smooth maps . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.4 The tangent space . . . . . . . . . . . . . . . . . . . . 3

1.1.5 The tangent bundle . . . . . . . . . . . . . . . . . . . 4

1.1.6 The differential of a smooth map . . . . . . . . . . . . 4

1.1.7 Velocity vectors . . . . . . . . . . . . . . . . . . . . . . 5

1.1.8 The tangent map . . . . . . . . . . . . . . . . . . . . . 6

1.1.9 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.10 Related vector fields . . . . . . . . . . . . . . . . . . . 7

1.1.11 Integral curves . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Banach-Lie groups . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Basic notations . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 Invariant vector fields . . . . . . . . . . . . . . . . . . 8

1.2.3 The exponential map . . . . . . . . . . . . . . . . . . . 9

1.2.4 The adjoint representation . . . . . . . . . . . . . . . . 10

1.2.5 Lie algebra-valued differential forms . . . . . . . . . . 10

1.2.6 The Maurer-Cartan differentials . . . . . . . . . . . . 11

1.3 Smooth actions . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.2 Fundamental (Killing) vector fields . . . . . . . . . . . 13

1.4 Banach vector bundles . . . . . . . . . . . . . . . . . . . . . . 14

1.4.1 The structure of a Banach vector bundle . . . . . . . . 14

1.4.2 Transition maps . . . . . . . . . . . . . . . . . . . . . 16

1.4.3 Morphisms of vector bundles . . . . . . . . . . . . . . 17

v

vi Contents

1.4.4 Some useful constructions and examples . . . . . . . . 19

1.4.5 Exact sequences . . . . . . . . . . . . . . . . . . . . . 24

1.4.6 The exact sequence associated to a vector bundle . . . 26

1.5 Connections on vector bundles . . . . . . . . . . . . . . . . . 27

1.5.1 General definitions . . . . . . . . . . . . . . . . . . . . 27

1.5.2 Linear connections . . . . . . . . . . . . . . . . . . . . 29

1.5.3 The Christoffel symbols of a linear connection . . . . . 30

1.5.4 Linear connections and covariant derivations . . . . . 32

1.5.5 Parallel displacement and holonomy groups . . . . . . 33

1.5.6 Related linear connections . . . . . . . . . . . . . . . . 34

1.6 Banach principal bundles . . . . . . . . . . . . . . . . . . . . 37

1.6.1 The structure of a principal bundle . . . . . . . . . . . 37

1.6.2 Morphisms of principal bundles . . . . . . . . . . . . . 39

1.6.3 Natural sections and transition maps . . . . . . . . . . 40

1.6.4 The pull-back of a principal bundle . . . . . . . . . . . 43

1.6.5 The frame bundle of a vector bundle . . . . . . . . . . 44

1.6.6 Associated bundles . . . . . . . . . . . . . . . . . . . . 46

1.6.7 The exact sequence associated to a principal bundle . 52

1.7 Connections on principal bundles . . . . . . . . . . . . . . . . 53

1.7.1 Principal bundle connections as splitting G-morphisms 54

1.7.2 Connection forms . . . . . . . . . . . . . . . . . . . . . 55

1.7.3 Local connection forms . . . . . . . . . . . . . . . . . 56

1.7.4 Connections on the frame bundle . . . . . . . . . . . . 58

1.7.5 Related connections on principal bundles . . . . . . . 59

1.7.6 Connections on associated bundles . . . . . . . . . . . 63

1.8 The curvature of a principal connection . . . . . . . . . . . . 64

1.8.1 Curvature forms . . . . . . . . . . . . . . . . . . . . . 64

1.8.2 Flat connections . . . . . . . . . . . . . . . . . . . . . 67

1.9 Holonomy groups . . . . . . . . . . . . . . . . . . . . . . . . . 68

1.10 Classification of flat bundles . . . . . . . . . . . . . . . . . . . 71

2 Frechet spaces 75

2.1 The topology of Frechet spaces . . . . . . . . . . . . . . . . . 75

2.2 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . 81

2.3 Frechet spaces as projective limits . . . . . . . . . . . . . . . 84

2.4 Differential equations in Frechet spaces . . . . . . . . . . . . . 102

Contents vii

3 Frechet manifolds 107

3.1 Smooth structures on Frechet manifolds . . . . . . . . . . . . 108

3.2 The tangent bundle of a plb-manifold . . . . . . . . . . . . . 113

3.3 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

3.4 Frechet-Lie groups . . . . . . . . . . . . . . . . . . . . . . . . 123

3.5 Equations with Maurer-Cartan differential . . . . . . . . . . . 130

3.6 Differential forms . . . . . . . . . . . . . . . . . . . . . . . . . 133

4 Projective systems of principal bundles 141

4.1 Projective systems and Frechet principal bundles . . . . . . . 142

4.2 Connections on limit principal bundles . . . . . . . . . . . . 153

4.3 Parallel translations and holonomy groups . . . . . . . . . . . 162

4.4 The curvature of a plb-connection . . . . . . . . . . . . . . . 167

4.5 Flat plb-bundles . . . . . . . . . . . . . . . . . . . . . . . . . 174

5 Projective systems of vector bundles 185

5.1 A particular Frechet group . . . . . . . . . . . . . . . . . . . . 185

5.2 Projective systems and Frechet vector bundles . . . . . . . . . 187

5.3 Morphisms of plb-vector bundles . . . . . . . . . . . . . . . . 195

5.4 The sections of plb-vector bundles . . . . . . . . . . . . . . . 200

5.5 The pull-back of plb-vector bundles . . . . . . . . . . . . . . . 202

6 Examples of projective systems of bundles 209

6.1 Trivial examples of plb-vector bundles . . . . . . . . . . . . . 209

6.2 Plb-vector bundles of maps . . . . . . . . . . . . . . . . . . . 210

6.3 The infinite jet bundle . . . . . . . . . . . . . . . . . . . . . . 214

6.4 The tangent bundle of a plb-bundle . . . . . . . . . . . . . . . 215

6.5 The generalized frame bundle . . . . . . . . . . . . . . . . . . 218

6.6 Generalized associated bundles . . . . . . . . . . . . . . . . . 221

7 Connections on plb-vector bundles 227

7.1 Projective limits of linear connections . . . . . . . . . . . . . 227

7.2 Parallel displacement and holonomy groups . . . . . . . . . . 234

7.3 Connections on plb-vector and frame bundles . . . . . . . . . 240

8 Geometry of second order tangent bundles 247

8.1 The (first order) tangent bundle in brief . . . . . . . . . . . . 248

8.1.1 Linear connections on manifolds . . . . . . . . . . . . 248

8.1.2 First order differentials . . . . . . . . . . . . . . . . . 249

8.1.3 Related linear connections on manifolds . . . . . . . . 250

viii Contents

8.2 Second order tangent bundles . . . . . . . . . . . . . . . . . . 2508.3 Second order differentials . . . . . . . . . . . . . . . . . . . . 2558.4 Connection dependence of second order tangent bundles . . . 2588.5 Second order Frechet tangent bundles . . . . . . . . . . . . . 2598.6 Second order frame bundles . . . . . . . . . . . . . . . . . . . 264

Appendix: Further study 275

Bibliography 283

List of Notations 297

Subject index 305

Preface

The aim of the authors is to lay down the foundations of the projectivesystems of various geometrical structures modelled on Banach spaces, even-tually leading to homologous structures in the framework of Frechet differ-ential geometry, by overcoming some of the inherent deficiencies of Frechetspaces. We elaborate this brief description in the sequel.

Banach spaces, combining a metric topology (subordinate to a norm),and a linear space structure (for representing derivatives as linear approx-imations to functions in order to do calculus), provide a very convenientsetting for many problems in functional analysis, which we need for han-dling calculus on function spaces, usually infinite dimensional. They area relatively gentle extension from experience on finite dimensional spaces,since many topological properties of spaces and groups of linear maps, aswell as many of the existence and uniqueness theorems for solutions of dif-ferential equations carry over to the infinite dimensional case.

Manifolds and fibre bundles modelled on Banach spaces arise from thesynthesis of differential geometry and functional analysis, thus leading toimportant examples of global analysis. Indeed, many spaces of (differen-tiable) maps between appropriate manifolds admit the structure of Banachmanifolds (see, for instance, J. Eells [Eel66, § 6]).

On the other hand, as mentioned also in [Eel66], Riemannian manifolds,represented as rigid maps on infinite dimensional function spaces, arise asconfiguration spaces of dynamical systems, with metrics interpreted as ki-netic energy. Much of the calculus of variations and Morse theory is con-cerned with a function space in differential geometry—the Euler-Lagrangeoperator of a variational problem is interpreted as a gradient vector field,with integral curves the paths of steepest ascent. Some eigenvalue prob-lems in integral and differential equations are interpretable via Lagrangianmultipliers, involving infinite dimensional function spaces from differentialgeometry—such as focal point theory and geometric consequences of theinverse function theorem in infinite dimensions.

ix

x Preface

However, in a number of situations that have significance in global anal-ysis and physics, for example, physical field theory, Banach space repre-sentations break down. A first step forward is achieved by weakening thetopological requirements: Instead of a norm, a family of seminorms is con-sidered. This leads to Frechet spaces, which do have a linear structure andtheir topology is defined through a sequence of seminorms.

Although Frechet spaces seem to be very close to Banach spaces, a num-ber of critical deficiencies emerge in their framework. For instance, despitethe progress in particular cases, they lack a general solvability theory ofdifferential equations, even the linear ones; also, the space of continuouslinear morphisms between Frechet spaces does not remain in the category,and the space of linear isomorphisms does not admit a reasonable Lie groupstructure.

The situation becomes much more complicated when we consider mani-folds modelled on Frechet spaces. Fundamental tools such as the exponentialmap of a Frechet-Lie group may not exist. Additional complications becomeparticularly noticeable when we try to collect Frechet spaces together to formbundles (over manifolds modelled on atlases of Frechet spaces), in order todevelop geometrical operators like covariant derivatives and curvature toact on sections of bundles. The structure group of such bundles, being thegeneral linear group of a Frechet space, is not a Lie group—even worse, itdoes not have a natural topological structure. Parallel translations do notnecessarily exist because of the inherent difficulties in solving differentialequations within this framework, and so on.

This has relevance to real problems. The space of smooth functionsC∞(I,R), where I is a compact interval of R, is a Frechet space. Thespace C∞(M,V ), of smooth sections of a vector bundle V over a compactsmooth Riemannian manifold M with covariant derivative ∇, is a Frechetspace. The C∞ Riemannian metrics on a fixed closed finite-dimensionalorientable manifold has a Frechet model space. Frechet spaces of sectionsarise naturally as configurations of a physical field. Then the moduli space,consisting of inequivalent configurations of the physical field, is the quotientof the infinite-dimensional configuration space X by the appropriate sym-metry gauge group. Typically, X is modelled on a Frechet space of smoothsections of a vector bundle over a closed manifold.

Despite their apparent differences, the categories of Banach and Frechetspaces are connected through projective limits. Indeed, the simplest exam-ple of this situation is the limiting real product space R∞ = limn→∞Rn.Taking notice of how R∞ arises from Rn, this approach extends to arbi-trary Frechet spaces, since always they can be represented by a countable

Preface xi

sequence of Banach spaces in a somewhat similar manner. Although carefulconcentration to the above example is salutary, (bringing to mind the storyof the mathematician drafted to work on a strategic radar project some 70years ago, who when told of the context said “but I only know Ohms Law!”and the response came, “you only need to know Ohms Law, but you mustknow it very, very well”), it should be emphasized that the mere propertiesof R∞ do not answer all the questions and problems referring to the morecomplicated geometrical structures mentioned above.

The approach adopted is designed to investigate, in a systematic way,the extent to which the shortcomings of the Frechet context can be workedround by viewing, under sufficient conditions, geometrical objects and prop-erties in this context as limits of sequences of their Banach counterparts,thus exploiting the well developed geometrical tools of the latter. In thisrespect, we propose, among other generalizations, the replacement of cer-tain pathological structures and spaces such as the structural group of aFrechet bundle, various spaces of linear maps, frame bundles, connectionson principal and vector bundles etc., by appropriate entities, susceptible tothe limit process. This extends many classical results to our framework and,to a certain degree, bypasses its drawbacks.

Apart from the problem of solving differential equations, much of ourwork is motivated also by the need to endow infinite-dimensional Lie groupswith an exponential map [a fact characterizing–axiomatically–the categoryof (infinite-dimensional) regular Lie groups]; the differential and vector bun-dle structure of the set of infinite jets of sections of a Banach vector bundle(compare with the differential structure described in [Tak79]); the need toput in a wider perspective particular cases of projective limits of mani-folds and Lie groups appearing in physics (see e.g. [AM99], [AI92], [AL94],[Bae93]) or in various groups of diffeomorphisms (e.g. [Les67], [Omo70]).

For the convenience of the reader, we give an outline of the presentation,referring for more details to the table of contents and the introduction toeach chapter.

Chapter 1 introduces the basic notions and results on Banach manifolds andbundles, with special emphasis on their geometry. Since there is not a sys-tematic treatment of the general theory of connections on Banach principaland vector bundles (apart from numerous papers, with some very funda-mental ones among them), occasionally we include extra details on specifictopics, according to the needs of subsequent chapters. With a few excep-tions, there are not proofs in this chapter and the reader is guided to theliterature for details. This is to keep the notes within a reasonable size;

xii Preface

however, the subsequent chapters are essentially self-contained.

Chapter 2 contains a brief account of the structure of Frechet spaces and thedifferentiability method applied therein. From various possible differentia-bility methods we have chosen to apply that of J.A. Leslie [Les67], [Les68],a particular case of Gateaux differentiation which fits well to the structureof locally convex spaces, without recourse to other topologies. Among themain features of this chapter we mention the representation of a Frechetspace by a projective limit of Banach spaces, and that of some particularspaces of continuous linear maps by projective limits of Banach functionalspaces, a fact not true for arbitrary spaces of linear maps. An applicationof the same representation is proposed for studying differential equations inFrechet spaces, including also comments on other approaches to the samesubject. Projective limit representations of various geometrical structuresconstitute one of the main tools of our approach.

Chapter 3 is dealing with the smooth structure, under appropriate condi-tions, of Frechet manifolds arising as projective limits of Banach manifolds,as well as with topics related to their tangent bundles. The case of Frechet-Lie groups represented by projective limits of Banach-Lie groups is also stud-ied in detail, because of their fundamental role in the structure of Frechetprincipal bundles. Such groups admit an exponential map, an importantproperty not yet established for arbitrary Frechet-Lie groups.

Chapter 4 is devoted to the study of projective systems of Banach principalbundles and their connections. The latter are handled by their connectionforms, global and local ones. It is worthy of note that any Frechet principalbundle, with structure group one of those alluded to in Chapter 3, is al-ways representable as a projective limit of Banach principal bundles, whileany connection on the former bundle is an appropriate projective limit ofconnections in the factor bundles of the limit. Here, related (or conjugate)connections, already treated in Chapter 1, provide an indispensable tool inthe approach to connections in the Frechet framework. We further note thatthe holonomy groups of the limit bundle do not necessarily coincide withthe projective limits of the holonomy groups of the factor bundles. This issupported by an example after the study of flat bundles.

Chapter 5 is concerned with projective limits of Banach vector bundles. Ifthe fibre type of a limit bundle is the Frechet space F, the structure of thevector bundle is fully determined by a particular group (denoted by H0(F)and described in § 5.1), which replaces the pathological general linear groupGL(F) of F, thus providing the limit with the structure of a Frechet vectorbundle. The study of connections on vector bundles of the present type is

Preface xiii

deferred until Chapter 7.

Chapter 6 contains a collection of examples of Frechet bundles realized asprojective limits of Banach ones. Among them, we cite in particular thebundle J∞(E) of infinite jets of sections of a Banach vector bundle E. Thisis a non trivial example of a Frechet vector bundle, essentially motivatingthe conditions required to define the structure of an arbitrary vector bundlein the setting of Chapter 5. On the other hand, the generalized bundle offrames of a Frechet vector bundle is an important example of a principalbundle with structure group the aforementioned group H0(F).

Chapter 7 aims at the study of connections on Frechet vector bundles thelatter being in the sense of Chapter 5. The relevant notions of paralleldisplacement along a curve and the holonomy group are also examined.Both can be defined, despite the inherent difficulties of solving equations inFrechet spaces, by reducing the equations involved to their counterparts inthe factor Banach bundles.

Chapter 8 is mainly focused on the vector bundle structure of the secondorder tangent bundle of a Banach manifold. Such a structure is alwaysdefined once we choose a linear connection on the base manifold, thus anatural question is to investigate the dependence of the vector bundle struc-ture on the choice of the connection. The answer relies on the possibilityto characterize the second order differentials as vector bundle morphisms,which is affirmative if the connections involved are properly related (conju-gate). The remaining part of the chapter is essentially an application of ourmethods to the second order Frechet tangent bundle and the corresponding(generalized) frame bundle.

We conclude with a series of open problems or suggestions for further appli-cations, within the general framework of our approach to Frechet geometry,eventually leading to certain topics not covered here.

These notes are addressed to researchers and graduate students of mathe-matics and physics with an interest in infinite-dimensional geometry, espe-cially that of Banach and Frechet manifolds and bundles. Since we have inmind a wide audience, with possibly different backgrounds and interests, wehave paid particular attention to the details of the exposition so that it isas far as possible self-contained. However, a familiarity with the rudimentsof the geometry of manifolds and bundles (at least of finite dimensions) isdesirable if not necessary.

It is a pleasure to acknowledge our happy collaboration, started overten years ago by discussing some questions of common research interestand resulting in a number of joint papers. The writing of these notes is

xiv Preface

the outcome of this enjoyable activity. Finally, we are very grateful toan extremely diligent reviewer who provided many valuable comments andsuggestions on an earlier draft, we have benefited much from this in the finalform of the monograph.

Manchester – Piraeus – Athens,February 2015

Chapter 1

Banach manifolds andbundles

The geometry of Banach manifolds and bundles has been greatly developedsince the 1960s and now there are many papers and a number of bookscovering a great variety of related topics. Here we intend to fix our notationand give a brief account of the basic results which will be used in the mainpart of the present work. Occasionally, some topics are dealt with in moredetail. These refer to subjects either not easily found in the literature ortheir methods have a particular interest and cover explicit needs of theexposition.

1.1 Banach manifolds

The main references for this section are [AMR88], [AR67], [Bou67], and[Lan99], where the reader may find the necessary details.

1.1.1 Ordinary derivatives in Banach spaces

Let E and F be two Banach spaces. We denote by L(E,F) the (Banach)space of continuous linear maps between E and F. In particular, we setL(E) := L(E,E), which is a Banach algebra. On the other hand, Lis(E,F)denotes the (open) set of invertible elements (viz. linear isomorphisms) ofL(E,F), while Lis(E) := Lis(E,E). The latter space, viewed as a groupunder the composition of automorphisms, is denoted by GL(E) and is calledthe general linear group of E.

A map f : U → F (U ⊆ E open) is called differentiable at x if there

1

2 Chapter 1. Banach manifolds and bundles

exists a map Df(x) ∈ L(E,F), the (Frechet) derivative of f at x, suchthat

limh→0

‖f(x+ h)− f(x)− [Df(x)](h)‖

h= 0 (h 6= 0).

The (total) derivative, or differential, of f is Df : U → L(E,F). If Dfis continuous, then we say that f is of class C1. Inductively, we set

Dkf = D(Dk−1f) : U −→ Lk(E,F) ≡ L(E,Lk−1(E,F)),

if the latter derivative exists. The map f will be called smooth, or (of class)C∞, if the derivatives Dk exist for every k and are continuous.

For an excellent treatment of the differential calculus in Banach spaceswe refer also to [Car67(a)].

1.1.2 Smooth structures

A Banach manifoldM is a smooth manifold whose differential structure isdetermined by local charts of the form (U, φ,B), where the ambient spaceor model B is a Banach space. If all the charts have the same model B (afact ensured in the case of a connected manifold), we say that B is modelledon B or it is a B-manifold. If there is no ambiguity about the model, thecharts will be simply denoted by (U, φ). The (maximal) atlas inducing thedifferential structure is denoted by A.

A Banach space B is a Banach manifold whose differential structure isdetermined by the global chart (B, idB).

For the sake of simplicity, unless otherwise stated, differentiability isassumed to be of class C∞, a synonym of smoothness. Usually, a Banachmanifold is assumed to be Hausdorff, equipped with smooth partitions ofunity.

1.1.3 Smooth maps

A map f : M → N is said to be smooth at x ∈M if there are charts (U, φ)and (V, ψ) of M and N , respectively, such that x ∈ U , f(U) ⊆ V , and thelocal representation or representative of f , with respect to the previouscharts,

(1.1.1) ψ f φ−1 : φ(U) −→ ψ(V )

is smooth at f(x) in the sense of ordinary differentiability in Banach spaces.Short-hand notations for (1.1.1) are fV U or fψφ. We also write fβα for thelocal representation of f with respect to the charts (Uα, φα) and (Uβ , φβ),with f(Uα) ⊆ Uβ.

1.1. Banach manifolds 3

1.1.4 The tangent space

A smooth curve at x ∈ M is a smooth map α : J → M with α(0) = x,where J is an open interval of R containing 0. Two curves α and β at x arecalled equivalent or tangent if there is a chart (U, φ) at x such that

(1.1.2) (φ α)′(0) = (φ β)′(0)

Here we have that

(1.1.3) (φ α)′(t) = [D(φ α)(t)](1),

for every t ∈ J such that α(t) ∈ U . Clearly, (1.1.2) is equivalent to

(1.1.2′) D(φ α)(0) = D(φ β)(0).

The equivalence classes of curves as above are denoted by [(α, x)] (or [α, x]for complicated expressions of curves) and are called tangent vectors atx. The set of all tangent vectors at x is the tangent space at x, denotedby TxM .

Considering any chart (U, φ) ≡ (U, φ,B) at x, we check that TxM is in abijective correspondence with B by means of the map

(1.1.4) φ : TxM −→ B : [(α, x)] 7→ (φ α)′(0).

Therefore, TxM becomes a Banach space and φ a continuous linear isomor-phism. The Banach structure of TxM is independent of the choice of thechart containing x. This is an immediate consequence of the following fact:If (U, φ,B) and (U,ψ,B′) are two charts at x, then the following diagram iscommutative:

TxM

BD(ψ φ−1)(φ(x))

-

φ

B′

ψ

-

Considering a Banach space B as a smooth manifold, the tangent spaceTbB, for every b ∈ B, is identified with B by means of idB (see § 1.1.2). Inparticular, TtR is an 1-dimensional vector space, with the natural basis

(1.1.5)d

dt

∣∣∣t

:= id−1R (1).


1.1.5 The tangent bundle

As usual, the tangent bundle of a (Banach) manifold M is determined bythe triple (TM,M, τM ), where

TM :=

.⋃

x∈M

TxM ≡⊔

x∈M

TxM

(disjoint union) is the total space and τM : TM → M the projection ofthe tangent bundle, with τM([(α, x)]) := x.

The total space TM is a Banach manifold, whose structure is inducedas follows: Given a local chart (U, φ) ≡ (U, φ,B), we define the map

(1.1.6) Φ: π−1(U) −→ φ(U) × B

by setting

(1.1.7) Φ(u) :=(τM(u), φ(u)

)=(x, (φ α)′(0)

),

if u = [(α, x)] ∈ TxM and x ∈ U . Then the collection of all pairs (π−1(U),Φ),obtained by running (U, φ) in the maximal atlas of M , determines a smoothatlas on TM , whose maximal counterpart induces the desired smooth struc-ture on TM .

1.1.6 The differential of a smooth map

The tangent spaces and the tangent bundle provide the appropriate frame-work for the development of a differential calculus on manifolds. Precisely:if f : M → N is a smooth map between two Banach manifolds, then thedifferential or tangent map of f at x is the map

(1.1.8) Txf : TxM −→ Tf(x)N,

given by

(1.1.9) Txf([(α, x)]) := [(f α, f(x))].

This is a well-defined continuous linear map, independent of the choice ofthe representatives of the tangent vectors.

In various computations, the differential Txf is handled by using localcharts and the derivative of the corresponding local representation (1.1.1)of f . More precisely, if (U, φ,E) and (V, ψ,F) are local charts of M and N ,

1.1. Banach manifolds 5

respectively, such that x ∈ U and f(U) ⊆ V (as ensured by the smoothnessof f at x), then one proves that the next diagram is commutative.

TxMTxf

- Tf(x)N

E

φ

?

D(ψ f φ−1)(φ(x))- F

ψ

?

In particular, differentiating the map φ of a chart (U, φ,B), we obtainthe following commutative diagram:

TxU ≡ TxMTxφ

- Tφ(x)B

B

idB

?

φ-

Frequently, omitting idB, we simply write

(1.1.10) φ ≡ Txφ.

1.1.7 Velocity vectors

Let α : J → M be a smooth curve. The tangent or velocity vector atα(t) (or, simply, at t) is the vector

(1.1.11) α(t) := Ttα( ddt

∣∣∣t

)∈ Tα(t)M.

In particular, if α passes through x, i.e. α(0) = x, then

(1.1.12) α(t) = [(α, x)].

If the curve has a more complicated form, e.g. f α, then the corre-

sponding velocity vector is denoted by (f α).(t) instead of˙

(f α)(t).


1.1.8 The tangent map

Let f : M → N be a smooth map. The tangent map or (total) differen-tial of f is obtained by gluing together the differentials Txf , for all x ∈M ;that is,

(1.1.13) Tf : TM −→ TN : Tf∣∣TxM

= Txf.

The following diagram is also commutative:

TMTf

- TN

M

τM

?

f- N

τN

?

Note: For the differentials of maps on manifolds we prefer to use the func-torial T instead of d, the latter been reserved for the exterior differential ofdifferential forms.

1.1.9 Vector fields

A vector field on M is a section of the tangent bundle; that is, a map ofthe form X : M → TM such that π X = idM . The set of smooth vectorfields on M is denoted by X (M). The bracket of vector fields determinesthe structure of a Lie algebra on X (M).

A vector field X induces a derivation of the algebra of smooth functionson M by X(f)(x) = Txf(Xx), for every x ∈ M . For the correspondencebetween vector fields and derivations of smooth functions or Banach spacevalued maps on M see, for instance, [AMR88].

Given a chart (U, φ,B) of M and the corresponding chart (π−1(U),Φ) ofthe tangent bundle (see § 1.1.5), the local representation of X ∈ X (M), withrespect to the previous charts, is the map Φ X φ−1 (see § 1.1.3) shownalso in the following diagram:

MX

- π−1(U)

φ(U)

φ

?

Φ X φ−1- φ(U)× B

Φ

?

1.2. Banach-Lie groups 7

Then, the (local) principal part of X (with respect to the above repre-sentation) is the map

(1.1.14) Xφ := pr2 Φ X φ−1 : φ(U) −→ B.

If we consider an indexed chart (Uα, φα,B), then we set

(1.1.14′) Xα := Xφα .

1.1.10 Related vector fields

Let f : M → N be a smooth map. Two vector fields X ∈ X (M) andY ∈ X (N) are f -related, if Tf X = Y f . Equivalently,

Txf(Xx) = Yf(x), x ∈M.

1.1.11 Integral curves

A smooth curve α : Jα → M (Jα: open interval containing 0) such thatα(0) = x and

X(α(t)) = α(t), t ∈ Jα

is called an integral curve of X ∈ X (M) with initial condition x. Lo-cally, the problem of finding α reduces to the determination of a smoothcurve β : Jβ → B such that β(0) = φ(x) and

(1.1.15) β′(t) = Xφ(β(t)), t ∈ Jβ

[recall also (1.1.14)]. The theory of differential equations in Banach spacesensures the existence and uniqueness of such a β. Thus α = φ−1 β is anintegral curve of X with initial condition α(0) = x.

If M is a Hausdorff manifold, then there is a unique integral curve αwith α(0) = x, defined on a maximal interval of R containing 0.

1.2 Banach-Lie groups

Beside the references given in the begining of § 1.1, here we add [Bou72] and[Mai62].


1.2.1 Basic notations

A Banach-Lie group G is a Banach manifold with a compatible groupstructure, i.e. the multiplication or product

γ : G×G −→ G : (x, y) 7→ γ(x, y) := xy ≡ x · y,

and the inversion

α : G −→ G : x 7→ α(x) := x−1

are smooth maps. γ comes from the Greek word γινόμενο meaning product.Observe the use of the bold typeface γ to distinguish the product from thenormal γ usually denoting a curve. α (bold typeface, again) is the firstletter of αντιστροφή, the Greek word for inversion. The unit (element) of Gis denoted by e.

The left translation by g ∈ G is the diffeomorphism

λg : G −→ G : x 7→ λg(x) := gx.

Similarly, the right translation by g ∈ G is

ρg : G −→ G : x 7→ ρg(x) := xg.

The differentials of γ and α, in terms of the translations, are given,respectively, by

T(x,y)γ(u, v) = Txρy(u) + Tyλx(v),(1.2.1)

Txα(u) = −Teλx−1 Txρx−1(u) = −Tx(λx−1 ρx−1)(u),(1.2.2)

for every x, y ∈ G and every u ∈ TxG, v ∈ TyG.

In the following subsections G will denote a Banach-Lie group.

1.2.2 Invariant vector fields

A vector field X ∈ X (G) is said to be left invariant if it is λg-related withitself, for every g ∈ G; that is,

Tλg X = X λg, g ∈ G;

equivalently,

Teλg(Xe) = Xg, g ∈ G.


The set of all left invariant vector fields on G forms a Lie subalgebra ofX (G), denoted by L(G) and called the Lie algebra of G.

L(G) is in bijective correspondence with TeG by means of the linearisomorphism

(1.2.3) h : L(G) ∋ X 7−→ Xe ∈ TeG

whose inverse is given by

(1.2.4) h−1(v) = Xv; v ∈ TeG,

where Xv ∈ L(G) is defined by

(1.2.5) Xv(x) = TeLx(v), x ∈ G.

Therefore, TeG becomes a Lie algebra by setting (same symbol of bracket !)

[u, v] := h([h−1(u),h−1(v)

]).

Equivalently, if u = Xe and v = Ye, for X,Y ∈ L(G), then

[Xe, Ye] = [X,Y ]e,

under the appropriate interpretation of the bracket in each side.

For convenience, sometimes, we shall denote by g the Lie algebra TeGwith the previous structure. As is the custom, we shall denote the Liealgebra of G by g and L(G) interchangeably, as a result of the identification(1.2.3).

1.2.3 The exponential map

The exponential map of G is the map

exp ≡ expG : TeG 7−→ G : v 7→ exp(v) := α(1),

where α is the integral curve of X = h−1(v) ∈ L(G) with initial conditionα(0) = e. Recall that the left invariant vector fields are complete, thus thedomain of α is R.


1.2.4 The adjoint representation

The adjoint representation of G is the map Ad: G→ Aut(g), with

Ad(g) := Te(ρg−1 λg) = Te(λg ρg−1).

It is a smooth map whose differential at e ∈ G,

TeAd(g) : TeG ≡ g −→ L(g),

is given by (TeAd(g)(X)

)(Y ) = [X,Y ]; X,Y ∈ g.

1.2.5 Lie algebra-valued differential forms

Let B be a Banach manifold and let G be a Banach-Lie group with Liealgebra g.

Heuristically, a g-valued differential form of degree k (g-valuedk-form, for short) on B is a smooth map ω assigning a k-alternating (anti-symmetric) map ωx ∈ Ak(TxB, g) to each x ∈ B. Formally, ω is a smoothsection of the vector bundle

Ak(TB, g) :=⋃

x∈B

Ak(TxB, g)

of the k-alternating maps described in detail in § 1.4.4(e) (see also § 1.4.1).The set of g-valued k-forms on B is denoted by Λk(B, g).

Important examples of g-valued forms are the Maurer-Cartan forms ona Lie group defined below, and the Maurer-Cartan differentials defined inthe next subsection. More specifically, the left Maurer-Cartan (or leftcanonical) form on G is the differential form ωl ∈ Λ1(G, g) given by

ωlg(v) := Tgλg−1(v); g ∈ G, v ∈ TgG.

Analogously, the right Maurer-Cartan form on G is the differential formωr ∈ Λ1(G, g) defined by

ωrg(v) := Tgρg−1(v); g ∈ G, v ∈ TgG.

The form ωl is left invariant, i.e. λ∗ωl = ωl. Likewise, ωr is rightinvariant, i.e. ρ∗ωr = ωr. The two forms satisfy the respective Maurer-Cartan equations:

dωl = −1

2

[ωl, ωl

]= −ωl ∧ ωl,

dωr =1

2

[ωr, ωr

]= ωr ∧ ωr.


For the exterior product, the bracket and the exterior differentiation of g-valued forms, we refer to the general theory of [Bou71, § 8.3], [Car67(b)] and[Nab00, § 4.2].

1.2.6 The Maurer-Cartan differentials

These differentials will be encountered in the study of local connection forms(see § 1.7.2 below).

Let B be a Banach manifold, G a Banach Lie group and f : B → Ga smooth map. Then the left Maurer-Cartan differential of f is thedifferential form Dlf ≡ f−1df ∈ Λ1(G, g) defined by

(Dlf

)x(v) ≡

(f−1df

)x

(v) :=(Tf(x)λf(x)−1 Txf

); x ∈ B, v ∈ TxB.

Analogously, the right Maurer-Cartan differential of f is the differentialform Drf ≡ df.f−1 ∈ Λ1(G, g) defined by

(Drf)x (v) ≡(df.f−1

)x

(v) :=(Tf(x)ρf(x)−1 Txf

); x ∈ B, v ∈ TxB.

It is immediate that

(1.2.6) f−1df = f∗ωl and df.f−1 = f∗ωr,

where ωl and ωr are the Maurer-Cartan forms of G defined in § 1.2.5.

Equations (1.2.6) justify our terminology. Other terms in use are leftand right differentials of f (N. Bourbaki [Bou72, Ch. III, § 3.17]), loga-rithmic derivatives (A. Kriegl and P. Michor [KM97, Ch. VIII, § 38.1]), ormultiplicative differentials (S.G. Krein and N.I. Yatskin, [KJ80, Ch. I,§ 3]). Another legitimate term is total left/right differentials since Dlf =Drf = Tf , for G = (E,+) and any smooth map f : B → E (see the termi-nology of § 1.1.8).

We list below a few properties of the Maurer-Cartan differentials, refer-ring for details to the aforementioned sources.

Drf−1 = −Dlf,

Dlf = Ad(f−1

)Drf,

Drf−1 = −Ad(f−1

)Drf,

Dr(f · h) = Drf + .Ad(f)Drh,

Dl(f · h) = Dlh+ Ad(h−1

)Dlf,


for all smooth maps f, h : B → G. We recall that f−1 : B → G is given byf−1(x) := f(x)−1, for every x ∈ B. On the other hand, Ad(f)Drh is the1-form given by

(Ad(f)Drh

)x(v) = Ad(f(x))

((Drh)x(v)

); x ∈ B, v ∈ TxB.

Analogously for the other expressions involving the adjoint representation.

We also have:

Drf = 0 = Dlf ⇔ f locally constant,

Drf = Drh ⇔ h = fC, C locally constant,

Dlf = Dlh ⇔ h = Cf, C locally constant.

Of particular interest are the equations with Maurer-Cartan differentials.For instance, let us consider the equation

(1.2.7) Drx = θ, where θ ∈ Λ1(M, g).

Let (x0, g0) ∈M ×G. Then:

Equation (1.2.7) admits a unique solution f : U → G (U : open neigh-borhood of x0) such that f(x0) = g0, if and only if dθ = 1

2 [θ, θ].

In this case θ is called integrable. If M is simply connected, then thereexist global solutions.

Lifting (1.2.7) to the universal cover M of M, we obtain the equation

(1.2.8) Drz = π∗θ,

where π : M → M is the natural projection. We fix an arbitrary x0 ∈ M .If θ is integrable, there is a global solution Fθ : M → G of (1.2.8), calledfundamental, with initial condition Fθ(x0) = e.

The monodromy homomorphism of (1.2.7) is the homomorphism

(1.2.9) θ# : π1(M) −→ G : [γ] 7→ θ#([γ]) := Fθ (x0 · [γ]) ,

where π1(M) ≡ π1(M, x0) is the fundamental group of M .

Analogous results hold for equations with the left Maurer-Cartan differ-ential, under the integrability condition dθ = −1

2 [θ, θ].

1.3. Smooth actions 13

1.3 Smooth actions

1.3.1 Definitions

A Banach-Lie group G acts (from the right) on a Banach manifold M ifthere is a smooth map δ : M ×G −→M , called action, such that:

δ(x, e) = x,(A.1)

δ(δ(x, g1), g2) = δ(x, g1g2),(A.2)

for every x ∈M and g1, g2 ∈ G. For convenience, we write x ·g or xg insteadof δ(x, g).

An action δ will be called effective if

xg = x ∀ x ∈M ⇔ g = e.

The action will be called free if

xg = x, for some x ∈M, ⇒ g = e.

Finally, δ is called transitive if

∀ (x, y) ∈M ×M ⇒ ∃ g ∈ G : y = xg.

If g ∈ G in the previous definition is uniquely determined, then the actionis called freely transitive.

For a g ∈ G, the partial map

δg : M −→M : x 7→ δg(x) := x · g

is a diffeomorphism. As a matter of fact, a smooth action is equivalentlydefined by requiring (A.2) and δg to be a diffeomorphism. The map δg isalso denoted by Rg (especially in the case of principal bundles, §1.6) andcalled the right translation of M by g. In the case of the (obvious) actionof a Lie group on itself, Rg = ρg, according to the notations of § 1.2.1.

1.3.2 Fundamental (Killing) vector fields

Let δ : M ×G→ M be a smooth action. Given a left invariant vector fieldX ∈ L(G), we set

X∗(x) := Teδx(Xe); x ∈M,


where, as usual, the partial map δx : G→M is given by δx(g) = δ(x, g), forevery g ∈ G. Since X∗(x) ∈ TxM , it follows that X∗ : M → TM is a smoothvector field of M . It is called the fundamental (or Killing) vector fieldon M corresponding to X ∈ L(G), with respect to the action δ. Obviously,the vector fields X and X∗ are δx-related, for every x ∈M (see § 1.1.10).

The integral curve β of X∗, with initial condition x ∈M , is given by

β(t) = x · αX(t) = x · exp(tX); t ∈ R,

where αX is the integral curve of X with initial condition e. Therefore, X∗

is a complete vector field.The map

∗ : L(G) ∋ X 7−→ X∗ ∈ X (M)

is a morphism of Lie algebras. If the action is effective, then the map ∗ isinjective. Moreover, if the action is free, then, for every X ∈ L(G) withXe 6= 0, it follows that X∗x 6= 0, for every x ∈M .

1.4 Banach vector bundles

We mainly follow [Bou67], [Lan99] and [AR67].

1.4.1 The structure of a Banach vector bundle

Let E,B be smooth manifolds, π : E → B a smooth map, and E a Banachspace. We also assume that Uα ⊂ B |α ∈ I is an open cover of B and, foreach α ∈ I, there is a diffeomorphism τα : π−1(U) → U × E satisfying thefollowing conditions:

(VB. 1) The diagram

π−1(Uα)τα

- Uα × E

Uα

pr1π-

is commutative and the restriction of τα to the fibre (over x) Ex = π−1(x),

(1.4.1) τα,x : Ex −→ x × E ∼= E

is a bijection for every x ∈ Uα.

1.4. Banach vector bundles 15

(VB. 2) For two pairs (Uα, τα) and (Uβ, τβ), with Uαβ = Uα ∩ Uβ 6= ∅, themap

τα,x τ−1β,x : E −→ E

is an isomorphism of Banach spaces, for every x ∈ Uαβ .

(VB. 3) For (Uα, τα) and (Uβ , τβ) as above, the map

(1.4.2) Tαβ : Uαβ ∋ x 7−→ Tαβ(x) := τα,x τ−1β,x ∈ L(E)

is smooth. As a matter of fact, Tαβ(x) ∈ GL(E).

We shall use the following terminology: (Uα, τα) is called a trivializa-tion of E with trivializing map τα. The collection C = (Uα, τα)α∈I isa trivializing cover of E.

Two trivializing covers of E are said to be equivalent if their unionsatisfies conditions (VB. 2) and (VB. 3). An equivalence class of trivializingcovers determines the structure of a Banach vector bundle of fibre typeE, with total space E, projection π, and base (space) B.

A vector bundle, as above, will be denoted by ℓ = (E,B, π). We refer toit either by ℓ or E if there is no ambiguity about its elements.

Because of (1.4.1), each fibre Ex admits the structure of a Banach spaceisomorphic to E, and

(1.4.3) τα(u) = (x, τα,x(u)), u ∈ Ex.

Clearly, τα,x = pr2 τ |π−1(x).

It is often useful to assume that the open sets Uα of the trivializing coverare the range of charts (Uα, φα) of the base B (this can always be intersectinga trivializing cover of E with the atlas of the smooth structure of B). Then,in analogy to the local structure of the tangent bundle of a manifold definedby (1.1.6) and (1.1.7), we may consider the map

(1.4.3′) Φα := (φα × idE) τα : π−1(Uα)→ φα(Uα)× E,

and the commutative diagram on the next page. The triple (Uα, φα,Φα)is called a vector bundle chart, or vb-chart for short, and (Uα,Φα), orsimply Φα, still a trivialization of E. As a matter of fact, a vector bun-dle structure is completely determined by a (maximal) atlas of compatiblevector bundle charts. Here the compatibility is expressed by means of an


isomorphism of local vector bundles (for details see also [AR67]).

π−1(Uα)Φα

- φα(Uα)× E

Uα

π

?

φα- φα(Uα)

pr1

?

A vector bundle chart (Uα, φα,Φα) induces an isomorphism of Banachspaces Φα,x : Ex → E such that the following analog of (1.4.3)

(1.4.4) Φα(u) = (φα,Φα,x(u)), u ∈ Ex.

holds true. Obviously,

(1.4.5) τα,x = Φα,x, x ∈ Uα.

1.4.2 Transition maps

The maps Tαβ defined by (1.4.2) are the transition maps or functions ofthe bundle ℓ = (E,B, π). They satisfy the cocycle condition

(1.4.6) Tαβ(x) = Tαγ(x) Tγβ(x), x ∈ Uαβγ := Uα ∩ Uβ ∩ Uα.

It follows that

(1.4.7) Tαα(x) = idE and Tβα(x) = Tαβ(x)−1,

for every x ∈ Uα and x ∈ Uαβ , respectively.The collection Tαβ is the cocycle of ℓ (with respect to the trivializing

cover (Uα,Φα)α,β∈I). More precisely, in the formal language of cohomol-ogy theory, Tαβ is a 1-cocycle, [DP97].

Given an open cover C = Uα | α ∈ I of a smooth manifold B, and acollection of smooth maps Tαβ : Uαβ → L(E), with Im(Tαβ) ⊂ GL(E), andsatisfying the cocycle condition (1.4.6), there exists a Banach vector bundleℓ = (E,B, π) with transition maps Tαβ. More precisely, E is obtained byquotienting the set ⋃

α∈I

(α × Uα × E)

by the equivalence relation

(β, y, k) ˜ (α, x, h) ⇔ y = x, k = Tβα(x)(h).


Then π : [(α, x, h)] 7→ x, and the trivializing maps τα : π−1(Uα) → Uα × E

are given by τα([(γ, z,m)]) := (z, Tαγ(z)(m)). The bundle ℓ is unique up toisomorphism (see the application in the next subsection).

It is often convenient to connect the transition maps Tαβ with vector bun-dle charts: If (Uα, φα,Φα) and (Uβ , φβ,Φβ) are two intersecting vb-charts,we define the smooth maps

(1.4.8) Gαβ : φβ(Uαβ) −→ GL(E) ⊂ L(E),

by setting

(1.4.9) Gαβ(φβ(x)) := Φα,x Φ−1β,x, x ∈ Uαβ.

As a result,

(1.4.10) Tαβ(x) = Gαβ(φβ(x)), x ∈ Uαβ .

1.4.3 Morphisms of vector bundles

Let ℓ1 = (E1, B1, π1) and ℓ2 = (E2, B2, π2) be two Banach vector bundles,of corresponding fibre types E1 and E2. A vector bundle morphism, orvb-morphism for short, between ℓ1 and ℓ2 is a pair of smooth maps (f, h),with f : E1 → E2 and h : B1 → B2 satisfying the following properties:

(VBM. 1) The next diagram is commutative

E1f

- E2

B1

π1

?

h- B2

π2

?

and the restriction of f to the fibres, namely fx := f |E1,x : E1,x → E2,h(x),is a continuous linear map, for every x ∈ B.

(VBM. 2) For each x0 ∈ B, there are trivializations

τ1 : π−11 (U1) −→ U1 × E1 and τ2 : π−12 (U2) −→ U2 × E2

with x0 ∈ U1, h(U1) ⊆ U2, and such that the map

U1 ∋ x 7−→ τ2,h(x) fx τ−11,x ∈ L(E1,E2)


is smooth. Using the corresponding vb-charts (U1, φ1,Φ1) and (U2, φ2,Φ2),condition (VBM. 2) is equivalent to the smoothness of

φ1(U1) ∋ φ1(x) 7−→ Φ2,h(x) fx Φ−11,x ∈ L(E1,E2).

A vb-morphism, as above, will be also denoted by (f, h) : ℓ1 → ℓ2.The composition of two vb-morphisms, as well as the notion of a vb-isomorphism are defined in the obvious way.

We shall mostly deal with vector bundles over the same base B and vb-morphism between them with h = idB . In this case we obtain the categoryVBB, in particular VBB(E) if the bundles have the same fibre type E. Amorphism between bundles in the latter category will be also denoted byf : E1 → E2.

As an application, we outline the following relationship between cocyclesand vb-isomorphisms: Assume that ℓ = (E,B, π) and ℓ′ = (E′, B, π′) areBanach vector bundles of the same fibre type E. By appropriate restrictions,we may take trivializations of ℓ and ℓ′ over the same open cover C = Uαα∈Iof B. Denote by Tαβ and T ′αβ the respective cocycles over C. Then:

There exists a vb-isomorphism (f, idB) of ℓ onto ℓ′ if and only if thecocycles Tαβ and T

′αβ are cohomologous.

The latter term means that there are smooth maps hα : Uα → Lis(E) suchthat

T ′αβ(x) = hα(x) Tαβ(x) hβ(x)−1; x ∈ Uαβ,

for all indices α, β ∈ I.Indeed, if there is a vb-isomorphism (f, idB), then we define hα by setting

hα(x) = τ ′α,x fx τ−1α,x. The smoothness of hα is ensured by (VBM 2).

Conversely, assume that the cocycles are cohomologous. We define themaps fα : EUα → E′Uα

with

fα(u) :=((τ ′α,x)−1 hα τα,x

)(u),

for every u ∈ EUα with π(u) = x. It is smooth because

fα = (τ ′α,x)−1 (π, ev (hα π,pr2 τα)

),

where pr2 : Uα × E→ E is the projection to the second factor and

ev : Lis(E)× E −→ E : (f, u) 7→ f(u)

is the evaluation map. The assumption implies that the collection fαdetermines a smooth bijection f : E → E′. It remains to see that (f, idB)


is a vb-morphism by verifying conditions (VBM. 1)–(VBM. 2). The first isobviously satisfied. For the second one, observe that the map Uα ∋ x 7→τα,x fx (τ ′α,x)−1, whose smoothness is required, is precisely hα, for everyα ∈ I. By the same token we prove that (f−1, idB) is also a vb-morphism,thus (f, idB) is a vb-isomorphism.

The previous arguments justify the uniqueness—up to isomorphism—ofthe vector bundle E constructed from a cocycle Tαβ, described in § 1.4.2.Indeed, if E′ is another bundle with the same cocycle, then E ∼= E′ bymeans of the vb-isomorphism f ≡ fα, where fα(u) =

((τ ′α,x)−1 τα,x

)(u),

for every u ∈ EUα with π(u) = x ∈ Uα, since now hα(x) = idE (constantly),for every x ∈ Uα.

Remark. In a more sophisticated way, the preceding relation between iso-morphic vector bundles and cohomologous cocycles leads to the followingcohomological classification:

Within an isomorphism, we obtain the equality

VBB(E)/˜ = H1(B,GL(E)),

where VBB(E)/˜ is the quotient of VBB(E) with respect to the equiv-

alence relation induced by vb-isomorphisms, and GL(E) is the sheaf ofgerms of smooth GL(E)-valued maps on B.

We recall that VBB(E) is the set of (Banach) vector bundles over B, offibre type E. The right-hand side of the identification is the 1st cohomologygroup of B with coefficients in GL(E). Briefly, H1(B,GL(E)) is the union ofH1(U,GL(E)), where U is running through the set of all proper open coversof B. Each set H1(U,GL(E)) consists of all the cohomologous 1-cocyclesgαβ : Uαβ → GL(E) (Uα, Uβ ∈ U) identified now with the sections of GL(E)over Uαβ .

For relevant details on the cohomological classification of fibred spaceswe refer to [Gro58] and [Hir66]. For the general theory of sheaves and sheafcohomology we refer also to [Dow62], [DP97], [God73] and [War83].

1.4.4 Some useful constructions and examples

a) Fibre product and direct sum

Let ℓk = (Ek, B, πk) ∈ VBB be vector bundles of fibre type Ek (k = 1, 2).Their fibre product is the vector bundle (E1 ×B E2, B, π), where

E1 ×B E2 := (u1, u2) ∈ E1 × E2 : π1(u1) = π2(u2) ,

π(u1, u2) := π1(u1) = π2(u2).


Clearly, (E1 ×B E2)x = E1,x×E2,x, for every x ∈ B. Moreover, by intersect-ing trivializing covers of ℓ1 and ℓ2, we may take the corresponding trivializingcovers (Uα, τ

1i )i∈I and (Uα, τ

2i )i∈I inducing the trivializations (Uα, τα)

of E1 ×B E2, where the maps

τα : π−1(Uα) −→ Uα × E1 × E2

are given by

τ(u1, u2) :=(π(u1, u2) = x, τ1α,x(u1), τ

2α,x(u2)

).

On the other hand, setting

E1 ⊕ E2 :=⋃

x∈B

E1,x ⊕ E2,x,

we obtain the direct or Whitney sum (E1 ⊕ E2, B, π), whose projectionand trivializations are defined as in the case of the fibre product.

Finite direct sums of vector bundles can be identified with their finitefibre products, as a result of the analogous identification of vector spaces.

b) The pull-back of a vector bundle

If Y → B is a smooth map, the pull-back of ℓ = (E,B, π) by f is thevector bundle f∗(ℓ) = (f∗(E), Y, f∗(π) ≡ π′), with

f∗(E) ≡ Y ×B E := (y, u) ∈ Y × E : f(y) = π(u) ,

f∗(π) ≡ π′ := pr1 |f∗(E) : f∗(E) −→ Y.

The fibres of f∗(ℓ) are identified with the fibre type E of ℓ, since

f∗(ℓ)y = y × Ef(y) ∼= Ef(y), y ∈ Y.

Moreover, if

f ′ ≡ π∗(f) := pr2 |f∗(E) : f∗(E)→ E,

then the pair (f ′, f) is a vb-morphism of f∗(ℓ) into ℓ.For later reference, we note that each trivialization (U, τ) of E, τ : EU =

π−1(U)→ U × E, induces the trivialization (f−1(U), τ∗) of f∗(E), with

τ∗ : (π′)−1(U) = f−1(U)×U EU −→ f−1(U)× E

defined by τ∗(y, u) := (y, τf(y)(u)) = (y, τ(f(y), u)). Accordingly, if Tαβis the cocycle of E, with respect to the trivializing cover (Uα, τα)α∈I


of E, the corresponding cocycle of f∗(E) is Tαβ f, with respect to(f−1(Uα), τ∗α

)α∈I

.The pull-back has the following universal property: For each vector

bundle ℓ = (E, π, Y ) and each vb-morphism (f , f) : ℓ→ ℓ, there is a uniquesmooth map π : E → f∗(E) such that (π, idY ) is a vb-morphism of ℓ intof∗(ℓ) and f ′ π = f . In fact, it suffices to take π = (π, f).

The universal property is depicted in the next diagram.

E

f∗(E)f ′

-

π

-

E

f

-

Y

π′

?

f-

π

-

B

π

?

c) Linear map bundles

Let ℓ = (E,B, π) and ℓ′ = (E′, B, π′) be vector bundles of fibre type E andE′, respectively. We define the set of linear maps (see also the notations of§ 1.1.1)

L(E,E′) :=⋃

x∈B

L(Ex, E′x)

and the projection

L ≡ Lπ,π′ : L(E,E′) −→ B : f 7→ L(f) := x, if f ∈ L(Ex, E′x).

Then the triple (L(E,E′), B, L) is a vector bundle, a particular case of alinear map bundle (more generally, we can take bundles over differentbases). For details we also refer to [AR67]. Here we only mention the vb-charts of L(E,E′): Choosing vb-charts (Uα, φα,Φα) and (Uα, φ

′α,Φ

′α) of ℓ

and ℓ′, respectively (over the same open cover Uαα∈I of B), with

Φα : π−1(Uα) −→ φα(Uα)× E, Φ′α : (π′)−1(Uα) −→ φ′α(Uα)× E′,

we obtain the vb-chart(L−1α (Uα), φα, Lα

), where the map

Lα : L−1(Uα) −→ φα(Uα)× L(E,E′)


is given by

Lα(f) := (φα(x), λα(x)) ; f ∈ L(Ex, E′x),

with λα(x) ∈ L(E,E′) defined in turn by

λα(x)(v) :=(pr2 Φ

′α f Φ−1α

)(φ(x), v), v ∈ E.

d) Multilinear map bundles

The previous construction extends to k-linear maps. More precisely: Letℓi = (Ei, B, πi) (i = 1, . . . , k) be vector bundles of fibre type Ei, and letℓ′ = (E′, B, π′) be a vector bundle of fibre type E′. The k-linear mapbundle consists of the triple

(Lk(E1 × · · · × Ek, E

′), B, Lk), where

Lk(E1 × · · · × Ek, E′) :=

⋃

x∈B

Lk(E1,x × · · · ×Ek,x, E′x)

[Lk(E1,x × · · · × Ek,x, E′x) is the space of continuous k-linear maps between

the indicated Banach spaces], and

Lk : Lk(E1 × · · · × Ek, E′) −→ B : f 7→ L(f) := x,

if f ∈ Lk(E1,x × · · · × Ek,x, E′x).

Choosing vb-charts (Uα, φα,Φiα, ) and (Uα, φ

′α,Φ

′α) of ℓi (i = 1, . . . , k)

and ℓ′, respectively, we define the vb-chart((Lk)−1α (Uα), φα, Lα

), with

Lkα : (Lk)−1(Uα) −→ φα(Uα)× Lk(E1 × · · · × Ek,E′),

given by

Lkα(f) :=(φα(x), λkα(x)

),

while λkα(x) ∈ Lk(E1 × · · · × Ek,E′) is defined by

λkα(x)(v1, . . . , vk) :=(

pr2 Φ′α f

(Φ−11,α × · · · × Φ−1k,α

)) ((φ(x), v1), . . . , (φ(x), vk)

).

e) Alternating map bundles

Analogously to the preceding bundle, for E1 = · · · = Ek = E, we constructthe k-alternating (antisymmetric) map bundle whose total space is

Ak(E,E′) :=

⋃

x∈B

Ak(Ex, E′x),


with Ak(Ex, E′x) denoting the space of continuous k-alternating maps of

Ex × · · · × Ex (k factors) into E′x.

A particular case, which will be frequently encountered, occurs whenE = TB and E′ = B × g, the latter being the total space of the trivialbundle over B with fibre the Lie algebra of a Banach Lie group G. Then

Ak(TB,B × g) ≡ Ak(TB,B × g) :=⋃

x∈B

Ak(TxB, g),

after the identification x × g ≡ g. The smooth sections of Ak(B, g) arethe g-valued differential k-forms on B, already discussed in § 1.2.5.

f) Jets of sections

Let (E,B, π) be a Banach vector bundle of fibre type E, over the Banachmanifold B of respective model B. By a (global) section of E we mean asmooth map ξ : B → E such that πξ = idB . We denote by Γ(E) ≡ Γ(B,E)the C∞(B,R)-module of smooth sections of E. Analogously, if U is an opensubset of B, the module of smooth sections of E over U is denoted byΓ(U,E).

Given a ξ ∈ Γ(E) and a vb-chart (U, φ,Φ) of ℓ, we define its local repre-sentation to be Φ ξ φ−1 : φ(U)→ φ(U)× E and the corresponding localprincipal part ξφ : φ(U)→ E with

(1.4.11)(Φ ξ φ−1

)(x) = (x, ξφ(x)) , x ∈ φ(U).

If (Uα, φα,Φα) is an indexed vb-chart, then we set ξα := ξφα ; hence,

(1.4.11′)(Φα ξ φ

−1α

)(x) = (x, ξα(x)) , x ∈ φα(Uα).

Our next goal, roughly speaking, is to partition the sections of a vectorbundle into a kind of equivalence classes and provide the quotient space witha vector bundle structure.

For our purpose we first introduce the following notations: If Lks(B,E)is the space of continuous symmetric k-linear maps of Bk into E, then

P k(B,E) := E× Ls(B,E)×L2s(B,E)× · · · × Lks(B,E)

is the Banach space of E-valued polynomials of degree k on B. Foran open A ⊆ B, an a ∈ A, and a smooth map f : A → E, we denote bypkf(a) ∈ P k(B,E) the polynomial

pkf(a) := (f(a),Df(a), . . . ,Dkf(a)).


Fix a vector bundle ℓ = (E,B, π) as above. Let x1, x2 ∈ B and ξ1, ξ2local sections of E whose domains contain the points x1 and x2, respectively.We define the following equivalence relation:

(ξ1, x1) ∼k (ξ2, x2) ⇔

∃ vb-chart (U, φ,Φ) :

x1 = x2 ∈ U, and

pkξ1,φ(φ(x1)) = pkξ2,φ(φ(x2))

By appropriate restrictions, we may assume that the domains of the sectionscoincide with the domain of the chart.

It is easily shown that ∼k is an equivalence relation, independent of thechoice of the vb-chart satisfying the above conditions. We denote by

jkxξ the equivalence class of (ξ, x),

Jk(E) the derived quotient space,

πk the projection Jk(E) −→ B : jkxξ 7→ x.

Then Jk(ℓ) := (JkE,B, πk) is a Banach vector bundle of fibre type P k(B,E),called the k-jet bundle of sections of ℓ. Its elements are the k-jets of(local) sections of ℓ.

If (U, τ) is a trivialization of ℓ, with corresponding vb-chart (U, φ,Φ), weobtain the trivialization (U, τk) of Jk(ℓ), where

τk :(πk)−1

(U) −→ U × P k(B,E) : jkxξ 7→(x, pkξφ(φ(x))

).

The corresponding vb-chart is (U, φk,Φ), with

Φk :(πk)−1

(U) −→ φ(U)× P k(B,E) : jkxξ 7→(φ(x), pkξφ(φ(x))

).

Details and additional material can be found in [AR67] and [Bou71].The latter source treats also the general case of jets of smooth maps.

1.4.5 Exact sequences

Let ℓk = (Ek, B, πk) (k = 1, 2) be Banach vector bundles of respective fibretype E1 and E2. If f : E1 → E2 is a vb-morphism, then the sequence

(1.4.12) 0 −→ E1f−−→ E2

is called exact if, for every x ∈ B, the map fx := f |E1,x : E1,x −→ E2,x isinjective and its image fx(E1,x) has a closed complement in E2,x.


Equivalently, there is an open cover C of B and, over each U ∈ C, thereare corresponding trivializations

τ1 : π−11 (U) −→ U × E1, τ2 : π−12 (U) −→ U × E2

such that E2 = E1 × F (: F Banach space) and the diagram

π−11 (U)f

- π−12 (U)

U × E1

τ1

?

inc- U × E1 × F

τ2

?

is commutative, with inc(x, u) := (x, u, 0) the inclusion map.Analogously, if ℓk = (Ek, B, πk) (k = 2, 3) are Banach vector bundles of

respective fibre type E2 and E3, and g : E2 → E3 is a vb-morphism, thenthe sequence

(1.4.13) E2g−−→ E3 −→ 0

is called exact if, for every x ∈ B, the map gx : E2,x −→ E3,x is surjectiveand its kernel ker(gx) has a closed complement in E2,x.

Equivalently, there is an open cover C of B and, over each U ∈ C, thereare corresponding trivializations

τ2 : π−12 (U) −→ U × E2, τ3 : π−13 (U) −→ U × E3,

such that E2 = E3 × F and the diagram

π−12 (U)g

- π−13 (U)

U × E3 × F

τ2

?

p- U × E3

τ3

?

is commutative, where p(x, u, v) := (x, u).In particular, the sequence

(1.4.14) 0 −→ E1f−−→ E2

g−−→ E3 −→ 0


is exact if the sequences (1.4.12) and (1.4.13) are exact and

Im(f) =⋃

x∈B

Im(fx) =⋃

x∈B

ker(gx) = ker(g).

Both Im(f) and ker(g) are Banach subbundles of E2. Recall that a set S ⊂ Eis a subbundle of (E,B, π) if there exists an exact sequence 0 → E′ → Esuch that S = f(E′).

1.4.6 The exact sequence associated to a vector bundle

Let ℓ = (E,B, π) be a Banach vector bundle. In the formalism of (lin-ear) connections, we shall encounter the following exact sequence of vectorbundles associated to ℓ:

(1.4.15) 0 −→ V Ej−−→ TE

Tπ!−−−−→ π∗(TB) −→ 0

where• V E is the vertical subbundle of the tangent bundle TE ≡ (TE,E, τE),whose fibres are given by

VuE := (V E)u = ker (Tπ!|TuE) = ker(Tuπ) = Tu(π−1(x)

),

for every u ∈ E with π(u) = x. The last equality is easily proved byconsidering Tu

(π−1(x)

)as a subspacce of TuE and using local trivializations

(as a matter of fact, this is particular case of an analogous result for thetangent spaces of the fibres of a submersion).

• The morphism j is the natural inclusion.

• π∗(TB) = E ×B TB (: the pull-back of TB by π; see § 1.4.4).

• Tπ! is the vb-morphism defined by the universal property of the pull-back,as pictured in the next diagram.

TE

π∗(TB)pr2 -

Tπ !

-

TB

Tπ

-

E

pr1

?

π-

τE

-

B

τB

?

1.5. Connections on vector bundles 27

The following vb-isomorphism is standard:

(1.4.16) V E ∼= E ×B E,

(see. for instance, [Die72, problem 11, p. 136]). Therefore, there is a canon-ical map r : V E → E as in the diagram

V E∼= - E ×B E

E

pr2r-

so that (r, π) is a vb-morphism between (V E,E|V E , τE) and (E,B, π).

1.5 Connections on vector bundles

Details of the material included here can be found mainly in [Eli67], [FK72]and [Vil67].

1.5.1 General definitions

Let ℓ = (E,B, π) be a Banach vector bundle of fibre type E. A connectionon ℓ is a splitting of the exact sequence (1.4.15); that is, an exact sequenceof vector bundles

(1.5.1) 0 −→ π∗(TB)C−−−→ TE

V−−−→ V E −→ 0

such that

Tπ! C = idπ∗(TB) and V j = idV E

(see the diagram on the next page).As is well known, it suffices to know either C or V . The splitting implies

the decomposition

(1.5.2) TE = V E ⊕HE,

where HE := Im(C) is the horizontal subbundle of TE.Given a connection as above, its connection map is defined to be

(1.5.3) K := r V : TE −→ E,


0

π∗(TB)?

0 - V Ej

- TE

C

? Tπ!- π∗(TB) -

idπ∗(TB)

-

0

V E

V

?

idV E-

0?

where r : V E → E is the canonical morphism defined in the end of § 1.4.6.Obviously, the pair (K,π) is a vb-morphism of (TE,E, τE) into (E,B, π).Clearly, a tangent vector u ∈ TE is horizontal, i.e. u ∈ HE, if and only ifK(u) = 0, thus

(1.5.4) HE = ker(K).

Let (Uα, φα,Φα) be a vb-chart of E [see (1.4.3′) end the ensuing defini-tion]. Following (1.1.7) with the appropriate modifications, the correspond-ing vb-chart of TE is

(τ−1E (π−1(Uα)),Φα, Φα

), where

(1.5.5)Φα : τ−1E

(π−1(Uα)

)−→ φα(Uα)× E× B× E :

X ≡ [(γ, u)] 7−→(Φα(u),Φα(X)

)=(Φα(u), (Φα γ)′(0)

),

if X ≡ [(γ, u)] ∈ TuE, u ∈ π−1(Uα), and γ is a smooth curve in E withγ(0) = u. Then the local representation of K is the smooth map

(1.5.6) Kα ≡ KUα : Φα K Φ−1α : φα(Uα)× E× B× E −→ φα(Uα)× E


given by

(1.5.7) Kα(x, λ, y, µ) = (x, µ+ κα(x, λ).y)

(recall that line dots as above replace parentheses), where

κα : φα(Uα)× E→ L(B,E)

is a smooth map, called the local component of K (relative to the chosencharts).

A map K : TE → E is the connection map of a connection on E if andonly if K is locally given by (1.5.7) (see [Vil67, Lemma 1]).

1.5.2 Linear connections

Since (K,π) is a vb-morphism between (TE,E, τE) and (E,B, π), the re-striction of K to the fibre TuE = τ−1E (u), for every u ∈ E, is a continuouslinear map. On the other hand, TE is also equipped with the vector bundlestructure T (ℓ) = (TE, TB, Tπ), obtained by applying the tangent functorto (E,B, π). In this case, we construct the diagram

TEV

- V Er

- E

I II

TB

Tπ

?

E

τE

?

π- B

π

?

where the sub-diagram I does not close in a natural way. Therefore, K =r V is not necessarily a vb-morphism with respect to the vector bundlestructure of T (ℓ), and the restrictions of K to the fibres of T (ℓ) are notnecessarily continuous linear maps.

A connection on ℓ = (E,B, π) will be called linear if the connectionmap K is linear on the fibres of T (E). Of course, even in this case, theabove sub-diagram I does not necessarily close.

If K is a linear connection, then each local component κα is continuouslinear with respect to the second variable, i.e.

κα(x, ·) ∈ L(E,L(B,E)), x ∈ φα(Uα).


1.5.3 The Christoffel symbols of a linear connection

Let K be a linear connection on E with local components κα, α ∈ I, definedin the preceding subsection. For each index α, K determines a smooth map,

(1.5.8) Γα : φα(Uα) −→ L(E,L(B,E)),

given by

(1.5.8′) Γα(x).λ := κα(x, λ), (x, λ) ∈ φα(Uα)× E.

The maps Γαα∈I are called the (local) Christoffel symbols (or Christof-fel maps) of K.

Because of the (Banach space) identification of L(E,L(B,E)) with thespace of continuous bilinear E-valued maps on B × E, L2(E,B;E)), (see,e.g., [Lan99, Proposition 2.4]), Γα identifies with a smooth map of the formφα(Uα)→ L2(E,B;E)). Also, applying the symmetry s : E×B→ B×E, weobtain the smooth map

(1.5.9) Γα : φα(Uα) −→ L2(B,E;E),

which is the local Christoffel symbol of K in the sense of [FK72], with

(1.5.9′) Γα(x)(y, λ) = (Γα(x).λ)(y),

for every (x, λ, y) ∈ φα(Uα)× E× B.Still, after the toplinear identification L2(B,E;E) ≡ L(B,L(E)), we may

consider the map

(1.5.10) Γα : φα(Uα) −→ L(B,L(E)),

such that

(1.5.10′)(Γα(x).y

)(λ) = Γα(x)(y, λ).

Therefore, a linear connection satisfies the equalities

Kα(x, λ, y, µ) =(x, µ+ (Γα(x).λ)(y)

)

=(x, µ+ Γα(x)(y, λ)

)

=(x, µ+

(Γα(x).y

)λ)

= (x, µ + κα(x, λ).y),

for every (x, λ, y, µ) ∈ φα(U)× E× B× E and every α ∈ I.


Since the preceding maps generalize the ordinary Christoffel symbols of alinear connection (viz. covariant derivation) on a finite-dimensional smoothmanifold (see, for instancce, [KN68]), we use for all of them the termChristoffel symbols, instead of the more appropriate Christoffel maps.

We shall use the same symbol Γα to refer to any one of (1.5.8), (1.5.9)and (1.5.10), clarifying each time the range of the symbol involved.

The use of a specific type of Christoffel symbol (map) will be dictated byconcrete needs.

Given two vb-charts (Uα, φα,Φα) and (Uβ , φβ ,Φβ) with Uαβ 6= ∅, weobtain the following compatibility condition of the local components of anarbitrary connection K [see also equalities (1.4.8)–(1.4.10)]

(1.5.11)κβ(x, λ).y = Gβα((φαβ)(x))

[DGαβ(x)(y, λ) +

+ κα(φαβ(x), Gαβ(x).λ

)(Dφαβ(x).y)

],

for every (x, λ, y) ∈ φβ(Uαβ)× E× B. Here, for the sake of convenience, wehave set

φαβ := φα φ−1β .

In particular, if K is linear, then the compatibility condition of the Christof-fel symbols e.g. Γα : φα(Uα)→ L2(B,E;E)α∈I [see (1.5.9)] is

(1.5.12)Γβ(x) = Gβα

(φαβ(x)

)[DGαβ(x) +

+ Γα(φαβ(x)

)(Dφαβ(x)×Gαβ(x)

)]

for every x ∈ φβ(Uαβ). Analogous relations hold for the other types ofChristoffel symbols.

Anticipating a later application in § 1.7.4 below, we transcribe (1.5.12) interms of the transition functions Tαβ : Uαβ → GL(E)α,β∈I of E [see (1.4.2)]in the following way:

(1.5.13)Γβ(φβ(x))

(φβ)

=(T−1αβ dTαβ

)x(v) +

+ Ad(T−1αβ (x)

)(Γα(φα(x)).φα(v)

),

now for every x ∈ Uαβ(!), v ∈ TxB, and after the natural identification

idGL(E) = idGL(E),1E : T1E(GL(E))≃−−→ L(E), in virtue of (1.1.4). Here we

view the Christoffel symbols as maps Γα : φα(Uα) → L(B,L(E)), α ∈I. Recall that the first summand in the right-hand side of (1.5.13) is theleft Maurer-Cartan differential of Tαβ (see § 1.2.6), while Ad is the adjointrepresentation of GL(E).


The proof of both (1.5.11) and (1.5.12) [or (1.5.13)] is based on elemen-tary computations.

Because of the bijective correspondence between linear connection K andfamilies of compatible smooth maps Γαα∈I , we write

K ≡ Γαα∈I ,

for whatever form of the Christoffel symbols.

1.5.4 Linear connections and covariant derivations

Before proceeding, we recall that X (B) is the set of smooth vector fields onB (see § 1.1.9), and Γ(E) is the set of smooth sections of a vector bundleℓ = (E,B, π) (§ 1.4.4(f)).

A linear connection induces a covariant derivation

∇ : X (B)× Γ(E) −→ Γ(E) : (X, ξ) 7→ ∇Xξ := K Tξ X.

As in the case of an ordinary covariant derivation on a finite-dimensionalmanifold, ∇ is C∞(B,R)-linear (hence, also R-linear) with respect to thefirst variable, whereas, with respect to the second variable, it is R-linear andsatisfies the Leibniz condition

(1.5.14) ∇X(fξ) = f∇Xξ +X(f) · ξ,

for every f ∈ C∞(B,R) and every ξ ∈ Γ(E). Moreover, the local principalpart of ∇, with respect to a vb-chart (Uα, φα,Φα), is given by

(1.5.15)(∇Xξ

)α(x) = Dξα(x)

(Xα(x)

)+ Γα(x)

(Xα(x), ξα(x)

).

Here Xα is the local principal part of X, with respect to the chart (Uα, φα)of B [see (1.1.14) and (1.1.14′)], and ξα is the local principal part of ξ, withrespect to (Uα, φα,Φα) [see (1.4.11) and (1.4.11′)]. The maps Γα are alsothe Christoffel symbols of ∇.

• Warning. It should be noted that, unlike the finite-dimensional case,an operator ∇, satisfying only the aforementioned linearity condition and(1.5.14), does not determine a linear connection in the sense of § 1.5.2; hence,linear connections and covariant derivations are not equivalent notions inthe infinite-dimensional framework. However, if an operator ∇ satisfies alsocondition (1.5.15) over every local trivialization, with compatible Γαα∈I[in the sense of (1.5.12)], then a covariant derivation identifies with a linearconnection.


1.5.5 Parallel displacement and holonomy groups

Let (E,B, π) be a Banach vector bundle endowed with a linear connectionK. If γ : [0, 1] → B is a smooth curve, then a section of E along γ is asmooth curve ξ : [0, 1] → E such that π ξ = γ. The set of such sectionsis denoted by Γγ(E). The choice of the interval [0, 1] is only for the sake ofconvenience and does not restrict of the generality.

A section ξ ∈ Γγ(E) is called parallel with respect to K if

(1.5.16) ∇γ ξ := K Tξ ∂ = 0,

with ∂ ≡d

dtdenoting the basic vector field of R. In virtue of (1.5.4) and

(1.5.15) [see also (1.1.11)], ξ(t) := Ttξ(∂t) ∈ Hξ(t)E; hence, ξ is a horizontalcurve in E. Locally, over a vb-chart (Uα, φα,Φα) of E, (1.5.16) leads to

(1.5.17) ξ′α(t) = −Γα(φα(γ(t))((φα γ)′(t), ξα(t)

),

where ξα : [0, 1]→ E now denotes the principal part of the local representa-tion Φαξ : [0, 1]→ φα(Uα)×E of ξ [compare with the general case of (1.4.11)and (1.4.11′)]. Differential equation (1.5.17) is linear of type x′ = A(t) · xwith A(t) = −Γα(φα(γ(t))

((φα γ)′(t), ·

)∈ L(E,E). By the general theory

of such equations (in Banach spaces) we prove that, for any u ∈ E, thereexists a unique parallel section ξu along γ, such that ξu(0) = u.

Under the previous notations, the parallel displacement/translationalong the curve γ is defined to be the map

(1.5.18) τγ : Eγ(0) −→ Eγ(1) : u 7→ ξu(1).

In virtue of the properties of the resolvent of (1.5.17), we find that

• τγ is an isomorphism of Banach spaces whose inverse is τ−1γ = τγ−1 ;

• τγ2∗γ1 = τγ2 τγ1 .

As usual, γ−1 is the inverse (or reverse) of γ and γ2 ∗ γ1 the product (orjuxtaposition, composition) of γ1 followed by γ2.

Accordingly, the holonomy group of the linear connection K, withreference point b ∈ B, denoted KΦb, is defined by

KΦb := τγ : Eb → Eb,

for all smooth curves γ : [0, 1] → B with γ(0) = γ(1) = b. Similarly, therestricted holonomy group of the linear connection K, with referencepoint b ∈ B, is

KΦ0b := τγ : Eb → Eb,


for all homotopic to zero smooth curves γ : [0, 1]→ B with γ(0) = γ(1) = b.

The left superscriptK is set in order to distinguish the present holonomygroups from those induced by connections on principal bundles, as discussedin later sections.

By appropriate identifications, KΦb and KΦ0b can be realized as Banach-

Lie subgroups of GL(E). For details we refer to [Max72] and the briefexposition of § 1.9 below.

1.5.6 Related linear connections

Let (f, h) be a vb-morphism between the vector bundles ℓ = (E,B, π) andℓ′ = (E′, B′, π′). Two connections K and K ′ on E and E′, respectively, aresaid to be (f, h)-related if

(1.5.19) K ′ Tf = f K;

in other words, the next diagram commutes.

TETK

- E

TE′

Tf

?

K ′- E′

f

?

For various applications, it is useful to express (1.5.19) locally. To thisend assume that the structures of ℓ and ℓ′ are determined by the familiesof vb-charts (Uα, φα,Φα)α∈I and (Vβ , ψβ ,Ψβ)β∈J , respectively, where(Uα, φα) ≡ (Uα, φα,B)α∈I and (Vβ , ψβ) ≡ (Vβ , ψβ ,B

′)β∈J are charts ofB and B′, respectively. For a pair of vb-charts (Uα, φα,Φα) and (Vβ , ψβ,Ψβ)with h(Uα) ⊆ Vβ (ensured by the definition of a vb-morphism), we obtainthe commutative diagram:

EUα = π−1(Uα)f

- (π′)−1(Vβ) = E′Vβ

φα(Uα)× E

Φα

?

Ψβ f Φ−1α

- ψβ(Vβ)× E′

Ψβ

?


As a result, the local representation of f takes the form

(1.5.20)(Ψβ f Φ−1α

)(x, λ) =

(hβα(x), f#βα(x).λ

),

for every (x, λ) ∈ φα(Uα)× E, where

hβα := ψβ h φ−1α ,(1.5.21)

f#βα : φα(Uα) −→ L(E,E′).(1.5.22)

The map f#βα, denoting the local principal part of f (with respect to theprevious local representation), is given by [see also (1.4.1) and (1.4.5)]

(1.5.23) f#βα(x) = Ψβ,h(b) fb Φ−1α,b = τ ′β,h(b) fb τ−1α,b ,

if π(b) = x. Moreover, differentiation of (1.5.20) yields

(1.5.24)D(Ψβ f Φ−1α

)(x, λ).(y, µ) =

(Dhβα(x).y, f#βα(x).µ + (Df#βα(x).y).λ

).

Similarly, using the vb-charts (Uα,Φα, Φα), and (Vβ,Ψβ, Ψβ) of the tan-gent bundles (TE,E, τE) and (TE′, E′, τE′), respectively [see (1.5.5)], wecheck that the corresponding local representation of Tf : TE → TE′, namely

(1.5.25) Ψβ Tf Φ−1α : φα(Uα)× E× B× E −→ ψβ(Vβ)× E′ × B′ × E′,

has, in virtue of (1.5.20) and (1.5.24), the expression

(1.5.26)(Ψβ Tf Φ−1α )(x, λ, y, µ) =

=(hβα(x), f#βα(x).λ,Dhβα(x).y, f#βα(x).µ +

(Df#βα(x).y

).λ),

for every (x, λ, y, µ) ∈ φα(Uα)× E× B× E.Now the equivalent form of (1.5.19), in terms of the local components of

the connections, is essentially found from the commutative diagram on thenext page. Indeed, evaluating the equality

K ′β (Ψβ Tf Φ−1α

)=(Ψβ f Φ−1α

)Kα

at any (x, λ, y, µ) ∈ φα(Uα) × E × B × E, and applying equalities (1.5.20),(1.5.26), together with (1.5.7), we obtain:

(1.5.27)κ′β(hβα(x), f#βα(x).λ

)(Dhβα(x).y) =

f#βα(κα(x, λ).y

)−(Df#βα(x).y

).λ,

for every (x, λ, y, µ) ∈ φα(Uα)× E× B× E

36C

hapter1.B

anachm

anifoldsand

bundles

φα(Uα)× E× B× EKα = Φα K Φ−1α - φα(Uα)× E

T (EUα)K

-

Φα

EUα

Φα

-

T (E′Vβ )

Tf

?

K ′- E′Vβ

f

?

ψβ(Vβ)× E′ × B′ × E′

Ψβ Tf Φ−1α

?

K ′β = Ψβ K′ Ψ−1β

-

Ψβ

ψβ(Vβ)× E′

Ψβ f Φ−1α

?

Ψβ

-

1.6. Banach principal bundles 37

In particular, if K and K ′ are linear connections, then (1.5.27) can beexpressed via the respective Christoffel symbols. For the sake of complete-ness, we write down the relatedness condition in terms of the Christoffelsymbols discussed in 1.5.3:

• If Γα : φα(Uα)→ L(E,L(B,E)), Γ′β : ψβ(Vβ)→ L(E′,L(B′,E′)), then

(1.5.28)

[Γ′β(hβα(x))

(f#βα(x).λ

)](Dhβα(x)) =

f#βα((Γα(x).λ).y

)−(Df#βα(x).y

).λ.

• If Γα : φα(Uα)→ L(B,L(E)), Γ′β : ψβ(Vβ)→ L(B′,L(E′)), then

(1.5.29)

(Γ′β(hβα(x))(Dhβα(x).y)

) f#βα(x) =

f#βα (Γα(x).y)−Df#βα(x).y.

• If Γα : φα(Uα)→ L2(B,E;E), Γ′β : ψβ(Vβ)→ L2(B′,E′;E′), then

(1.5.30) Γ′β(hβα(x)) (Dhβα(x)× f#βα

)= f#βα Γα(x)−Df#βα(x).

Equalities (1.5.28)–(1.5.30) hold for every (x, λ, y) ∈ φα(Uα)× E× B.When we deal with bundles over the same base B and vb-morphisms

(f, idB), all the preceding local expressions have considerably simplified vari-ants. Note that, in this case, we can always find vb-charts (Uα, φα,Φα) and(Uα, φα,Φ

′α) over the same open cover Uαα∈I of B. Thus hβα = idφα(Uα)

and, for simplicity, we write

(1.5.31) f#α := f#αα : φα(Uα) −→ L(E,E′).

1.6 Banach principal bundles

Basic material can be found in [Bou67], [KM97]. For finite-dimensionalprincipal bundles, which are very similar to the Banach case (and are treatedin many books), we refer e.g. to [KN68], [Nab00].

1.6.1 The structure of a principal bundle

A principal bundle is a quadruple ℓ = (P,G,B, π), where P and B aresmooth Banach manifolds, π : P → B a smooth map, and G a Banach-Liegroup acting on P (from the right), such that: For every x ∈ B, there isan open U ⊂ B, with x ∈ U , and a diffeomorphism Ψ: U × G → π−1(U)satisfying the following properties:


(PB. 1) π Ψ = pr1, in other words, the next diagram is commutative

U ×GΨ

- π−1(U)

U

π

?

pr1-

(PB. 2) For every y ∈ U and g, g′ ∈ G,

Ψ((y, g) · g′

)= Ψ(y, g · g′) = Ψ(y, g) · g′,

that is, Ψ is an equivariant map with respect to (the action of) G. Briefly,Ψ is a G-equivariant map.

In the preceding equality, (y, g) · g′ denotes the obvious action of G onthe right of U×G. The same equality means that, Ψ(y, g) ·g′ = Ψ(y, g ·g′) ∈π−1(U), for every g′ ∈ G. Therefore, for every p ∈ π−1(U) and g ∈ G, itfollows that p · g ∈ π−1(U). If we set

Φ := Ψ−1,(1.6.1)

then Φ is also G-equivariant; namely,

Φ(p · g) = Φ(p) · g; p ∈ π−1(U), g ∈ G.(1.6.2)

A pair of the form (U,Φ) or (U,Ψ) determines a (local) trivializationof P . A family C = (Uα,Φα)α∈I , of local trivializations, where (Uαα∈Iis an open cover of B, will be called a trivializing cover of P .

In a standard terminology, P is the total space, B the base, π the pro-jection and G the structure group of the bundle.

If there is no danger of confusion, we refer to a principal bundle ℓ =(P,G,B, π) either by ℓ or P .

Immediate consequences of the definitions are the following fundamentalproperties of a principal bundle:

• The projection π is a submersion.

• The fibres π−1(x), x ∈ B, are non empty regular submanifolds of P .In particular, if B is a Hausdorff space, then every π−1(x) is a closedsubmanifold of P .


• For every x ∈ B, π−1(x) = p·G, if p is any element of P with π(p) = x.

• If p ∈ π−1(x) and g ∈ G, then p · g ∈ π−1(x), thus G acts on the rightof each fibre.

• The action of G on P is free.

• The action of G on the fibres is freely transitive.

In particular, if (U,Φ) is a local trivialization of P , then

• Φ(π−1(x)

)= x ×G, for every x ∈ U .

• The map

(1.6.3) Φx := pr2 Φ∣∣π−1(x)

: π−1(x) −→ G

is a G-equivariant diffeomorphism.

• Φ−1x (g) = Ψ(x, g) for every g ∈ G.

• By means of Φx, the fibre π−1(x) has the structure of a Banach-Liegroup and Φx becomes an isomorphism of Lie groups.

1.6.2 Morphisms of principal bundles

A morphism between the principal bundles ℓi = (Pi, Gi, Bi, πi) (i = 1, 2)(pb-morphism, for short) is a triple (f, ϕ, h) where f : P1 → P2 andh : B1 −→ B2 are smooth maps and ϕ : G1 → G2 a morphism of Banach-Liegroups, satisfying the following conditions:

(PBM. 1) π2 f = h π1;

in other words, the diagram

P1f

- P2

B1

π1

? h- B2

π2

?

is commutative, and

(PBM. 2) f(p · g) = f(p) · ϕ(g), (p, g) ∈ P1 ×G1.;


that is, f is equivariant with respect to (the actions of) G and G′.

The map h : B1 → B2 is completely determined by f and ϕ.

A pb-morphism (f, ϕ, h) is an isomorphism if f, h are diffeomorphismsand ϕ is an isomorphism of Banach-Lie groups, If B1 = B2 = B and h = idB ,then (f, ϕ, idB) is called a B-morphism. If G1 = G2 = G and ϕ = idG, wecall (f, idG, h) a G-morphism. Finally, if B1 = B2 = B, G1 = G2 = G,h = idB and ϕ = idG, then (f, idG, idB) is said to be a G-B-morphism.

Every G-B-morphism is a pb-isomorphism.

1.6.3 Natural sections and transition maps

The sections of a principal bundle are defined in the usual way. We denoteby Γ(U,P ) the set of smooth sections of a principal bundle ℓ = (P,G,B, π)over an open U ⊂ B.

If C = (Uα,Φα) |α ∈ I is a trivializing cover of ℓ = (P,G,B, π), thenatural sections of P , with respect to C, are the maps [see also (1.6.1)]

(1.6.4) sα : Uα −→ P : x 7→ Ψα(x, e) = Φ−1α (x, e).

An arbitrary section s ∈ Γ(U,P ) induces a trivialization (U,Φ), withrespect to which s is the corresponding natural. Indeed, it suffices to set

(1.6.5) Ψ(x, g) := s(x) · g, (x, g) ∈ U ×G.

Therefore, Φ := Ψ−1 is given by

(1.6.5′) Φ(p) := (π(p), g); p ∈ π−1(U),

where g is determined by the equality p = s(π(p)) · g.As a result, there is a bijection between trivializations and smooth sec-

tions of a principal bundle. In particular, a principal bundle admits globalsections if and only if it is trivial, i.e. isomorphic to the trivial bundle(B ×G,G,B,pr1).

A useful tool is the map connecting elements of the same fibre. Moreprecisely, let us consider a principal bundle P as before. In analogy to thefibre product of vector bundles defined in § 1.4.4(a), we define the followingparticular fibre product

P ×B P = (p, q) ∈ P × P : π(p) = π(q) ,

and the smooth map

(1.6.6) k : P ×B P −→ G : q = p · k(p, q), (p, q) ∈ P ×B P ;


in other words, since p and q belong to the same fibre of P , k(p, q) is theunique element of G such that q = p ·k(p, q). The smoothness of k is checkedlocally by observing that, with respect to a trivialization (U,Φ) of P ,

k(p, q) =(

pr2(Φ(p))−1·(

pr2(Φ(q)),

for every (p, q) ∈ π−1(U)×U π−1(U).

Given a trivializing cover C = (Uα,Φα)α∈I of ℓ = (P,G,B, π), thetransition maps or functions of ℓ (with respect to C) are the smoothmaps

(1.6.7) gαβ : Uαβ −→ G : x 7→(Φα,x Φ−1β,x

)(e),

for all α, β ∈ I and Uαβ 6= ∅.The following equations provide equivalent ways to define the transition

functions:

(Φα Φ−1β

)(x, g) = (x, gαβ(x) · g) ,(1.6.8)

sβ(x) = sα(x) · gαβ(x),(1.6.9)

for every (x, g) ∈ Uαβ ×G.The transition functions form a cocycle (more precisely, a 1-cocycle);

that is,

(1.6.10) gαγ = gαβ · gβγ

which means that

gαγ(x) = gαβ(x) · gβγ(x), x ∈ Uαβγ .

It follows that

(1.6.11) gαα = e and gβα = g−1αβ .

The last equality means that gβα(x) = gαβ(x)−1, for every x ∈ Uαβ [comparewith the cocycle of a vector bundle (1.4.7)].

As in the case of a vector bundle (see § 1.4.2), a principal bundle iscompletely determined by its cocycles. We recall that, given a cocyclegαβ : Uαβ → G |α ∈ I over an open cover Uαα∈I of B, we considerthe set

S =⋃

α,β∈I

(α × Uα ×G) ,


and define the equivalence relation:

(α, x, g) ∼ (β, x′, g′) ⇔ x = x′ and g′ = gβα(x) · g.

Setting P = S/∼, we obtain the principal bundle ℓ = (P,G,B, π), withπ([(α, x.g)]) := x, and local trivializations Φα : π−1(Uα)→ Uα ×G given by

Φα[(β, x, g)] := (x, gαβ(x) · g).

The bundle thus constructed is unique up to isomorphism. This a particularcase of the following general result (see the vb-analog in § 1.4.3):

Two principal bundles (P,G,B, π) and (P ′, G,B, π′) are G-B-isomor-phic if and only they have cohomologous cocycles gαβ and g

′αβ,

respectively, over an open cover Uαα∈I of B.

Two cocycles, as above, are said to be cohomologous if there is a family ofsmooth maps hα : Uα → G |α ∈ I such that

g′αβ = hα · gαβ · h−1β on Uαβ.

Indeed, assume first that the bundles are G-B-isomorphic. If sα ands′α are the natural sections of P and P ′, respectively (over a common opencover Uα of B), then we define hα as the unique smooth map satisfyingf sα = s′α · hα. The equivariance of f and equality (1.6.9) now imply thatthe cocycles are cohomologous.

Conversely, assume that the cocycles are cohomologous. We definef : P → P ′ by setting

f(p) := s′α(x) · hα · gα(p),

for every p ∈ P , with π(p) = x ∈ Uα, where gα(p) is the unique elementof G determined by p = sα(x) · gα(p). The assumption ensures that f is awell-defined map such that π′ f = π. Its smoothness follows from equalityf |π−1(Uα) = (s′α π) · (hα π) ·gα and the smoothness of gα resulting, in turn,from equality gα = k (idP , sα π), where k is the smooth map (1.6.6).

In particular, if two bundles as above have the same cocycles, then theyare G-B-isomorphic by means of the map f , given by f(p) = s′α(x) · gα(p),if π(p) = x ∈ Uα.

Analogously to the classification of vector bundles, discussed towards theend of § 1.4.3, we have the identification

PB(G) ≡ H1(B,G),

where PB(G) is the set of equivalence classes of principal bundles over B,with structure group G, and G is the sheaf of germs of G-valued smoothmaps on B.


1.6.4 The pull-back of a principal bundle

Let ℓ = (P,G,B, π) be a principal bundle and h : B′ → B a smooth map.The pull-back of ℓ by h is the principal bundle h∗(ℓ) = (h∗(P ), G,B′, π∗),where

h∗(P ) = B′ ×B P := (x′, p) ∈ B′ × P : h(x′) = π(p),

π∗ := pr1 |h∗(P ) : h∗(P ) −→ B′,

h∗ := pr2 |h∗(P ) : h∗(P ) −→ P.

The action of G on the total space h∗(P ) is given by

δ∗ : h∗(P )×G −→ h∗(P ) : ((x′, p), g) 7→ (x′, p · g).

The fibres of h∗(ℓ) are isomorphic to the fibres of ℓ (and both isomorphicto G), while (h∗, idG, h) is a G-morphism of h∗(ℓ) into ℓ. We add that eachlocal trivialization (Uα,Φα) of P determines the trivialization

(f−1(Uα),Φ∗α

)

of h∗(P ), where

Φ∗α : h−1(Uα)×Uα π−1(Uα) −→ Uα ×G : (b′, p) 7→ Φα,h(b′)(p)

[see (1.6.3)]. As a result, the corresponding cocycle g∗αβα,β∈I of f∗(P ) isgiven by g∗αβ = gαβ hα,β∈I .

The universal property in the category of principal bundles now reads:If ℓ1 = (P1, G,B

′, π1) is a principal bundle and (f, idG, h) : ℓ1 → ℓ a pb-morphism, then there is a uniqueG-B′-(iso)morphism (π, idG, idB′) : ℓ1 → ℓ∗

such that f = h∗ π. We obtain now the following diagram:

P1

h∗(P ) = B′ ×B Ph∗

-

π

-

P

f

-

B′

π∗

? h-

π1

-

B

π

?

Clearly, π∗ and h∗ are the projections to the first and second factor, respec-tively.


1.6.5 The frame bundle of a vector bundle

Let ℓ = (E,B, πE) be a vector bundle of fibre type the Banach space E.Following [Bou67, no 7.10.1], the set

P (E) := (x, f) : x ∈ B, f ∈ Lis(E, Ex)

is an open submanifold of the linear map bundle L(B×E, E) [see § 1.4.4(c)].GL(E) acts on the right of P (E) by setting

(x, f) · g := (x, f g); (x, f) ∈ P (E), g ∈ GL(E).

If πP : P (E)→ B is the map with πP (x, f) := x, then the quadruple

ℓ(E) := (P (E),GL(E), B, πP )

is a principal bundle, called the frame bundle of E. The term bundle(of linear) frames is also in use. Equivalently, we may write

P (E) :=⋃

x∈B

Lis(E, Ex).

The local structure of P (E) is as follows: Let (Uα, τα) be a local trivi-alization of E with τα : π−1E (Uα)→ Uα × E. Then we obtain a local sectionσα : Uα → P (E) of P (E) by setting [recall the notation of (1.4.3)]

(1.6.12) σα(x) := (x.τ−1α,x), x ∈ U.

In virtue of (1.5.7) and (1.6.5′), equality (1.6.12) determines the local trivi-alization of P (E)

(1.6.13)Ψα : Uα ×GL(E) −→ π−1P (Uα) :

Ψα(x, g) = σα(x) · g = (x, τ−1α,x) · g = (x, τ−1x g),

whose inverse Φα : π−1P (Uα)→ Uα ×GL(E) is given by

(1.6.14) Φα(x, f) = (x, τα,x f), if f ∈ Lis(E, Ex).

Clearly, σα can be thought of as the natural section of P (E) with respectto (Uα,Ψα).

The preceding local structure of P (E), derived from that of E, impliesthat the transition functions of P (E) and E coincide; that is,

(1.6.15) gαβ = Tαβ : Uαβ −→ GL(E); α, β ∈ I.


where Tαβ is the cocycle of E, with respect to the trivializing cover(Uα, τα)α,β∈I (of E), and gαβ is the cocycle of P (E), with respect tothe corresponding trivializing cover (Uα,Φα)α,β∈I (we refer also to § 1.4.2and § 1.6.3).

The initial vector bundle E is related with P (E) in the following way,which is a particular case of an associated bundle discussed in the nextsubsections. Namely, we define on P (E)× E the equivalence relation

(x′, f ′, u′) ∼ (x, f, u) ⇔

x′ = x and ∃ g ∈ GL(E) : (f ′, u′) = (f g, g−1(u))

[g is uniquely determined by the free action of GL(E) on P (E)]. Let

E := P (E)× E/∼,

π : E −→ B : π([(x, f, u)]) := x.

Then (E, B, π) is a vector bundle isomorphic to (E,B, πE). In fact, if weare given a trivializing cover (Uα,Φα) of P (E), with corresponding naturalsections σα, we define the trivializations [see also (1.6.12)]

(1.6.16)τ−1α : Uα × E −→ π−1(Uα) :

τ−1α (x, u) := [(σα(x), u)] = [(x, τ−1α,x, u)];

thus, by simple computations,

(1.6.17) τα([(x, f, u)]

)=(x, (τα,x f)(u)

).

The desired vb-isomorphism is provided by the map

(1.6.18) F : E −→ E : [(x, f, u)] 7→ f(u).

It is well-defined for if [(x, f, u)] = [(x′, f ′, u′)], then x = x′ and f ′ = f g,u′ = g−1(x), for a (uniquely determined) g ∈ GL(E). Therefore f ′(u′) =f(u). Moreover,

1. πE F = π.2. F is injective: Let [(x, f, u)], [(x′, f ′, u′)] ∈ E with f ′(u′) = f(u).

Since

x′ = πP ([(x′, f ′, u′)]) = πE(f ′(u′)) = πE(f(u)) = πP ([(x, f, u)]) = x,

it follows that f, f ′ : E→ Ex. Setting g := f−1 f ′, we check that

[(x′, f ′, u′)] = [(x, f g, g−1(u))] = [(x, f, u)].


3. F is surjective: Let an arbitrary h ∈ Ex. If τα,x : Ex → E is theBanach space isomorphism induced by a trivialization (Uα, τα) of E, withx ∈ Uα, then [(x, τ−1α,x, τα,x(h))] is mapped to h by F .

It remains to show that F (and analogously F−1) satisfies condition(VBM. 2) of § 1.4.3. Indeed, for an arbitrary x0 ∈ B, we consider a trivial-ization (Uα, τα) of E, x0 ∈ Uα and the trivialization (Uα, τα) given by (1.6.17)(also ultimately determined by (Uα, τα), according to the local structure ofP (E) and E). Then, F (π−1(Uα)) ⊆ π−1E (Uα) and

(τα F τ−1α )(x, u) = (x, u), (x, u) ∈ U × E.

Therefore, restricted to the fibres over x ∈ U , we see that the map

U ∋ x 7−→ τα,x F τ−1α,x = idE

is smooth, thus proving the claim.

1.6.6 Associated bundles

Given a principal bundle, we intend to associate to it principal and vectorbundles by means of appropriate Lie group homomorphisms.

a) Principal bundles associated by Lie group morphisms

Let ℓ = (P,G,B, π) be a Banach principal bundle, and let ϕ : G → H be amorphism of Banach Lie groups. Then G acts on the left of H by

G×H −→ H : (g, h) 7→ g · h := ϕ(g) · h,

and on the right of P ×H by setting

(p, h) · g :=(p · g, ϕ(g−1) · h

); (p, h) ∈ P ×H, g ∈ G.

The previous action induces the following equivalence relation on P ×H:

(p′, f ′) ∼ (p, h) ⇔ ∃ g ∈ G : (p′, h′) =(p · g, ϕ(g−1) · h

).

The resulting quotient space, denoted by P×GH, has a differential structuremaking the natural projection

κ : P ×H −→ P ×G H : (p, h) 7→ [(p, h)]

a submersion. Thus, a map g : P×GH → Y (Y : smooth manifold) is smoothif and only if so is g p.


Regarding the smooth structure of the above quotient space, we refer to[Bou67, nos 6.6.1, 6.5.1], in conjunction with [Bou67, no 5.9.5], the lattercontaining more results on the structure of quotient manifolds induced byequivalence relations. Detailed proofs (in a more general setting) can befound in [KM97, § 37.12] and [Die72, §§ 16.10.3, 16.14.7].

Another way to define a smooth structure on P ×G H, by gluing localdata, will be described below. First, let H act on the right of P ×G H by

[(p, h)] · h′ = [(p, h · h′)]; p ∈ P, (h, h′) ∈ H ×H,

and define the projection

πH : P ×G H −→ B : [(p, h)] 7→ πH([(p, h)]) := π(p).

Then the quadruple ϕ(ℓ) := (P ×G H,H,B, πH) is, by definition, the prin-cipal bundle associated to ℓ by the Lie group morphism ϕ : G→ H.

The local structure of ϕ(ℓ) is derived from that of ℓ as follows: Let(Uα,Φα)α,β∈I be a trivializing cover of P , with Φα : π−1(Uα) → Uα × G,and natural sections sα : Uα → P given by sα(x) = Ψα(x, e) := Φ−1α (x, e).Then we define the trivializations

(1.6.19) Ψα : Uα ×H −→ π−1H (Uα) : (x, h) 7→ Ψ(x, h) := [(sα(x), h)]).

It is easy to check that Ψα is an equivariant (with respect to G and H)bijection, whose inverse Φα : π−1H (Uα)→ Uα ×H is given by

(1.6.20) Φα([(p, h)]) := (x, ϕ(g) · h),

with x = π(p), and g ∈ G determined by p = sα(x) · g.The previous local structure induces also a smooth structure on P ×GH:

Each Ψα determines a smooth structure on π−1H (Uα) by transferring that ofUα × H. Since the local smooth structures coincide on the overlappings,we obtain a smooth structure on the quotient by gluing the local structurestogether.

The relations between the local trivializations of ϕ(ℓ) and ℓ lead to thefollowing equalities, connecting the respective cocycles and the local sectionsof the aforementioned bundles:

gαβ = ϕ gαβ : Uαβ −→ H(1.6.21)

sα(x) = [(sα(x), e)]; x ∈ Uα,(1.6.22)

for all indices α, β ∈ I.


The bundles ℓ and ϕ(ℓ) are also related by the natural map

(1.6.23) κ : P −→ P ×G H : p 7→ κ(p) := [(p, e)],

which, in fact, determines the B-morphism (κ, ϕ, idB) between them. Then(1.6.22) is rewritten as

(1.6.22′) sα(x) = [(sα(x), e)], x ∈ Uα.

It is worth adding that P×GH (together with κ) has the following univer-sal property: If (F,ϕ, idB) is a B-morphism of ℓ into another principal bun-dle ℓ′ = (P ′,H,B, π′), then there a unique H-B-isomorphism (θ, idH , idB)of ϕ(ℓ) onto ℓ′, such that F = θ κ, as pictured also in the next diagram:

Pκ- P ×G H

P ′

∼= θ

?

F-

Actually, θ is given by

(1.6.24) θ([(p, h)]) = F (p) · h, [(p, h)] ∈ P ×G H.

Its smoothness is a consequence of the manifold structure on the quotientspace mentioned earlier. The other properties of θ are clear.

b) Vector bundles associated by representations of the structure groupLet again ℓ = (P,G,B, π) be a Banach principal bundle, and let ϕ be arepresentation of G into a Banach space E, i.e. ϕ : G→ GL(E) is a morphismof Banach-Lie groups. We construct a vector bundle associated to ℓ byspecializing the process of the foregoing case a) as follows:

We first define a left action of GL(E) on E

GL(E)× E : (g, u) 7→ g · u := ϕ(g)(u),

and the action of GL(E) on the right of P × E by

(p, u) · g :=(p · g, ϕ(g−1)(u)

); (p, u) ∈ P × E, g ∈ G.

We obtain the quotient space E := P ×G E and the map πE : E → B withπE([(p, u)]) := π(p). Then (E,B, πE) is the vector bundle associated toP by ϕ : G→ GL(E), of fibre type E.


The local structure of E is defined as follows: As in case a), we considera trivializing cover (Uα,Φα)α∈I of P , with natural sections sα : Uα → P .Then we define the maps

(1.6.25) τ−1α : Uα × E −→ π−1E (Uα) : (x, u) 7→ τ−1α (x, u) := [(sα(x), u)]);

thus (Uα, τα)α∈I is trivializing cover of E. We clarify that τα is definedanalogously to (1.6.20); namely,

(1.6.26) τα([(p, u)]) = (x, ϕ(g)(u)),

where x = π(p), and g ∈ G is determined by the equality p = sα(x) · g.Moreover, the corresponding cocycle Tαβ of E is connected with the

cocycle gαβ of P by

(1.6.27) Tαβ = ϕ gαβ : Uαβ −→ GL(E); α, β ∈ I.

Applying the previous results to the particular case of the frame bundleP (E) of a vector bundle E, and the morphism idGL(E) : GL(E) → GL(E),

we see that the vector bundle P (E)×GL(E)E, associated to P (E) by idGL(E),coincides with the bundle E defined in § 1.6.5. Therefore, by (1.6.18), E isisomorphic to P (E)×GL(E) E.

c) Interrelations

We consider, once again, a Banach principal bundle (P,G,B, π) and aBanach-Lie group morphism (representation) ϕ : G → GL(E). Then, invirtue of case a), we obtain the principal bundle

(Pϕ := P ×G GL(E),GL(E), B, πϕ

)

(for convenience, we set πϕ = πGL(E)), and, in virtue of case b), the vectorbundle (

Eϕ := P ×G E, B, π).

In turn, the latter determines the principal bundle of frames (see § 1.6.5)

(P (Eϕ),GL(E), B, π).

We wish to connect the three principal bundles P , Pϕ and P (Eϕ). Al-ready, we know that P , Pϕ are related by the B-morphism (κ, φ, idB) deter-mined by (1.6.23), now taking the form κ(p) = [(p, idE)]. On the other hand,if gαβ is the cocycle of P , then equalities (1.6.21), (1.6.25) and (1.6.15)imply that the corresponding cocycles of Pϕ, Eϕ and P (Eϕ) coincide with


ϕ gαβ. Therefore Pϕ and P (Eϕ) are GL(E)-B-isomorphic. However,based on the discussion at the end of § 1.6.6(b), we can single out a concreteisomorphism, because there is also a natural B-morphism of P into P (Eϕ).With this in mind, we first define the map

(1.6.28) F : P −→ P (Eϕ) : p 7→ (x, p),

where x := π(p) and p : E → Eϕ,x is the Banach space isomorphism givenby p(u) := [(p, u)] (recall that [(p, u)] ∈ Eϕ = P ×G E).

The smoothness of F is checked locally: If p0 is an arbitrary point in Pwith π(p0) = x0, we choose a local trivialization (U,Φ) of P with x0 ∈ U , andconsider the corresponding trivializations (U, τ) of Eϕ and (U,Φ) of P (Eϕ),given by the analogs of (1.6.25) and (1.6.14), respectively (for conveniencewe drop the index α from the latter). Since F (π−1(U)) ⊆ π−1ϕ (U), we obtainthe local representation Φ F Φ−1, as in the diagram

π−1(U)F

- π−1ϕ (U)

U ×G

Φ

?

Φ F Φ−1- U ×GL(E)

Φ

?

Therefore, for every (x, g) ∈ U × E,

(Φ F Φ−1

)(x, g) = (Φ F )(Ψ(x, g)) =

= (Φ F )(s(x) · g) =(x, τx s(x) · g

).

Because, for every u ∈ E,

(τx s(x) · g

)(u) = τx([(s(x) · g, u)])

= (pr2 τ)([(s(x), ϕ(g)(u))])

= pr2(x, ϕ(g)(u)) = ϕ(g)(u),

it follows that(Φ F Φ−1

)(x, g) = (x, ϕ(g)), from which we deduce the

smoothness of F at (an arbitrary) p0 ∈ P .On the other hand, it is immediate that π F = π, while

F (p · g) =(π(p · g), p · g

)=(π(p), p ϕ(g)

)= F (p) · ϕ(g),


for every p ∈ p and g ∈ G; that is, F is equivariant with respect to theactions of G and GL(E) on P and P (Eϕ). As a consequence of the previousarguments, (F,ϕ, idB) is a B-morphism of P into P (Eϕ).

The desired GL(E)-B-isomorphism between Pϕ and P (Eϕ) is providedby the universal property of the quotient manifold; namely, according to(1.6.24), (1.6.28) and the action of GL(E) on P (Eϕ),

(1.6.29) θ : Pϕ∼=−−→ P (Eϕ) : [(p, g)] 7→ θ([(p, g)]) := (π(p), p g).

Summarizing, we obtain the following diagram [in analogy to the onegiven in part a)]

Pκ- Pϕ = P ×G GL(E)

P (Eϕ)

∼= θ

?

F-

where κ, F and θ are given, respectively, by (1.6.23), (1.6.28) and (1.6.29).For later use, let us relate also the natural sections of the previously men-

tioned bundles, over corresponding trivializations. To this end, we denoteby

• sα the natural sections of P , with respect to the trivializing cover(Uα,Φα)α∈I ;

• sϕα the natural sections of Pϕ, with respect to the trivializing cover(Uα,Φα)α∈I , defined as in case a).

• σϕα the natural sections of P (Eϕ), with respect to the trivializing cover(Uα,Φα)α∈I , derived from the study of the smoothness of(1.6.28).

Then, adapting (1.6.22) to the present settings, we have that

(1.6.30) sϕα(x) = [sα(x), idE] = κ(sα(x)); x ∈ Uα, α ∈ I.

Taking into account the earlier definition of Φα (where the subscript α wasthen omitted), we check that

(1.6.31) σϕα(x) =(x, sα(x)

)= F (sα(x)); x ∈ Uα, α ∈ I,

where sα : E→ (P×GE)x is the Banach space isomorphism given by sα(u) =[(sα, u)]. Therefore, applying θ to the preceding equality, we also have

(1.6.32) σϕα(x) = θ(sϕα(x)

); x ∈ Uα, α ∈ I.


The latter equality is in accordance with the fact that the cocycles of Pϕand P (Eϕ) coincide, as explained before introducing the map (1.6.28).

1.6.7 The exact sequence associated to a principal bundle

Given a principal bundle ℓ = (P,G,B, π), we associate to it the followingexact sequence of vector bundles (over P ), used to define connections on ℓ(see § 1.7.1 below):

(1.6.33) 0 −→ P × gν−−→ TP

Tπ!−−−−→ π∗(TB) −→ 0

where

• P×g is the trivial vector bundle, g denoting the Lie algebra of G, identifiedwith TeG.

• The vb-morphism ν is defined by

ν(p,X) := X∗p , (p,X) ∈ P × g.

We recall that X∗ is the fundamental (Killing) vector field associated toX ≡ Xe by the action δ : P × G → G (see § 1.3.2). The morphism ν is animmersion and ν(P ×g) = V P . Here V P is the vertical subbundle of thetangent bundle (TP,P, τP ), whose fibres are given by

VpP := (V P )p = ker(Tpπ!|TpP

)= ker(Tpπ) = Tp(π

−1(x)),

for every p ∈ P with π(p) = x. Clearly, the restriction of ν to the fibrep× g ≡ g gives the linear isomorphism

νp : g∼=−−→ VpP ;

as a result,

(1.6.34) νp(X) = X∗p = Teδp(X), X ∈ g.

We notice that the partial map δp : G → π−1(x), x = π(p) is a diffeo-morphism, whose inverse is given by δ−1p (q) = k(p, q) [see (1.6.6) for thedefinition of k].

• π∗(TB) = P ×B TB (: the pull-back of TB by π : P → B; see § 1.4.4).

1.7. Connections on principal bundles 53

• Tπ! is the vb-morphism defined by the universal property of the pull-back,as pictured below.

TP

π∗(TB)π∗ = pr2

-

Tπ !

-

TB

Tπ

-

P

τ∗B

= pr1

?

π-

τP

-

B

τB

?

A crucial property of the vector bundles involved in (1.6.33) is that Gacts naturally on them. As a matter of fact, we have for every g ∈ G:

(1.6.35)

(p,Xe) · g :=(p · g,Ad(g−1)(Xe)

); (p,Xe) ∈ P × g,

u · g := TRg(u); u ∈ TP,

(p, v) · g := (p · g, v); (p, v) ∈ π∗(TB).

We recall that Rg : P → P is the right translation of P by g ∈ G (see § 1.3.1),i.e. Rg(p) = p · g ≡ pg, and Ad is the adjoint representation of G defined in§ 1.2.4.

Taking into account the previous actions and the fact that Tπ! = (τP , τB),it follows that the vb-morphisms ν and Tπ! are also G-equivariant:

(1.6.36)ν((p,Xe) · g) = TpRg

(ν(p,Xe)

)= ν(p,Xe) · g,

Tπ!(u · g) =(p · g, Tπ(u)

)=(p, Tπ(u)

)· g = Tπ!(u) · g,

for every (p,Xe) ∈ P × g, (p, v) ∈ π∗(TB) and every g ∈ G. Note that fromthe first of (1.6.36), the actions (1.6.35) and the definition of X∗ (§1.3.2),we see that

TgRg(X∗p ) =

(Ad(g−1)(Xe)

)∗p·g,

for every g ∈ G and p ∈ P .

1.7 Connections on principal bundles

Connections on principal bundles can be handled in many equivalent ways.Here we define connections as splittings of the exact sequence (1.6.33), as


g-valued connection forms on the total space of the bundle, or as a family ofg-valued local connection forms over a (trivializing) open cover of the basespace. The previous approaches will be used interchangeably.

1.7.1 Principal bundle connections as splitting G-morphisms

A connection on the principal bundle ℓ = (P,G,B, π) is a G-splitting ofthe exact sequence (1.6.33). This means that there is an exact sequence ofvector bundles

(1.7.1) 0 −→ π∗(TB)C−−−→ TP

V−−−→ P × g −→ 0

such that C, V are G-equivariant morphisms, and

Tπ! C = idπ∗(TB) and V ν = idV E

(see also [Pen69]). The definition is illustrated in the following diagram.

0

π∗(TB)?

0 - P × gν

- TP

C

? Tπ!- π∗(TB) -

idπ∗(TB)

-

0

P × g

V

?

idP×g-

0?

The reader may have noticed that the morphisms of (1.7.1) are denotedby the same symbols used in the splitting sequence (1.5.1). From the context


it will be understood which of the two sequences or connections we arereferring to.

As in the case of connections on vector bundles (discussed in § 1.5.1), itsuffices to know either C or V . The splitting implies the decomposition

(1.7.2) TP = V P ⊕HP,

where HE := Im(C) is the horizontal subbundle of TP . The verticalsubbundle of TP has already been defined in § 1.6.7. It follows that

HP = C (π∗(TB)) = ker V,(1.7.3)

TpRg(HpP ) = HpgP,(1.7.4)

for every p ∈ P and g ∈ G.On the other hand, the equalities Tπ!C = idπ∗(TB) and Tπ! = (τP , Tπ)

imply that

(1.7.5) uh = C(τP (u), Tπ(u)); u ∈ TpP ),

where uh is the horizontal component of u after the decomposition (1.7.2).

1.7.2 Connection forms

Global connection forms provide a useful tool to handle connections on aprincipal bundle. There is no essential difference between the finite andinfinite-dimensional case.

Let ℓ = (P,G,B, π) be a principal bundle and let L(TP,P × g) be thelinear map bundle, whose fibre over a p ∈ P is the space of continuous linearmaps L(TpP, g) [see § 1.4.4(c)]. The smooth sections of the previous bundleare called g-valued 1-forms on P and their set is denoted by Λ1(P, g).

A connection form of ℓ is a form ω ∈ Λ1(P, g) satisfying the followingconditions:

ω(X∗) = X; X ∈ g,(ω. 1)

R∗gω = Ad(g

)ω; g ∈ G.(ω. 2)

The preceding equalities, evaluated at any p ∈ P give, respectively:

ωp(X∗p ) = X ≡ Xe,

ωpg(TpRg(u)) = Ad(g−1).ωp(u),


for every X ∈ g, g ∈ G and u ∈ TpP . The line dot inserted above replacesobvious parentheses and should not be confused with center dots indicatingmultiplication or action of a group.

A connection form ω is related with the G-equivariant morphism V ofthe splitting sequence (1.7.1) by

ωp(u) = (pr2 V )(u); p ∈ P, u ∈ TpP.

Therefore, (1.7.3) implies that

(1.7.6) HpP = kerωp, p ∈ P.

Let u ∈ TpP with u = uv+uh where uv denotes the vertical componentof u. If we set ωp(u) = A, then (1.7.6) implies that ωp(u

v) = A. Since,ωp(A

∗p) = A, it follows that ωp(A

∗p−u

v) = 0, or VpP ∋ A∗p−u

v ∈ HpP , thusA∗p = uv. In other words,

ωp(u) identifies with the element of the Lie algebra g of G whose corre-sponding fundamental vector field coincides at p with the vertical com-ponent of u.

This is another way to define ω (see [KN68]). Moreover, using (1.7.5), wefind that

(1.7.7) C(p, Tpπ(u)) = u− νp(ωp(u)); p ∈ P, u ∈ TpP.

Equivalently,

(1.7.8) C(p, v) = u− νp(ωp(u)); (p, v) ∈ P ×B TB,

where u ∈ TpP is any vector such that Tpπ(u) = v.

1.7.3 Local connection forms

Local connection forms are particularly useful because they involve only thebase space and the structure group of the bundle.

Let ℓ = (P,G,B, π) be a principal bundle with trivializing cover C =(Uα,Φα) |α ∈ I and the corresponding natural sections sα defined by(1.6.4). Assume that ℓ is equipped with a connection whose connection formis ω ∈ Λ1(P, g). Then the 1-forms

(1.7.9) ωα := s∗αω ∈ Λ1(Uα, g), α ∈ I

are called the local connection forms of the given connection (≡ ω), withrespect to the trivializing cover C. Therefore,

ωα,x(v) = ωsα(x)(Txsα(v)); x ∈ Uα, v ∈ TxUα ≡ TxB.


The local connection forms satisfy the compatibility condition

(1.7.10) ωβ = Ad(g−1αβ)ωα + g−1αβdgαβ ; α, β ∈ I,

over Uαβ . The second summand on the right-hand side of (1.7.10) is theleft Maurer-Cartan differential of gαβ ∈ C

∞(Uαβ , G) defined in § 1.2.6. Moreexplicitly, (1.7.10) evaluated at any x ∈ Uαβ and v ∈ TxB, yields:

ωβ,x(v) = Ad(gαβ(x)−1

).ωα,x(v) + Tx

(λgαβ(x)−1 gαβ

)(v).

Recall that λg is the left translation of G and the line dot replaces paren-theses.

Conversely, let C = (Uα,Φα) |α ∈ I be a trivializing cover of a principalbundle ℓ. A family of 1-forms ωα ∈ Λ1(Uα, g) |α ∈ I, satisfying thecompatibility condition (1.7.10), determines a unique connection form ω ∈Λ1(P, g), whose local connection forms coincide with the given ωα. Indeed,for each α ∈ I, we define the map gα : π−1(Uα) → G given by gα(p) =(pr2 Φα)(p). It follows that gα is a smooth map such that

p = sα(π(p)) · gα(p), p ∈ π−1(Uα)

(the preceding equality can be also used to define gα). Then, for every p ∈ Pwith π(p) ∈ Uα, and every u ∈ TpP , we set

(1.7.11) ωp(u) := Ad(gα(p)−1

).(π∗ωα)p(u) +

(g−1α dgα

)p

(u).

Condition (1.7.10) ensures that ω is a well-defined g-valued smooth 1-formon P . It turns out that ω is a connection form.

For the analog of (1.7.11) in the case of a Lie group G acting on the leftof P see [SW72, p. 129]. We refer also to [Ble81, pp. 32–33], [KN68, p. 66]and [Pha69, pp. 227–228] for other ways to define ω from ωα.

Since ω is completely known by its local connection forms, we may write

ω ≡ ωαα∈I .

The proof of (1.7.10) and the fact that (1.7.11) is well-defined are basedon certain arguments and computations which will be also used later. Moreprecisely, assume that σ and s are two sections of P over the same opensubset U of B. Then there is a unique smooth map g : U → G such thatσ = s · g = δ (s, g), where δ is the action of G on P . Then, for every x ∈ Uand v ∈ TxB,

(1.7.12)

Txσ(v) = Ts(x)δg(x)(Txs(v)) + Tg(x)δs(x)(Txg(v))

= Ts(x)Rg(x)(Txs(v)) +(Teδs(x)·g(x) Tg(x)λg(x)−1

)(Txg(v))

= Ts(x)Rg(x)(Txs(v)) + Teδσ(x)((g−1dg)x(v)

).


Setting (g−1dg)x(v) = A ∈ g, we have that

Teδσ(x)((g−1dg)x(v)

)= νσ(x)

((g−1dg)x(v)

)= A∗σ(x);

hence,

(1.7.13) Txσ(v) = Ts(x)Rg(x)(Txs(v)) +A∗σ(x).

Applying ω to the latter, we obtain

(1.7.14)

ωσ(x)(Txσ(v)) =(R∗g(x)ω)s(x)(Txσ(v)

)+ ωσ(x)

(A∗σ(x)

)

= Ad(g(x)−1

).ωs(x)(Txs(v)) +A

= Ad(g(x)−1

).ωs(x)(Txs(v)) +

(g−1dg

)x(v).

Equivalently, for every x ∈ U and v ∈ TxB,

(σ∗ω)x(v) = Ad(g(x)−1

).(s∗ω)x(v) +

(g−1dg

)x(v).

In summary [see also the comments following (1.7.10)],

(1.7.15) σ∗ω = Ad(g−1)(s∗ω) + g−1dg,

for every σ, s ∈ Γ(U,P ) with σ = s · g.

1.7.4 Connections on the frame bundle

Let ℓ = (E,B, πE) be a vector bundle of fibre type E and the correspondingprincipal bundle of frames ℓ(E) = (P (E),GL(E), B, πP ), defined in § 1.6.5.Let also (Uα, φα)α∈I be an atlas of B over which we define the localtrivializations of E and P (E).

Assume first that E admits a linear connection K ≡ Γαα∈I withChristoffel symbols viewed as smooth maps of the form Γα : φα(Uα) →L(B,L(E)) [see (1.5.10′)]. For every x ∈ Uα and every v ∈ TxB, we set

(1.7.16) ωα,x(v) := Γα(φα(x))(φα(v)

),

where φα ≡ φα,x : TxB → B is the isomorphism (1.1.4). More explicitly, forevery x, v as before, and every u ∈ E,

(1.7.17) (ωα,x(v)) (u) =(Γα(φα(x)).φα(v)

)(u).

It turns out that ωα ∈ Λ1(ωα,L(E)), for every α ∈ I. Since the transitionfunctions of E and P (E) coincide [see (1.6.5)], the compatibility condition of


the Christoffel symbols (1.5.13) implies the compatibility condition (1.7.10),thus ωαα∈I determine a connection (form) ω on P (with local connectionforms ωα).

Conversely, assume that P (E) admits a connection ω ≡ ωα. We set

(1.7.18) Γα(z).y := [(ψ∗αωα)z](y) ≡ [(ψ∗αωα)z](idB,z(y)

)

for every z ∈ φα(Uα), y ∈ B, with ψα = φ−1α and idB,z : TzB≃−−→ B. Now

(1.7.10) implies (1.5.13), thus Γαα∈I determine a linear connection K,with Christoffel symbols the previous family.

It is obvious that (1.7.8) is the inverse of (1.7.7). Moreover, the associa-tion K ↔ ω, by the described procedure, establishes a bijective correspon-dence between linear connections on E and connections on the principalbundle of frames P (E).

It is also possible to relate the linear connections on E with the con-nections on P (E) using the splittings C of the exact sequences (1.5.1) and(1.7.1) (see, for instance, [Pen69] and [Vas82]). However, this approach willnot be pursued here.

1.7.5 Related connections on principal bundles

Let ℓ = (P,G,B, π) and ℓ′ = (P ′, G′, B′, π′) be principal bundles endowedwith the connections ω and ω′, respectively. If (f, ϕ, h) is a pb-morphism ofℓ into ℓ′, then the connections ω and ω′ are said to be (f, ϕ, h)-related ifone of the following equivalent conditions hold (see also [Vas78(a)]):

(1.7.19)

f∗ω′ = ϕ · ω,

Tf(uv) = (Tf(u))v′

,

T f(uh) = (Tf(u))h′

,

V ′ Tf = C (f × ϕ),

C ′ (f × Th) = Tf C.

More explicitly: The first condition means that

(f∗ω′)p(u) = ω′f(p)(Tpf(u)) = ϕ(ωp(u)); p ∈ P, u ∈ TpP.

Here ϕ is the Lie algebra morphism induced by ϕ, identified with Teϕ [invirtue of (1.2.3)]. Recall that C, V are the splittings of the exact sequence(1.7.1) corresponding to ω. The superscripts v and h indicate, respectively,the vertical and horizontal components of u ∈ TP , after the decomposition(1.7.2). The dashed quantities refer to the bundle (P ′, G′, B′, π′).


Another equivalent condition in terms of parallel displacements is givenby (1.9.5).

Assume now that ℓ = (P,G,B, π) and ℓ′ = (P ′, G′, B, π′) are principalbundles over the same base, equipped with the respective connections ω andω′. Taking local trivializations over the same open cover C = Uα |α ∈ Iof B, we consider the natural local sections sαα∈I and s′αα∈I of P andP ′, respectively, as well as the local connection forms ωαα∈I and ω′αα∈Icorresponding to ω and ω′.

We prove two propositions, referring also to [Vas13] for further resultson related connections and their applications.

Proposition 1.7.1. Let (f, ϕ, idB) be a pb-morphism of ℓ = (P,G,B, π)into ℓ′ = (P ′, G′, B, π′). Two connections ω and ω′ on ℓ and ℓ′, respectively,are (f, ϕ, idB)-related if and only if

(1.7.20) ϕωα = Ad(h−1α )ω′α + h−1α dhα; α, β ∈ I,

where hα : Uα → G′ |α ∈ I are smooth maps defined by

f(sα(x)) = s′α(x) · hα(x), x ∈ Uα.

Proof. Assume first that ω and ω′ are (f, ϕ, idB)-related. Then, for everyx ∈ Uα and v ∈ TxB,

[s∗α(f∗ω′)]x(v) = [(f sα)∗ω′]x(v)

or, applying (1.7.15),

(1.7.21) [s∗α(f∗ω′)]x(v) = Ad(hα(x)−1

).ωα,x(v) +

(h−1α dhα

)x

(v).

Similarly,

(1.7.22) [s∗α(ϕω)]x(v) = ϕ ((s∗αω)x(v)) = ϕ (ωα,x(v)) .

In virtue of the assumption, equalities (1.7.21) and (1.7.22) lead to (1.7.20).

Conversely, assume that condition (1.7.20) holds. To proceed, we shallneed the following two equalities whose verification is immediate:

ϕ Ad(gα(p)−1

)= Ad

((ϕ gα)(p)−1

) ϕ,(1.7.23)

ϕ(g−1α dgα

)= (ϕ gα)−1d(ϕ gα),(1.7.24)

for any smooth map gα : π−1(Uα)→ G.


Now, on π−1(Uα) we determine a smooth map gα : π−1(Uα) → G suchthat p = sα(π(p)) · gα(p), for all p ∈ π−1(Uα). Then, for every p ∈ π−1(Uα)and u ∈ TpP , (1.7.23) and (1.7.24) applied to (1.7.11) imply that

ϕ(ωp(u)) = ϕ Ad(gα(p)−1

).(π∗ωα)p(u) + ϕ

(g−1α dgα

)p(u)

= Ad(ϕ(gα(p))−1

) ϕ.(π∗ωα)p(u) +

((ϕ gα)−1d(ϕ gα)

)p(u)

= Ad(ϕ(gα(p))−1

) ϕ.ωα,x(Tpπ(u)) +

((ϕ gα)−1d(ϕ gα)

)p(u),

or, by (1.7.20) and setting π(p) = x,

(1.7.25) ϕ(ωp(u)) =

Ad(ϕ(gα(p))−1

)[Ad(hα(x)−1).ω′α,x(Tpπ(u))

+ (h−1α dhα)p(u]

+((ϕ gα)−1d(ϕ gα)

)p(u).

On the other hand, (f∗ω′)p(u) = ω′f(p)(Tp(u). To express ω′ by local

connection forms [analogously to (1.7.11)], we define g′α : π′−1(Uα)→ G′ suchthat p′ = s′α(π′(p′)) · g′α(p′), for every p′ ∈ π′−1(Uα). Then, for p′ = f(p),the latter equality yields

f(p) = s′α(π′(f(p)) · g′α(f(p)) = s′α(x) · g′α(f(p)).

Since also

f(p) = f(sα(x) · gα(p)) = f(sα(x)) · ϕ(gα(p))) = s′α(x) · hα(x) · ϕ(gα(p)),

it follows that g′α(f(p)) = hα(x) · ϕ(gα(p)), or

(1.7.26) g′α f |π−1(Uα) = (hα π) · (ϕ gα)

over π−1(U). Therefore,

(f∗ω′)p(u) = ω′f(p)(Tpf(u))

= Ad(g′α(f(p))−1

).(π′∗ω′α)f(p)(Tpf(u)) +

(g′−1α dg′α

)p(Tpf(u))

= Ad(g′α(f(p))−1

).(π′∗ω′α)f(p)(Tpf(u)) +

((g′α f)−1d(g′α f)

)x(u).

Applying (1.7.26), together with the Maurer-Cartan differential for the prod-


uct of maps, we transform the last series of equalities into

(f∗ω′)p(u) = Ad(ϕ(gα(p))−1

)Ad

(hα(x)−1

).ω′α,x(Tpπ(u))

+[(

(hα π) · (ϕ gα))−1

d((hα π) · (ϕ gα)

)]p(u)

= Ad(ϕ(gα(p))−1

)Ad

(hα(x)−1

).ω′α,x(Tpπ(u))

+((ϕ gα)−1d((ϕ gα))

)p(u)

+ Ad(ϕ(gα(f(p))−1

).((hα π)−1d(hα π)

)p(u)

= Ad(ϕ(gα(p))−1

)[Ad(hα(x)−1

).ω′α,x(Tpπ(u)) + (h−1α dhα)x(u)

]

+((ϕ gα)−1d((ϕ gα))

)p(u).

Comparing the preceding with (1.7.25), we finally obtain the first of (1.7.19),which proves the statement.

Remark. Proposition 1.7.1 can be easily extended to principal bundles ℓand ℓ′ over diffeomorphic bases B and B′, respectively. In this case we mayconsider morphisms of the form (f, ϕ, h), where h : B → B′ is a diffeomor-phism.

The following result will be systematically used in the next subsection.

Proposition 1.7.2. Let (f, ϕ, idB) be a pb-morphism of ℓ = (P,G,B, π)into ℓ′ = (P ′, G′, B, π′). If ω is a connection on ℓ, then there exists a uniqueconnection ω′ on ℓ′, (f, ϕ, idB)-related with ω.

Proof. As before, we consider trivializations over the same open cover C =Uαα∈I of B. The connection (form) ω is completely known from its lo-cal connection forms ωα∈I . If hαα∈I are the smooth maps of Proposi-tion 1.7.1, we define the local forms

(1.7.27) ω′α := Ad(hα).(ϕωα)− dhαh−1α , α ∈ I.

By quite lengthy computations, in the spirit of the proof of Proposition 1.7.1,based also on the interplay between left and right Maurer-Cartan differen-tials, we prove that

ω′β = Ad(g′αβ−1

)ω′α + g′αβ−1dg′αβ ; α, β ∈ I.

This is precisely the analog of (1.7.10), ensuring the existence of a connection(form) ω′ on ℓ′, as we have described in the second part of § 1.7.3.

Since (1.7.27) transforms into

ϕωα = Ad(h−1α )ω′α + h−1α dhα; α ∈ I,


Proposition 1.7.1 implies that ω and ω′ are (f, ϕ, idB)-related.It remains to show that ω′ is unique. The easiest way to see this is to use

the corresponding splittings C and C ′ of the connections. More explicitly,since ω and ω′ are (f, ϕ, idB)-related, by the equivalent conditions (1.7.19)we obtain that

C ′ (f × idB) = Tf C.

Analogously, if there is a connection ω ≡ C, also (f, ϕ, idB)-related with ω,

C (f × idB) = Tf C,

thus C (f × idB) = C ′ (f × idB); that is,

(1.7.28) C(f(p), v) = C ′(f(p), v); (p, v) ∈ P ×B TB.

Let now any (p′, u) ∈ P ′ ×B TB. If x := π′(p′) = τB(u), we choose anarbitrary p ∈ π−1(x), thus there is a g′ ∈ G′ such that p′ = f(p) · g′. As aresult, in virtue of the action of G′ on (π′)∗(TB) and the G′-equivariance ofC ′ and C [see also equalities (1.6.35)], as well as in conjunction with (1.7.28),we obtain the following equalities concluding the proof:

C(p′, u) = C(f(p) · g′, u) = C(f(p), u) · g′

:= Tf(p)·g′R′g′(C(f(p), u)

)= Tf(p)·g′R

′g′(C ′(f(p), u)

)

= C ′(f(p) · g′, u) = C ′(p′, u).

For further details on related connections and their applications we referto [Vas13].

1.7.6 Connections on associated bundles

We fix a principal bundle ℓ = (P,G,B, π) and a connection ω on it. Ifϕ : G→ H is a morphism of (Banach-) Lie groups, then as in § 1.6.6(a), weobtain the associated principal bundle ϕ(ℓ) = (P ×G H,H,B, πH) and thecanonical morphism (κ, ϕ, idB) defined by (1.6.23). Therefore, by Propo-sition 1.7.2, ϕ(ℓ) admits a uniquely defined connection, say, ωϕ, which is(κ, ϕ, idB)-related with ω; that is κ∗ωϕ = ϕω. The latter condition, interms of local connection forms, is equivalent with ϕωα = ωϕα , α ∈ I. Thisis the case, because the local connection forms ωϕα of ωϕ are induced bythe natural local sections of ϕ(ℓ) given by (1.6.22′), thus the maps hα ofProposition 1.7.1 are now identified with the unit of H.

Specializing to the particular case of a representation of G into a Ba-nach space E, ϕ : G → GL(E), we obtain the analogous associated bundle


ϕ(ℓ) =(Pϕ := P ×G GL(E),GL(E), B, πϕ

)[see § 1.6.6(c)]; hence, ω induces

a connection ωϕ as in the previous general case. The corresponding equalityof local connection forms is obtained by applying (1.6.30).

The same representation determines the principal bundle (of frames)ℓ = (P (Eϕ),GL(E), B, π) along with the canonical morphism (F,ϕ, idB),where F is given by (1.6.28). We obtain a unique connection, say, ω, suchthat F ∗ω = φω. Equivalently ϕωα = ωα, because again hα ≡ idE ∈ GL(E),as a consequence of equality (1.6.31).

Finally we see that ωϕ and ω are (θ, idGL(E), idB)-related [see (1.6.29) forthe definition of θ] since ωϕα = ωα.

1.8 The curvature of a principal connection

We review a few facts about the curvature of a connection on a principalfibre bundle needed in our treatment.

1.8.1 Curvature forms

Let ω be a connection on ℓ = (P,G,B, π). The curvature form of ω is theg-valued 2-form Ω on P , i.e. Ω ∈ Λ2(P, g), defined by

(1.8.1) Ω = Dω := dω (h× h),

where h : TP → HP is the vb-morphism assigning to each tangent vectorof P its horizontal component. Ω satisfies Cartan’s (second) structureequation

(1.8.2) Ω = dω +1

2[ω, ω].

Equalities (1.8.1) and (1.8.2) are equivalent.Clearly, Ω is horizontal, i.e. Ω(X,Y ) = 0, if one of the vector fields X,

Y of P is vertical, and

(1.8.3) R∗gΩ = Ad(g−1)Ω

(G-equivariance of Ω). The last two properties characterize Ω as a tensorialform of adjoint type.

On the other hand, if X and Y are horizontal vector fields of P , thenthe annihilation of ω on the horizontal subbundle and (1.8.2) imply that

Ω(X,Y ) = dω(X,Y ) +1

2[ω(X), ω(Y )]

= X(ω(Y ))− Y (ω(X)) − ω([X,Y ]) = −ω([X,Y ]),

1.8. The curvature of a principal connection 65

which shows that Ω is the obstruction to the integrability of the horizontalsubbundle of TP .

Moreover, Ω satisfies the Bianchi identity

(1.8.4) dΩ = [Ω, ω],

equivalently written in the form

(1.8.4′) DΩ = 0,

with D being defined as in (1.8.1).Given a trivializing cover C = (Uα,Φα) |α ∈ I of P and the cor-

responding natural sections sα defined by (1.6.4), the local curvatureforms of Ω (with respect to C) are

(1.8.5) Ωα := s∗αΩ ∈ Λ1(Uα, g), α ∈ I.

An immediate consequence of the structure equation (1.8.2) and theproperties of the pull-back are the local structure equations

(1.8.6) Ωα = dωα +1

2[ωα, ωα], α ∈ I.

On the other hand, the compatibility condition

(1.8.7) Ωβ = Ad(g−1αβ )Ωα; α, β ∈ I,

holds over Uαβ . Indeed, for every x ∈ Uαβ and every u, v ∈ TxB,

Ωβ,x(u, v) = Ωsβ(x)

(Txsβ(u), Txsβ(v)

).

Since sβ = sα · gαβ , by the formulas leading to (1.7.15), we obtain

Txsβ(u) = Tsα(x)Rgαβ(x)(Txsα(u)) +A∗sβ(x),

and similarly for Txsβ(v). Therefore, the horizontality and G-equivarianceof Ω imply that

Ωβ,x(u, v) = Ωsα(x)·gαβ(x)

(Tsα(x)Rgαβ(x)(Txsα(u)), Tsα(x)Rgαβ(x)(Txsα(v))

)

=(R∗gαβ(x)

Ω)sα(x)

(Txsα(u), Txsα(v))

= Ad(gαβ(x)−1

).Ωsα(x)(Txsα(u), Txsα(v))

= Ad(gαβ(x)−1

).Ωα,x(u, v),


which proves (1.8.7).

As is expected, Ω is completely known by the local forms Ωα. Moreprecisely, let ω ≡ ωαα∈I be a connection on P , and θα ∈ Λ1(Uα), gα∈Ia family of 2-forms satisfying

θβ = Ad(g−1αβ )θα,(1.8.8)

θα = dωα +1

2[ωα, ωα].(1.8.9)

Then θα determine a 2-form Θ ∈ Λ1(P, g which coincides with the cur-vature Ω of ω.

To see this we need the following (general) formulas satisfied by theMaurer-Cartan differentials. Namely, with the notations of § 1.2.6,

(1.8.10) d(Ad(g)θ) = [Drg,Ad(g)θ] + Ad(g). dθ,

or, equivalently,

(1.8.11) d((g−1)θ)

= −[Dlg,Ad(g−1)θ

]+ Ad

(g−1).dθ,

for every smooth map g : U → G (U ⊆ B open) and every θ ∈ Λ1(U, g).The proof of both formulas is a bit technical and involves the differentiationof the exterior product of the Aut(g)-valued 0-form Ad(g) and the g-valued1-form θ. A very detailed proof, regardless the dimension of the manifoldsat hand, can be found in the monograph [KJ80], whereas [Nic95] contains aproof valid for finite-dimensional g-valued forms.

The second formula needed is

(1.8.12) d(Dlg

)= −

1

2

[Dlg,Dlg

].

This is a direct consequence of the expression of the (left) Maurer-Cartandifferential as the pull-back of the (left) canonical Maurer-Cartan form onG by g and the corresponding Maurer-Cartan equation.

Going back to the proof of our claim, we define the form Θ|π−1(Uα) bysetting Θ(p) := Ad(g−1α (p)).(π∗θα)p, where gα : π−1(Uα)→ G is the smoothmap determined by p = sα(x) · gα(p), with x = π(p) [see the analogous con-struction of a connection ω by the local connection forms ωα and comparewith (1.7.11)].

Equalities (1.8.8) and sβ = sα · gαβ imply that Θ is well-defined. Thenext step is to show that Θ = dω+ 1

2 [ω, ω]. Indeed, writing, for convenience,(1.7.11) in the form

(1.8.13) ω|π−1(Uα) = Ad(g−1α).(π∗ωα) +Dlgα,

1.8. The curvature of a principal connection 67

and omitting restrictions, we have

(1.8.14)dω = d

(Ad(g−1α).π∗ωα +Dlgα

)

= d(Ad(g−1α).π∗ωα

)+ d(Dlgα

).

In virtue of (1.8.11) and (1.8.13), the first summand in the right-hand sideof (1.8.14) transforms into

d(Ad(g−1α).π∗ωα

)=

= −[Dlgα,Ad

(g−1α ).π∗ωα

]+ Ad

(g−1α).d(π∗ωα)

= −[Dlgα,Ad(g−1α

).π∗ωα

]+ Ad

(g−1α). π∗(θα −

1

2[ωα, ωα]

)

= −[Dlgα, ω −D

lgα]

+ Ad(g−1α).π∗θα −

1

2

[Ad(g−1α).π∗ωα,Ad

(g−1α).π∗ωα

]

= −[Dlgα, ω

]+[Dlgα,D

lgα]

+ Θ−1

2

[ω −Dlgα, ω −D

lgα]

= Θ−1

2

[ω, ω

]+

1

2

[Dlgα,D

lgα].

Substituting (1.8.12) and the preceding in (1.8.14), we obtain

Θ = dω +1

2[ω, ω] = Ω,

as claimed.

1.8.2 Flat connections

As usual, a connection ω on ℓ = (P,G,B, π) is called flat if Ω = 0. Someuseful equivalent conditions are the following:

i) The horizontal subbundle HP of TP is (completely) integrable.

ii) For every x ∈ B, there is an (open) neighbourhood U of x and a trivial-ization Φ: π−1(U)→ U ×G of P such that ω|π−1(U) = Φ∗ωo|U×G, where ωo

is the canonical flat connection on B ×G. We recall that

ωo(x,g)(u, v) := Tgλg−1(v) = (pr∗2 α)(x,g)(u, v),

for every (x, g) ∈ B × G and (u, v) ∈ TxB × TgG, if α denotes the leftMaurer-Cartan form of G.

iii) For every x ∈ B, there is a neighbourhood U of x and a (smooth) sections : U → P of P such that s∗ω = 0.


iv) P reduces to a bundle with structure group Gd (the group G consideredwith the discrete structure).

The following properties are also well known:

a) If B is 1-dimensional, then every connection on P is flat.

b) If B is simply connected and P admits a flat connection ω, then P istrivial. Moreover, there is a (global) section s : B → P such that s∗ω = 0;in other words, s is a horizontal section.

For relevant proofs, valid for finite-dimensional and Banach bundles, werefer to [Dup78], [KN68], [Kos60] and [Pha69].

1.9 Holonomy groups

Let ℓ = (P,G,B, π) be a principal bundle endowed with a connection ω. Ifα : I = [0, 1]→ B is a smooth curve, a horizontal lifting of α is a smoothcurve α : I → P projecting to α, with horizontal tangent (velocity) vectors;that is, according to the notation (1.1.11),

(1.9.1)π α = α,

.

α(t) ∈ Hα(t)P, equivalently, ωα(t)( .α(t)

)= 0.

For convenience we also write ω(α(t))( .

α(t)), even simpler ω

( .α(t)

), in place

of ωα(t)( .

α(t)).

We shall be mainly concerned with the horizontal lifts of piece-wisesmooth curves.

Given a piece-wise smooth curve α : I → B and any p ∈ P , there existsa unique (piece-wise smooth) horizontal lift αp : I → P with αp(0) = p.

A way to prove this is the following (see, e.g., [Kos60], [Pha69]): Denotingalso by α : J → B the smooth curve extending the initial α, where J is anopen interval containing I, we consider the pull-back of P by the former α

α∗(P ) = J ×B Ppr2 - P

J

pr1

?

α- B

π

?

1.9. Holonomy groups 69

and the pull-back connection ω := pr∗2 ω on α∗(P ). By the properties a)and b) listed in § 1.8.2, ω is flat and α∗(P ) trivial, thus there is a (global)horizontal section σ : J → α∗(P ). The curve α := pr2 σ|I is a horizontallift of α. If α(0) = p, then αp = α. If α(0) 6= p, there is a unique g ∈ G suchthat p = α(0) · g; hence, αp = α · g.

Another way, using equations with total differentials, goes as follows: Ifα is contained in an open set U ⊆ B over which P is trivial, then there isa section σ : U → P such that β = σ α is a lift of α with β(0) = p. Sinceβ is not necessarily horizontal, we look for a smooth curve g : I → G suchthat g(0) = e and γ(t) := β(t) · g(t) is a horizontal curve of P . Therefore,differentiating the latter equation and applying the horizontality conditionω(γ(t)) = 0 [see the analogous computations preceding (1.7.15)], we are leadto

(1.9.2)g(0) = e,

−Tg(t)ρg(t)−1(g(t)) = ω(β(t)),

or, in the notations of § 1.2.6,

(1.9.3)g(0) = e,

Drg = −β∗ω.

Equation (1.9.3) has a unique solution on I, provided that the form −β∗ω isintegrable; that is, d(β∗ω)+ 1

2 [β∗ω, β∗ω] (see the comments following (1.2.7)and the detailed proof of [Bou72, Ch. III, §6, Proposition 15]). This is thecase, for if we take the pull-back bundle α∗(P ) and the flat connection ω =pr∗2 ω considered earlier, we check that the curve δ(t) := (t, β(t)) ∈ α∗(P ),with t ∈ J , satisfies the equality δ∗ω = β∗ω. Therefore,

0 = δ∗Ω = δ∗(dω +

1

2[ω, ω]

)= d(β∗ω) +

1

2[β∗ω, β∗ω],

as desired.

For the solution of (1.9.2) in the finite-dimensional case see also [KN68].We notice that (1.9.2) is a particular case of the following result (see [Pen67,Proposition 1.5]), stated here for the sake of completeness (after adaptingthe original statement to the present setting and the final notation of § 1.1.7):

Let G be a Banach-Lie group with algebra Lie g. If X is a smoothmanifold and f : X → C∞(I, g) a smooth map, then there exists a unique


smooth map g : X → C∞(I,G) such that

g(x)(0) = e,

Tg(x)(t)ρ(g(x)(t))−1

(g(x).(t)

)= f(x)(t),

for every (x, t) ∈ X × I.

To conclude our discussion on the existence of horizontal lifts, we addthat if the curve α : I → B is not entirely contained in a single open set ofB defining a local trivialization of the bundle, then we cover I by a finitefamily Ui of such sets, and we solve the corresponding equations, taking asinitial condition of the i-th equation the final point of the (i−1)-th solution.

Having defined the horizontal lifts of a curve α, we obtain the diffeomor-phism

τα : π−1(α(0)) −→ π−1(α(1)) : τα(p) := αp(1),

called the parallel displacement/translation along α. It is shown thatτβ∗α = τβ τα and τ−1α = τα−1 , for appropriate curves (see the vb-analogin § 1.5.6).

Denoting by Cx the set of all piecewise smooth closed curves startingand ending at x (loop group at x), and by C0

x ⊂ Cx the group of 0-homotopic loops, the holonomy group with reference point x ∈ Bis Φx := τα |α ∈ Cx, while the restricted holonomy group withreference point x ∈ B is Φ0

x := τα |α ∈ C0x.

To realize these groups as subgroups of G, we choose an arbitrary pointp ∈ P with π(p) = x. Then the map kp : τα |α ∈ Cx → G, given byτα(p) = p · kp(τα) determines a group homomorphism. Accordingly, theholonomy group of ω with reference point p ∈ P is defined to be

Φp := kp(τα) |α ∈ Cx, x = π(p).

Equivalently, Φp consists of all g ∈ G such that p and p · g can be joined bya (piecewise) smooth curve.

Analogously, the restricted holonomy group of ω with referencepoint p is given by

Φ0p =

kp(τα) |α ∈ C0

x

.

The latter is a normal subgroup of Φp and there is a natural homomorphism

(1.9.4) h : π1(B) −→ Φp/Φ0p,

where π1(B) ≡ π1(B,x), if x = π(p) (B is assumed to be connected). Forrelevant proofs and other properties of the holonomy groups in the finite-dimensional case, we refer to [KN68].

1.10. Classification of flat bundles 71

Regarding now the holonomy groups in the Banach framework, the fol-lowing result has been proved by L. Maxim ([Max72]):

Theorem 1.9.1. i) Φp and Φ0p are Banach-Lie subgroups of G. In partic-

ular, Φ0p is the identity component of Φp.

ii) If the base B is connected and paracompact, then Φp/Φ0p is countable.

iii) The structure group G reduces to Φp.iv) Φp = e if and only if P is trivial.v) Φ0

p = e if and only if the connection is flat.

A few comments are necessary here: In i), saying that Φp is a Banach-Lie subgroup of G we mean that Φp is a Banach-Lie group such that thenatural injection i : Φp → G is a smooth morphism and Tei : L(Φp) →L(G) ≡ g is 1–1 (see [Laz65] and [Mai62]). Therefore, Φp here is neither aregular nor an embedded submanifold of G (compare [Bou71] and [Lan99]).Similar remarks apply to Φ0

p. Also, iii) means that (P,G,B, π) reduces tothe principal bundle (P [p],Φp, B, π

′), where P [p] is the holonomy bundleat p, consisting of all the points of P joined with p by a horizontal curve.Φp acts on P [p] in a natural way.

Another useful result, concerning the parallel displacements of relatedconnections is the following:

Two connections ω and ω′ on (P,G,B, π) and (P ′, G′, B′, π′), respectively,are (f, ϕ, h)-related if and only if

(1.9.5) f τγ = τ ′hγ∣∣π−1(γ(0))

.

Therefore, (1.9.5) is equivalent to the conditions of (1.7.5).

1.10 Classification of flat bundles

In this section we are dealing with principal bundles with a fixed connectedbase B and a fixed structure group G.

A flat bundle is a pair (P, ω), where (for simplicity) P ≡ (P,G,B, π)and ω is a flat connection on P . Two flat bundles (P, ω) and (P ′, ω′) arecalled equivalent if there is a G-B-isomorphism (f, idG, idB) of P onto P ′

such that ω = f∗ω′. The set of equivalence classes thus obtained is denotedby H(B,G).

On the other hand, in virtue of (1.9.4) and property v) of the holon-omy groups (Theorem 1.9.1), every flat bundle (P, ω) determines the grouphomomorphism

(1.10.1) hω : π1(B) −→ G : [α] 7→ kp(τα),


called the holonomy homomorphism of (P, ω), thus hω(π1(B)) = Φp,where π1(B) ≡ π1(B,xo), for a fixed xo ∈ B, and p ∈ π−1(xo) arbitrarilychosen. Recall that τα(p) = p · kp(τα). The homomorphism hω is thoughtof as a Lie group homomorphism under the usual discrete smooth structureon π1(B).

Two (arbitrary) homomorphisms h, h′ : π1(B) → G are called similar(or conjugate) if they differ by an inner automorphism of g, i.e. there is a g ∈G such that h′([α]) = g ·h([α])·g−1, for every [α] ∈ π1(B). Briefly, h′ = I(g)h (the inner)automorphism I(g) is also denoted by ad(g)). The terminologycomes from the similarity of matrices in the case of G = GL(n,R). Theset of equivalence classes of similar homomorphisms as above is denoted byS(B,G).

With the previous notations,

H(B,G) = S(B,G), within a bijection.

For a coherent proof, based on elementary properties of connections andJ. Milnor’s association of flat bundles with covering spaces (see [Mil58]), werefer to [Vas83]. We outline the main ideas of the proof:

We saw before that a flat bundle determines the corresponding holon-omy homomorphism hω. Conversely, let h : π1(B) → G be an arbitraryhomomorphism. If (B, π1(B), B, p) is the principal bundle determined bythe universal covering B of B, then h associates to B the principal bundle

ℓ :=((B ×G)/π1(B), G,B, π

),

with π given by π([(b, s)]) = p(b). The bundle ℓ admits a flat connectionω such that ωo = κ∗ω, where ωo is the canonical flat connection of thetrivial bundle ℓo = (B × G, B,pr1), and κ : B × G → (B × G)/π1(B) isthe canonical map [not be confused with that given by (1.6.23)]. Obviously,(κ, idG, p) is a principal bundle morphism between ℓo and ℓ, while ωo and ωare (κ, idG, p)-related.

The previous arguments raise now the natural question: If we start with aflat bundle (P, ω) and its holonomy homomorphism hω : π1(B)→ G, and weconsider the flat bundle

((B×G)/π1(B), ω

)induced by the same h, how are

the two bundles related? It turns out that they are equivalent by means of anappropriate G-B-isomorphism, determined as follows: If π1(B) ≡ π1(B,x0),we fix two arbitrary points x0 ∈ p

−1(x0) and q0 ∈ π−1(x0). Then we define

the map 0 : B ×G → P associating to a pair (x, g) the element τα(q0) · g,where τα is the parallel displacement (with respect to ω) along the curveα = p α, if α is any piece-wise smooth curve joining x with x0. This is a

1.10. Classification of flat bundles 73

well-defined smooth map satisfying the equality

0((x, g) · [γ]

)= 0(x, g); (x, g) ∈ B ×G, [γ] ∈ π1(B).

The action of π1(B) on the right of B × G is defined by means of h, i.e.(x, g)·[γ] = (x·[γ], h([γ])−1 ·g). As a result, the map : (B×G)/π1(B)→ P ,with ([(x, g)]) = 0(x, g) is a G-B-isomorphism such that ω = ∗ω.

The classification stated above relies on the existence of the isomorphism, for each flat bundle, along with the fact that similar homomorphismshi : π1(B)→ G (i = 1, 2) determine equivalent flat bundles (ℓi, ωi) and viceversa.

Chapter 2

Frechet spaces

Starting with a brief summary of the topology of Frechet spaces and varioususeful examples, we then discuss the differentiability method adopted here,which is due to J.A. Leslie. It is valid for arbitrary topological vector spaces,while it remains closer to more classical methods, without recourse to anyparticular topologies.

Since projective systems of geometrical structures are our central theme,the main part of the present chapter is devoted to the representation of aFrechet space as the projective limit of (a countable projective system) ofBanach spaces, and questions related with the differentiability of projectivesystems of maps between such spaces. Of particular interest is the construc-tion of certain spaces of continuous linear maps between Frechet spaces soas to remain in the same category of spaces, a fact not in general true. Afunctional space of this kind eventually replaces (in subsequent chapters)the pathological general linear group GL(F) of a Frechet space F.

Linear differential equations in the same framework is the first applica-tion of the projective limit approach expounded from this chapter onwards.It should be noted that there is not a general solvability theory for differen-tial equations, even linear ones, in non-Banach spaces.

2.1 The topology of Frechet spaces

In a number of cases that have significance in global analysis and physicalfield theory, Banach space representations break down and we need Frechetspaces, which have weaker requirements for their topology. To see howthis happens we shall look at some examples, but first we need some def-initions and our main references for this are R.S. Hamilton [Ham82] and

75

76 Chapter 2. Frechet spaces

K-H. Neeb [Nee06], cf. also L.A. Steen and J.A. Seebach Jnr. [SS70]. Inthis section we give a brief description of the topology of Frechet spaces andpoint out the main differences from that of Banach spaces. In what followsX denotes a real vector space.

Definition 2.1.1. A seminorm on X is a real valued map p : X → R suchthat

p(x) ≥ 0,(i)

p(x+ y) ≤ p(x) + p(y),(ii)

p(λx) = |λ| p(x),(iii)

for every x, y ∈ X and λ ∈ R.

The notion of a seminorm seems to be a modest generalization of a normby being just one step away: the seminorm of a non trivial vector may bezero. However, this difference has a serious topological consequence, namelythe topology induced on X by p is not necessarily Hausdorff, as we explainbelow.

Definition 2.1.2. A family of seminorms Γ = pαα∈I on X defines aunique topology TΓ compatible with the vector space structure of X. Theneighbourhood base of TΓ is determined by the family

BΓ = S(∆, ε) : ε > 0 and ∆ a finite subset of Γ ,

whereS(∆, ε) = x ∈ X : p(x) < ε, ∀ p ∈ ∆ .

The basic properties of the previous topology are summarized in the nextstatement. For details we refer, e.g., to J. Dugundji [Dug75], H.H. Schaef-fer [Sch80]), R. Meise-D. Vogt [MV97].

Proposition 2.1.3. i) (X,TΓ) is a topological vector space. In particular,TΓ is the finest topology on X making all the seminorms of Γ continuous.

ii) The topological vector space (X,TΓ) is locally convex. Conversely, atopology on X is locally convex only if it is defined by a family of seminorms.

iii) (X,TΓ) is generally not a Hausdorff space. The Hausdorff propertyis ensured under the following additional condition:

x = 0 ⇔ p(x) = 0, for every p ∈ Γ.

iv) If a topological vector space (X,TΓ) is Hausdorff, then it is alsometrizable if and only if the family of seminorms Γ is countable.

2.1. The topology of Frechet spaces 77

v) The convergence of a sequence (xn)n∈N in X is controlled by all theseminorms of Γ; that is,

xn −→ x ⇔ p(xn − x) −→ 0, ∀ p ∈ Γ.

vi) X is complete (with respect to TΓ) if and only if every sequence (xn)n∈Nin X, such that

limn.m→∞

p(xn − xm) = 0; ∀ p ∈ Γ,

converges in X. Customarily such an (xn)n∈N is called a Cauchy se-quence.

We recall that a topological vector space is locally convex if each pointhas a fundamental system of convex neighbourhoods.

The spaces carrying all the above properties form the category of Frechetspaces with morphisms the continuous maps. Precisely:

Definition 2.1.4. A Frechet space is a topological vector space F that islocally convex, Hausdorff, metrizable and complete.

Some typical examples of Frechet spaces are listed below.

Examples 2.1.5.

1. Every Banach space E is a Frechet space where the family of seminormscontains only one element, the norm defining the topology of E.

2. The space R∞ =∏n∈N

Rn, endowed with the cartesian topology, is a

Frechet space with corresponding family of seminorms

pn(x1, x2, ...) = |x1|+ |x2|+ ...+ |xn|

n∈N

.

Metrizability can be established by setting

(2.1.1) d(x, y) =∑

i

|xi − yi|

2i(1 + |xi − yi|).

In R∞ the completeness is inherited from that of each copy of the real line.For if x = (xi) is a Cauchy sequence in R∞, then, for each i, (xmi ), withm ∈ N, is a Cauchy sequence in R; hence, it converges, say, to Xi and (Xi) =X ∈ R∞, with d(x,Xi) → 0 as i → ∞. We note that R∞ is separable inconsequence of the countable dense subset of elements having finitely manyrational components and the remainder zero; second countability comes frommetrizability.


3. More generally, every countable cartesian product of Banach spaces F =∏n∈N En is a Frechet space with topology defined by the seminorms (qn)n∈N,

given by

qn(x1, x2, ...) =n∑

i=1

‖xi‖i ,

where ‖ · ‖i denotes the norm of the i-factor Ei.4. The space of continuous functions C0(R,R) is a Frechet space withseminorms (pn)n∈N defined by

pn(f) = sup| f(x) |, x ∈ [−n, n]

.

5. The space of smooth functions C∞(I,R), where I is a compact intervalof R, is a Frechet space with seminorms defined by

pn(f) =

n∑

i=0

sup| Dif(x) |, x ∈ I

.

In this respect see also [MV97].6. The space C∞(M,V ), of smooth sections of a vector bundle V over acompact smooth Riemannian manifold M with covariant derivative ∇, is aFrechet space with

(2.1.2) ||f ||n =n∑

i=0

sup x | ∇if(x) |, for n ∈ N.

7. Frechet spaces of sections arise naturally as configurations of a physicalfield. Then the moduli space, consisting of inequivalent configurations of thephysical field, is the quotient of the infinite-dimensional configuration spaceX by the appropriate symmetry gauge group. Typically, X is modelled on aFrechet space of smooth sections of a vector bundle over a closed manifold.For example, see H. Omori [Omo70], [Omo97]. Lie groups and their algebrasare important in geometry and physics for symmetry groups of diffeomor-phisms, but for these difficulties quickly arise in the infinite dimensionalsetting. H.G. Dales [Dal00] gives a comprehensive account of the Banachcase and for recent algebraic results in the Frechet case see, for example,[CEO09, Nee09, Pir09, Tka10].

8. For a compact Riemannian manifold M with g = X (M) the Lie algebraof smooth vector fields on M, X (M) is a Frechet space. It is (topologically)regular since disjoint points and closed sets can be separated by disjointopen sets, and has an exponential function but in general it induces no local

2.1. The topology of Frechet spaces 79

diffeomorphism of a 0-neighbourhood in X (M) onto a 1-neighbourhood inG = Diff(M). R.S. Hamilton [Ham82] and K-H. Neeb [Nee06] provide detailsof the construction to circumvent this difficulty, including explicit study ofthe crucial case Diff(S1), where rotations illustrate the difference from theBanach case.

9. As we might anticipate from the prototype Frechet model R∞, which isthe projective limit of a countable collection of copies of R, a Frechet spacecan always be represented as a projective limit of Banach spaces, whichwe discuss further in § 2.3.1; for more details cf. [DZ84], [FW96], [Nee06],[Pal68], [Val89], as well as [BDH86], [BMM89] and [Wen03]. S. Agethenet al. [ABB09] consider locally convex spaces which are intersections of asequence of unions of sequences of Banach spaces of continuous functionswith weighted sup-norms, by clarifying when such spaces satisfy the principleof uniform boundedness.

Some important properties of Frechet spaces are summarized in the fol-lowing statement (see also [BP75], [Dug75], [Jar81], [MV97], [Sch80]):

Proposition 2.1.6. i) Every closed subspace of a Frechet space is also aFrechet space under the same family of seminorms.

ii) The open mapping theorem as well as the Hahn-Banach theorem holdtrue in Frechet spaces.

iii) The continuity of linear and bilinear maps between Frechet spaces ischecked via seminorms. More precisely, let F1, F2 and F3 be Frechet spaceswith topologies defined by the families of seminorms pnn∈N, qnn∈N andrnn∈N, respectively. Then a linear map f : F1 → F2 is continuous if andonly if, for every seminorm qn of F2, there exists a seminorm pm of F1 anda positive real M such that

qn(f(x)) ≤Mpm(x),

for every x ∈ F1. Analogously, a bilinear map g : F1×F2 → F3 is continuousif and only if, for every seminorm rn of F3, there exist seminorms pm of F1,qk of F2 and a positive real L such that

rn(g(x, y)) ≤ Lpm(x) qk(y),

for every x ∈ F1 and y ∈ F2 .

Although a Frechet space may be thought of as a short step away froma Banach space because of the Proposition 2.1.6, nevertheless importantdifferences between the two categories of spaces occur:


• The space of continuous linear maps L(F1,F2) between two Frechetspaces F1 and F2 is not necessarily a Frechet space. In particular, if pnn∈Nand qnn∈N are the seminorms of F1 and F2, respectively, then L(F1,F2) is aHausdorff locally convex topological vector space whose topology is derivedfrom the family of seminorms | · |n,B given by

(2.1.3) | f |n,B = sup qn(f(x)) x ∈ B ,

where n ∈ N and B is any bounded subset of F1 containing the zero element.This topology is complete but not metrizable since the family (2.1.3) is notcountable (see also [Hye45]).• Whereas the dual of a Banach space is a Banach space, the dual of aFrechet space that is not Banach is never a Frechet space. The dual of R∞

consists of sequences having only finitely many nonzero elements, the dualof C∞(M,R) for a compact manifold M is the space of distributions on M ,and none of these duals is a Frechet space, though they are complete locallyconvex topological vector spaces. This is a major source of difficulty, as weshall see in the sequel.• The inverse function theorem is not valid in general. However, rele-vant modifications of it have been proposed to deal with special cases ofFrechet spaces and maps (see, e.g., [Ham82], [Nee06], [Omo70]). For exam-ple, a Frechet space is graded if its topology can be defined by a collectionof increasing seminorms; this can be ensured by adding to each seminormall those below it. A tame linear map between graded spaces satisfies auniform linear growth constraint through the grading [Ham82, p. 135]. Agraded space is tame if it is a tame direct summand of a space of expo-nentially decreasing sequences in some Banach space. These constructionsturn out to cover the cases for all nonlinear partial differential operators andmost of their inverses, including those for elliptic, parabolic, hyperbolic andsubelliptic operators.

Nash-Moser inverse function theorem for the category of tameFrechet spaces (cf. [Ham82, pp. 67, 171]) states roughly that: If thederivatives DP (f)h = k of an operator P in the category have solutionsin the category, then the operator P has a local inverse in the category.

The elaboration of several examples is given in [Ham82] and further struc-tural results are given in [AO09], [BB03], [Dub79], [KLT09], [KM90], [KM97],[KS09], [LT09], [MV85], [PV95], [Vog10], [Wol09]. In addition, the interestedreader may consult the following (not exhaustive) list of more specializedaspects of the structure theory of Frechet spaces: [Vog77], [Vog79], [Vog83],[Vog87], [Vog10] and [VW80], [VW81].

2.2. Differentiability 81

• The general linear group GL(F) of a Frechet space F, i.e. the group ofall linear isomorphisms of F, does not admit a non trivial topology compat-ible with its group structure (see, e.g., [Ham82], [Les67], [Les68], [Nee06],[Omo70]).• There is no general solvability theory for differential equations in Frechetspaces analogous to that developed for finite-dimensional or Banach spaces.Since differential equations are very important in the framework of differen-tial geometry, § 2.4 is devoted to a detailed study of them within the categoryunder consideration.

2.2 Differentiability

The topological deviations between Frechet and Banach spaces, discussedin the previous section, have direct repercussions on the issue of differen-tiability. In the last 40 years or so, the ordinary differentiation of finite-dimensional or Banach spaces ([Car67(a)], [Lan99]) has been extended tomore general topological vector spaces, in a variety of ways, according tothe particular problems and applications each author has in mind (see,e.g., [Ham82], [Lem86], [Les67], [Les68], [KM97], [Nee06], [Omo70], [Pap80],[VerE83] and [VerE85]. The latter two sources contain a remarkably exten-sive bibliography).

In the present work, the differentiation method proposed by J.A. Leslieis adopted since it fits well to the requirements of our differential geomet-ric framework, without using any particular topology. In what follows weoutline the basic definitions and properties of this approach. For full detailsthe reader is referred to [Les67] and [Les68].

Let F1 and F2 be two Hausdorff locally convex topological vector spaces,and let U be an open subset of F1.

Definition 2.2.1. A continuous map f : U → F2 is said to be differen-tiable at x ∈ U if there exists a continuous linear map Df(x) : F1 → F2

such that

R(t, v) :=

f(x+ tv)− f(x)−Df(x)(tv)

t, t 6= 0

0, t = 0

is continuous at every (0, v) ∈ R× F1. The map f is said to be differen-tiable if it is differentiable at every x ∈ U . We call Df(x) the differential(or derivative) of f at x. Clearly, this is a special case of the Gateauxderivative.


As in the classical (Frechet) differentiation, Df(x) is uniquely deter-mined.

Definition 2.2.2. A map f : U → F2, as before, is called C1-differentiableif it is differentiable at every point x ∈ U , and the (total) differential or(total) derivative

Df : U × F1 −→ F2 : (x, v) 7→ Df(x)(v)

is continuous.

It is worth noticing here a non trivial difference between the above defini-tion of differentiability and the classical one for Banach spaces: The total dif-ferential Df does not involve the space of continuous linear maps L(F1,F2),thus avoiding the possibility of dropping out of the working framework incase where F1 and F2 are Frechet spaces (see the comments following Propo-sition 2.1.6).

The notion of Cn-differentiability (n ≥ 2) can be defined by induction:

Definition 2.2.3. A map f : U → F2 is Cn-differentiable on U if thefollowing conditions hold true:

1) f is Cn−1-differentiable.

2) For every x ∈ U , there exists a symmetric and continuous n-linear mapDnf(x) : Fn1 → F2 such that

R(t, v) :=

=

f(x+tv)−f(x)−Df(x)(tv)− 12!D2f(x)(tv,tv)−···− 1

n!Dnf(x)(tv,...,tv)

tn, t 6= 0

0, t = 0

is continuous at every (0, v) ∈ R× F1.

3) The differential (of order n)

Dnf : U × Fn1 −→ F2 :

(x; v1, v2..., vn) 7→ Dnf(x)(v1, v2..., vn)

is continuous.

The definition of C∞-differentiability is now obvious. Occasionally, aCn-map in the sense of Definition 2.2.3 will be called Leslie Cn.

2.2. Differentiability 83

Remarks 2.2.4. 1) Assuming that F1 and F2 are Banach spaces, then(see also [Gal96]) the ordinary (Frechet) Cn-differentiability implies Leslie’sCn-differentiability, but the converse is not always true. The previous in-compatibility disappears in the case of smooth functions, namely:

f is Leslie C∞ if and only if it is C∞ in the ordinary sense

Of course, Leslie’s differentiation coincides with the ordinary differentiationwithin the framework of finite dimensional topological vector spaces.

2) Linear and bilinear maps behave as in the classical case:

• Every continuous linear map f : F1 → F2 is Leslie C∞ with

Df = f and Dnf = 0 (n ≥ 2).

• Analogously, every continuous bilinear map g : F1×F2 → F3 is LeslieC∞ with

Dg(x, y)(a, b) = g(x, b) + g(a, y),

D2g(x, y)(a, b, c, d) = g(a, d) + g(c, b),

Dng = 0 (n ≥ 3),

for every x, a, c ∈ F1 and y, b, d ∈ F2.

3) From Definition 2.2.1, it follows that the differential at x satisfies thestandard relation

Df(x)(h) = limt→0

f(x+ th)− f(x)

t.

4) If the domain of a Cr-map f is a complete Hausdorff locally convextopological space, then Dsf is a uniquely determined Cr−s-map, for everypositive integer s ≤ r.

5) The chain rule holds for all differentiable maps f : U ⊂ F1 → F2 andg : V ⊂ F2 → F3, with f(U) ⊂ V and F1, F2 as in case 2); that is,

D(g f)(x) = Dg(f(x)) Df(x),

for every x ∈ U .

7) Clearly, the above considerations hold a fortiori for Frechet spaces.

For the sake of simplicity, throughout this work differentiability will beassumed to be of class C∞, unless something different is explicitly stated.


2.3 Frechet spaces as projective limits

As already mentioned in the previous sections, Frechet spaces have funda-mental differences from Banach spaces. However, many problems arising inthe Frechet category of spaces can be reduced to their Banach counterparts,because both categories are intimately related. As a matter of fact, everyFrechet space can be represented as a projective limit of Banach spaces.We give here a brief account of this approach, which will be systematicallyapplied throughout this book.

Definition 2.3.1. Let I be a directed set, and let Eii∈I be a family oftopological vector spaces mutually connected by the continuous linear mapsρji : Ej −→ Ei (j ≥ i) satisfying the conditions

ρik ρji = ρjk; j ≥ i ≥,

in other words, the following diagram commutes.

Ejρji

- Ei

Ek

ρik

?

ρjk-

The family Ei, ρjii∈I is called a projective system with factors thespaces Eii∈I and connecting morphisms the maps ρji. The (projec-tive) limit of this system is the subspace of the cartesian product

∏i∈I E

i

lim←−

Ei :=

(xi)i∈I : ρji(xj) = xi, ∀ j ≥ i.

For simplicity, we write (xi) instead of (xi)i∈I , if there no ambiguity aboutthe index set I.

The construction of a projective limit implies the existence of the canon-ical projections

ρi : lim←−

Ei −→ Ei : (xi)i∈I 7→ xi.

They are related with the connecting morphisms by

ρji ρj = ρi; j ≥ i,

2.3. Frechet spaces as projective limits 85

pictured also in the following commutative diagram:

lim←−

Eiρj

- Ej

Ei

ρji

?

ρi-

The above definitions extend to all categories of topological spaces pro-vided that the connecting maps are morphisms of the given category; in thecategory sense, projective limits are called left limits, inverse limits orjust limits of diagrams, see for example [Dod88].

Projective limits inherit the topological and linear structure of the carte-sian product with the following additional properties (see also [Dug75]):

Proposition 2.3.2. Let lim←−

Ei be the projective limit of a projective systemof topological vector spaces. Then:

i) The topology of the limit is the weakest (coarsest) topology making thecanonical projections ρi continuous.

ii) If each factor Ei is a Hausdorff space, then so is the limit.iii) If all the connecting morphisms ρji are injective (resp. bijective),

then so are all the canonical projections ρi.

Projective limits can be used to describe several different spaces or struc-tures. Some basic but characteristic examples are listed below.

Examples 2.3.3.

1. Let Ei (i ∈ N) be a countable family of topological vector spaces thatform a descending sequence of subspaces:

E1 ⊇ E2 ⊇ · · · ⊇ En ⊇ En+1 ⊇ · · ·

It determines a projective system with connecting morphisms the naturalembeddings

ρji : Ej → Ei (j ≥ i).

Thenlim←−

Ei =⋂

i∈N

Ei.

The topology of the limit now coincides with the relative topology of theintersection as a subspace of each factor Ei.


2. The cartesian product∏i∈NE

i of a countable family of topological vec-tor spaces coincides, both from the topological and the algebraic point ofview, with the limit of the projective system formed by the partial products∏ik=1E

k, with connected morphisms

ρji :

j∏

k=1

Ek −→i∏

k=1

Ek : (x1, x2, ..., xj) 7→ (x1, x2, ..., xi), j ≥ i.

In this case

∏

i∈N

Ei ≡ lim←−i∈N

i∏

j=1

Ei

;

therefore,

(xi) =(xi)i∈N≡(x1,(x1, x2

), . . . ,

(x1, x2, . . . , xi

), . . .

).

Projective limits of topological vector spaces naturally form a categorywhose morphisms are maps compatible with the connecting morphisms andthe canonical projections of the limit.

Definition 2.3.4. Let Ei; ρjii,j∈I and F i;ϕjii,j∈I be two projectivesystems with limits E = lim

←−Ei and F = lim

←−F i, respectively. A family

f i : Ei → F ii∈I forms a projective system of maps if

ϕji f j = f i ρji j ≥ i.

Schematically, this amounts to the commutativity of the following diagram.

Ejf j

- F j

Ei

ρji

?

f i- F i

ϕji

?

In this case, the projective limit of (f i) is defined by

f := lim←− fi : E −→ F : (xi)i∈I 7→

(f i(xi)

)i∈I.


One easily checks that lim←−

f i is the unique map related with the canonical

projections of the limits lim←−

Ei and lim←−

F i as in the following commutativediagram:

lim←−Ei

lim←− fi

- lim←−Fi

Ei

ρi

?

f i- F i

ϕi

?

in other wordsϕi f = f i ρi.

The uniqueness of lim←− fi essentially proves the following very frequently

used conclusion.

Proposition 2.3.5. With the assumptions of the preceding definition, con-sider a map g : lim←−E

i → lim←−Fi. Then

g = lim←− fi if and only if ϕi g = f i ρi, ∀ i ∈ N.

The next result is an immediate consequence of the definitions.

Proposition 2.3.6. With the same assumptions as before, we have:i) If every f i is linear, then f is linear.ii) If every f i is continuous, then f is continuous.iii) If every f i is bijective, then f is bijective.

The categories of projective limits and Frechet spaces are closely relatedas it will be shown below. A first important result towards this direction isthe following.

Proposition 2.3.7. LetEi, ‖·‖Ei

i∈N

be a countable family of Banach

spaces forming a projective system with connecting morphisms ρji : Ej → Ei,where j ≥ i. Then the corresponding limit lim

←−Ei is a Frechet space.

Proof. Setting

(2.3.1) pn((xi)i∈N

)=

n∑

i=1

∥∥xi∥∥Ei ; n ∈ N,


we obtain a (countable) family of seminorms on lim←−

Ei. The topology in-duced by this family coincides with the projective limit topology. Indeed,each pn is continuous, with respect to the projective limit topology, as thecomposite of the canonical projections and the norms of the factors, i.e.

(2.3.2) pn =

n∑

i=1

(‖·‖Ei ρi

).

On the other hand, the inequality

∥∥ρn((xi)i∈N

)∥∥En ≤ pn

((xi)i∈N

)

ensures that all the canonical projections are continuous with respect to theseminorm topology. Therefore, the two topological structures coincide.

Next we show that the space thus obtained is Hausdorff. Indeed, if

pn((xi)i∈N

)= 0

for all n ∈ N, then

n∑

i=1

∥∥xi∥∥Ei = 0 ⇒

∥∥xi∥∥Ei = 0, ∀ i ∈ N.

Taking into account that each factor Ei is a Hausdorff space, it followsdirectly that

((xi)i∈N

)= 0.

Moreover, lim←−Ei is metrizable as derived from a countable family ofseminorms (see Proposition 2.1.2) and complete. To check completeness weconsider a Cauchy sequence (xn)n∈N in lim←−Ei, where xn = (xin)i∈N. Then

n∑

i=1

∥∥xin − xim∥∥Ei −→

n,m0;

therefore, ∥∥xin − xim∥∥Ei −→

n,m0, ∀ i ∈ N.

In other words, every sequence(xin)n∈N

is Cauchy in the Banach space Ei,

thus it convergence to an element xi ∈ Ei. The continuity of the connectingmorphisms ρji, and the fact that every Ei is a Hausdorff space imply thatρji(xj) = xi, for every j ≥ i, or x = (xi)i∈N ∈ lim←−Ei. Now the initial

sequence (xn) converges to x since ρi(xn) = xin converges to xi, for everyi ∈ N.


What makes the theory of projective limits really efficient in the study ofFrechet spaces is that the converse of Proposition 2.3.7 is also true.

Theorem 2.3.8. Every Frechet space can be identified with a projectivelimit of Banach spaces.

Proof. Let F be a Frechet space with topology generated by a family ofseminorms (pi)i∈N. Without loss of generality, we may assume that thesequence of seminorms inducing the topology of F is increasing, i.e.

p1 ≤ p2 ≤ · · · . ≤ pi ≤ pi+1 ≤ · · ·

If this is not the case, we can consider the family

qi := p1 + p2 + · · ·+ pi ; i ∈ N,

which defines the same topology on F. We denote by Ei the completion ofthe quotient space F/ ker pi (i ∈ N), and by ρji the connecting morphisms

ρji : Ej −→ Ei : [x+ ker pj ]j 7→ [x+ ker pi]i ; j ≥ i,

where the bracket [ ]i stands for the corresponding equivalence class. ThenF coincides with the projective limit of Ei; ρjii,j∈I by means of the iso-morphism

Φ: F −→ lim←−Ei : x 7→(

[x+ ker pi]i)i∈N

.

Remarks 2.3.9. 1) As a byproduct of the preceding identification, thecanonical projections

ρi : F ≡ lim←−

Ei −→ Ei : x 7→ [x+ ker pi]i ; i ∈ N,

are isometries in the sense that

pi(x) =∥∥ [x+ ker pi]i

∥∥Ei =

∥∥ρi(x)∥∥Ei .

2) The construction of the above particular projective limit yielding F

allows us to consider the connecting morphisms and the canonical projec-tions of the system

Eii∈N

as surjective morphisms, a fact not true in thecase of arbitrary projective systems.

The representation of Frechet spaces as projective limits is very advan-tageous: Questions arising within the Frechet framework can be reducedto their counterparts in the Banach factors. In this way, obstacles set by


the very structure of Frechet spaces and obstructing the solution of manyproblems, can be surmounted.

A first important application of this representation is dealing with thespace of continuous linear maps L(F1,F2) between two Frechet spaces. Asmentioned earlier, it drops out of the Frechet category. However, if the mapsunder consideration can be realized also as projective limits, then L(F1,F2)can be replaced by a new space within the Frechet framework.

More precisely, we assume that F1 ≡ lim←−Ei1 and F2 ≡ lim←−Ei2 are Frechet

spaces, where Ei1; ρji1 i,j∈N and Ei2; ρji2 i,j∈N are projective systems. of

Banach spaces. Then we prove the following:

Theorem 2.3.10. The space of all continuous linear maps between F1 andF2 that can be represented as projective limits,

(2.3.3) H(F1,F2) :=

(f i) ∈

∏

i∈N

L(Ei1,Ei2) : lim←−

f i exists

,

is a Frechet space. Moreover, H(F1,F2) is also represented as the projectivelimit of appropriate Banach functional spaces, and the map

(2.3.4) ε : H(F1,F2) −→ L(F1,F2) : (f i) 7→ lim←−

f i

is continuous linear.

Proof. To prove that H(F1,F2) is a Frechet space, it suffices (by Proposi-tion 2.1.6) to show that H(F1,F2) is a closed subspace of

∏∞i=1 L(Ei1,E

i2),

since the latter is a Frechet space according to Example 2.1.5(3). Indeed,if (an)n∈N is a sequence of elements in H(F1,F2), with an = (f1n, f

2n, . . .),

such that limn an = (f1, f2, . . .) ∈∏∞i=1 L(Ei1,E

i2), then limn f

in = f i, for

every i ∈ N. Equivalently, limn ‖fin − f

i‖i = 0, where ‖‖i is the norm of theBanach space L(F1,F2). Therefore, for every j ≥ i,∥∥ρji2 f j − f i ρ

ji1

∥∥ ≤∥∥ρji2 (f j − f jn)

∥∥L(Fj

1,Fi2)

+∥∥(f in − f

i) ρji1∥∥L(Fj

1,Fi2)

≤∥∥ρji2

∥∥ ·∥∥f j − f jn

∥∥j

+∥∥f in − f i

∥∥i·∥∥ρji1

∥∥.

Since the right-hand side of the inequality tends to 0, it follows that ρji2 fj =

f i ρji1 for every i, j ∈ N with j ≥ i. Hence, limn an = (f i)i∈N ∈ H(F1,F2),which proves the claim.

For each i ∈ N, we define the set

(2.3.5) Hi(F1,F2) :=

(f j) ∈i∏

j=1

L(Ej1,Ej2) :

ρjk2 fj = fk ρjk1 , j ≥ k.


Working as before, we show that each Hi(F1,F2) is a Banach space as aclosed subspace of

∏ij=1L(Ej1,E

j2). On the other hand, the map

(2.3.6)hji : Hj(F1,F2) −→ Hi(F1,F2) :

(f1, . . . , f j) 7→ (f1, . . . , f i) (j ≥ i)

is continuous linear. Linearity is obvious, whereas continuity is a conse-quence of

∥∥hji(f1, . . . , f j)∥∥Hi(F1,F2)

=∥∥(f1, . . . , f i)

∥∥Hi(F1,F2)

=∥∥f1

∥∥1

+ · · ·+∥∥f i∥∥i

≤∥∥f1

∥∥1

+ · · ·+∥∥f j∥∥j

=∥∥(f1, . . . , f j)

∥∥Hj(F1,F2)

.

It is clear that hjk = hik hji holds for every i, j, k ∈ N with j ≥ i ≥ k,thus

Hj(F1,F2);hji

is a projective system of Banach spaces, and (see

Proposition 2.3.7) lim←−Hj(F1,F2) is a Frechet space.

Next, we define the maps

hk : H(F1,F2) −→ Hi(F1,F2) :(f i)i∈N7→(f1, . . . , fk

), k ∈ N.

They are continuous linear and satisfy the equality hi = hji hj , for everyj ≥ i. As a result, we obtain the continuous linear map

h := lim←−

hi : H(F1,F2) −→ lim←−Hi(F1,F2).

More explicitly,

(2.3.7)h((f i)i∈N

)= h

(f1, f2, . . .

)

=((f1), (f1, f2), (f1, f2, f3), . . .

)=((f1, . . . , f i)

)i∈N

.

We check that h is injective: Indeed,

h(f1, f2, . . .) = h(g1, g2, . . .)

⇒ hk(f1, f2, . . .) = hk(g1, g2, . . .), k ∈ N

⇒ (f1, f2, . . . , fk) = (g1, g2, . . . , gk), k ∈ N

⇒ (f1, f2, . . .) = (g1, g2, . . .).

Also, h is surjective: Let any a =(ai)i∈N∈ lim←−Hi(F1,F2), where

ai =(f1i , . . . , f

ii

)∈ Hi(F1,F2) ⊆

i∏

j=1

L(Ej1,Ej2).


By (2.3.5) and (2.3.6),

fki ∈ L(Ek1 ,Ek2); k ≤ i,

ρlk2 fli = fki ρ

lk1 ; k ≤ l ≤ i,

hji(f1j , . . . , f

jj

)=(f1i , . . . , f

ii

); i ≤ j.

The last equality implies that(f1j , . . . , f

ij

)=(f1i , . . . , f

ii

), thus f ij = f ii , for

every j ≥ i. Consequently,

f11 = f12 = f13 = · · · =: f1 ∈ L(E11,E

12),

f22 = f23 = · · · =: f2 ∈ L(E21,E

22),

...

f ii =: f i ∈ L(Ei1,Ei2).

Moreover,

ρlk2 fl = ρlk2 f

ll = fkl ρ

lk1 = fk ρlk1 ; k ≤ l,

thus(f1, f2, . . .

)∈ H(F1,F2), and

h(f1, f2, . . .

)=(hk(f1, f2, . . .

))i∈N

=

=((f1, f2, . . . , fk

))k∈N

=((f1i , f

2i , . . . , f

kk

))k∈N

= a.

The open mapping theorem (see also Proposition 2.1.6) now ensures that his an isomorphism of Frechet spaces; hence,

(2.3.8) H(F1,F2)h≡ lim←−H

i(F1,F2).

Finally, we immediately check that ε is linear. For its continuity wetake an arbitrary seminorm | · |(n,B) of L(F1,F2) given by (2.1.3); thatis, | f |(n,B) = sup qn(f(x)) x ∈ B, where B is a bounded subset of F1

containing 0, and qn =∑n

i=1

(‖ · ‖Fi

2 ρi2

)is a seminorm of F2 = lim←−Fi2

[recall equalities (2.3.1)–(2.3.2)]. Analogously, we consider the seminormspn =

∑ni=1

(‖ · ‖Fi

1 ρi1

)and rn =

∑ni=1

(‖ · ‖Hi(F1,F2

hi)

of F1 = lim←−Fi1

and H(F1,F2) ≡ lim←−Hi(F1,F2), respectively. Therefore, for every (f i) =


(f i)i∈N

, we obtain:

∣∣ε((f i)

)∣∣(n,B)

=∣∣ lim←− (f i)

∣∣(n,B)

= supqn((f i(xi))i∈N

), x = (xi) ∈ B

= sup

n∑

i=1

∥∥f i(xi)∥∥Fi2, x = (xi) ∈ B

≤ sup

n∑

i=1

∥∥f i∥∥Hi(F1,F2)

· ‖xi‖Fi1, x = (xi) ∈ B

≤ sup

(n∑

i=1

∥∥f i∥∥Hi(F1,F2)

)· pn(x), x ∈ B

= suprn((f i)

)· pn(x), x ∈ B

= suppn(x), x ∈ B

· rn((f i)

),

which proves the continuity of ε and concludes the proof.

As a byproduct of equalities (2.3.7) and (2.3.8), we obtain the followinguseful identification:

(2.3.9)(f1, f2, f3, . . .

) h≡((f1), (f1, f2), (f1, f2, f3), . . .

).

or, in a condensed form,

(2.3.9′) lim←−(f i) =(f i)i∈N

h≡ lim←−

(f1, . . . , f i

).

We connect now the projective limits with the differentiation method weadopted in § 2.2.

Proposition 2.3.11. Let F1 and F2 be two Frechet spaces as in the previousstatement. Let also U i ⊆ Ei1 (i ∈ N) be open sets and f i : Ui → Ei2. Weassume that U := lim

←−U i, f := lim

←−f i : U → F2 exist, and U is an open

subset of F1. Then

i) If each f i is differentiable on U i, then so is f , and

Df(x) = lim←−

Df i(xi), x = (xi) ∈ U.

ii) If each f i is Ck, then so is f .


Proof. i) Since the ordinary differentiability in Banach spaces is equivalentto that of Leslie (see Remarks 2.2.4), the assumptions imply that, for everyi ∈ N, there exists a continuous linear map Df i(xi) : Ei1 → Ei2 such that thefunctions

Ri(t, v) =

f i(x+ tv)− f i(x)−Df i(x)(tv)

t, t 6= 0

0, t = 0

are continuous at every (0, v) ∈ R× Ei1. On the other hand, the fact that fis the projective limit of (f i), yields

ρji2 fj = f i ρji1 and ρi2 f = f i ρi1,

for every i, j ∈ N with j ≥ i, where ρjik : Ejk → Eik (k = 1, 2) are theconnecting morphisms of the two projective systems producing F1 and F2,respectively. Taking into account that the connecting morphisms and thecanonical projections are continuous linear maps and differentiating the lastformulas at any point x = (xi) ∈ U , we have

(2.3.10)ρji2 Df

j(xj) = D(ρji2 f

j)(xj) = D

(f i ρji1

)(xj)

= Df i(ρji1 (xj)) = Df i(xi)j (j ≥ i).

As a result, the projective limit operator lim←−Dfi(xi) can be defined and is

continuous linear. Moreover, if

R(t, v) =

f(x+ tv)− f(x)−Df(x)(tv)

t, t 6= 0

0, t = 0

we check that

(ρji2 R

j)(t, v) = ρji2

(f j(xj + tv)− f j(xj)−Df j(xj)(tv)

t

)

=

(ρji2 f

j)(xj + tv)−

(ρji2 f

j)(xj)−

(ρji2 Df

j(xj))(tv)

t

=f i(xi + tρji1 (v)

)− f j(xi)−Df i(xi)

(tρji1 (v)

)

t

=(Ri

(idR×ρ

ji1

))(t, v),


for all j ≥ i. Analogously,(ρi2 R

)(t, v) =

(Ri (idR × ρ

i1))(t, v), i ∈ N.

Therefore, lim←−

Ri can be defined and coincides with R, making the latter a

continuous function. This means that f is Leslie differentiable at x = (xi),and its differential is the projective limit of the differentials of its factors, asin the formula of assertion i).

Assertion ii) is now a direct consequence of equality Df = lim←−Dfi and

the fact that projective limits respect continuity. Recall thatDf i : U i×Ei1 →Ei2 and Df : U×F1 → F2 are the (total) differentials of f i and f , respectively,as in Definitions 2.2.2 and 2.2.3.

By similar arguments, we can extend the assertions of Proposition 2.3.11to the case of Ck-differentiable maps.

If f : U ⊆ F1 → F2 is an arbitrary smooth map between Frechet spaces(U ⊆ F1 open), the total differential Df : U × F1 → F2 does not necessarilyimply that the map

U ∋ x 7−→ Df(x) ∈ L(F1,F2)

is smooth, as is the case of the (Frechet) derivative in Banach spaces. Incontrast, if we are dealing with projective limits of smooth maps, we obtainthe following result, which will be applied later on.

Proposition 2.3.12. With the assumptions of Proposition 2.3.11, the map

U ∋ x 7−→ Df(x) = lim←−

Df i(x) ∈ L(F1,F2); x =(xi)i∈N

,

is smooth.

Proof. We define the maps

Ri : U i −→ L(Ei1,Ei2) : xi 7→ Df i(xi)

(not to be confused with Ri : R× Ei1 → Ei2 of the previous proof), and

F i : U i −→ Hi(F1,F2) : xi 7→(R1(ρi11 (xi)), R2(ρi21 (xi)), . . . , Ri(xi)

).

Each Ri is smooth because of the smoothness of f i in Banach spaces. More-over, for every i, j, k with i ≥ j ≥ k,

ρjk2 Rj(ρij1 (xi)

)= ρjk2 Df

j(ρij1 (xi)

)

= Dfk(ρjk1 (ρij1 (xi)

)[by the analog of (2.3.10)]

= Rk(ρik1 (xi)

).


The maps F i do take values in Hi(F1,F2), for all i ∈ N. Indeed, for everyxj ∈ U j and j ≥ i, (2.3.6) implies that

(hji F j

)(xj) = hji

(R1(ρj11 (xj)

), R2

(ρj21 (xj)

), . . . , Rj(xj)

)

=(R1(ρj11 (xj)

), R2

(ρj21 (xj)

), . . . , Ri(ρji1 (xj)

)

=(R1(ρi1(ρji1 (xj)

)), R2

(ρi21 (ρji1 (xj))

), . . . , Ri

(ρii1(ρji1 (xj)

)))

=(F i ρji1

)(xj);

hence, the map

lim←−Fi : U = lim←−U

i −→ lim←−Hi(F1,F2) = H(F1,F2)

exists and is smooth, as a consequence of Theorem 2.3.10 and Proposi-tion 2.3.11. Moreover, taking into account (2.3.4) and the identifications(2.3.9),(2.3.9′), we have that

(ε lim←−

F i)

(x) = ε(lim←−

F i(xi))

= ε(R1(x1),

(R1(x1), R2(x2)

),(R1(x1), R2(x2), R3(x3)

), . . .

)

≡ ε(R1(x1), R2(x2), R3(x3), . . .

)= lim←−

Ri(xi)

= lim←−Dfi(xi) = Df,

for every x =(xi)i∈N

. Since ε lim←−

F i is smooth, we conclude the proof.

We close this section with a discussion on the composition and evaluationmaps, whose use in the context of Frechet spaces and their applications toFrechet manifolds and bundles is often critical. However, their continuity isunder question because spaces of continuous linear maps between Frechetspaces drop out of the category of these spaces.

A way out of this difficulty is achieved by the two results proved below.For the first of them, let F and G be two Frechet spaces whose topology isdefined by the families of seminorms pnn∈N and qnn∈N, respectively. Werecall that the space of continuous linear maps L(F,G) is a Hausdorff locallyconvex topological vector space, with topology defined by the non-countablefamily of seminorms | · |n,B given by

∣∣g∣∣n,B

= sup qn(g(x)), x ∈ B ,

where n ∈ N, and B is any bounded subset of F containing the zero element.L(F,G) fails to be a Frechet space since it is not metrizable.


On the other hand, the subspace

(2.3.11) LI(F) =

f ∈ L(F) : sup

pn(f(x))

pn(x), pn(x) 6= 0

<∞

of L(F) = L(F,F) becomes a Frechet space with corresponding seminorms

|f |n = sup

pn(f(x))

pn(x), pn(x) 6= 0

.

Setting, for the sake of simplicity [see also Proposition 2.3.7 and (2.3.3)],

(2.3.12) H(F) := H(F,F),

we obtain the following result, connecting the spaces (2.3.11) and (2.3.12):

Proposition 2.3.13. Let f : F → F be a continuous linear map. Thenf ∈ LI(F) if and only if it is a projective limit of continuous linear mapsbetween Banach spaces.

Proof. Assume first that f ∈ LI(F). If

Ci := sup

pi(f(x))

pi(x), pi(x) 6= 0

; i ∈ N,

then, following the analysis of Theorem 2.3.8 leading to the realization ofthe Frechet space F as a projective limit of Banach spaces, we define themaps

f i : Ei −→ Ei :[x+ ker pi

]7→[f(x) + ker pi

], i ∈ N.

Each, f i is a continuous linear map, since∥∥f i(u)

∥∥Ei ≤ Ci · ‖u‖Ei , for every

u ∈ Ei. Moreover,(f i)i∈N

is a projective system, since

(ρji f j

) ([x+ ker pj

])= ρji

([f(x) + ker pj

])=[f(x) + ker pi

]

= f i([x+ ker pi

])=(f i ρji

) ([x+ ker pj

]),

whereas, each f i (i ∈ N) is the i-th projection of f (by the correspondingcanonical map of the projective system), i.e.

(f i ρi

)(x) = f i

([x+ ker pi

])=[f(x) + ker pi

]= ρi(f(x)).

Therefore, f = lim←− fi.


Conversely, assume that f = lim←−

f i, with f i ∈ L(Ei). Then, for every

x = ([x+ ker pi])i∈N ∈ F, we have that

pi(f(x)) =

i∑

j=1

∥∥f j([x+ ker pi

])∥∥Ej

≤i∑

j=1

∥∥f j∥∥L(Ej)

· pj(x) ≤

i∑

j=1

∥∥f j∥∥ · pi(x).

As a result, f ∈ LI(F).

Based on (2.3.11), we prove the continuity of the following form of com-position, needed in § 3.6:

Proposition 2.3.14. The composition map

comp: L(F,G)× LI(F) −→ L(F,G) : (g, f) 7→ f g

is continuous.

Proof. Since LI(F) is a Frechet space, and therefore metrizable, it suffices toshow that, for any sequence (fn)n∈N in LI(F) and every net (gi)i∈I in L(F,G)converging to f and g, respectively, it follows that comp(gi, fn) = gi fnconverges to g f . Before this, recall that

B ⊆ F is bounded

⇔ B is bounded with respect to all seminorms

⇔ ∀ k ∈ N ∃Mk > 0 : |x|k ≤Mk, ∀ x ∈ B.

Moreover, for (fn)n∈N as above, the set

D = fn(x) |n ∈ N, x ∈ B

is also bounded, because, for every k-seminorm (for convenience the semi-norms of all spaces involved are denoted by | · |a with appropriate indices a),we have that ∣∣fn(x)

∣∣k≤∣∣fn∣∣k· |x|k ≤

∣∣f∣∣k·Mk,

since fn → f and∣∣fn∣∣k≤∣∣fn − f

∣∣k

+ |f |k ≤ |f |k.


Now, as a first step to the proof of our claim, we consider the casef = g = 0. Then gi fn → 0. Indeed,

∣∣gi fn∣∣n,B

= sup∣∣gi(fn(x))

∣∣k, x ∈ B

≤ sup∣∣gi(y)

∣∣k, y ∈ D

=∣∣gi∣∣k,D−→ 0.

In the general case, where fn → f and gi → g, with f , g not necessarilyzero, we see that((gi − g) (fn − f)

)(x) = gi(fn(x))− gi(f(x))− g(fn(x)) + g(f(x)) =

=(gi(fn(x))− g(f(x))

)+(g(f(x))− gi(f(x))

)+(g(f(x))− g(fn(x))

),

or

gi(fn(x)− g(f(x)) =((gi − g) (fn − f)

)(x) +

(gi(f(x))− g(f(x)

)

+(g(fn(x))− g(fn(x))

),

thus

(2.3.13)

∣∣(gi fn)− (g f)∣∣k,B

= sup∣∣gi(fn(x)− g(f(x))

∣∣k, x ∈ B

≤ sup∣∣(gi − g) (fn − f)

∣∣k, x ∈ B

+ sup∣∣(gi − g)(f(x))

∣∣k, x ∈ B

+ sup∣∣g(fn(x)− f(x))

∣∣kx ∈ B

.

Since (gi − g)→ 0 and (fn − f)→ 0, the first case implies that

(2.3.14)sup

∣∣(gi − g) (fn − f)∣∣k, x ∈ B

=

∣∣(gi − g) (fn − f)∣∣k,B−→ 0.

Also, by the convergence of g,

(2.3.15) sup∣∣(gi − g)(f(x))

∣∣k, x ∈ B

=∣∣gi − g

∣∣k,f(B)

−→ 0,

whereas, by the convergence of f and the continuity of g,

fn −→ f ⇒ fn(x) −→ f(x) ⇒ g(fn(x)− f(x)

)−→ 0, ∀ x ∈ B;

hence,

(2.3.16) sup∣∣g(fn(x)− f(x))

∣∣kx ∈ B

−→ 0.

As a result, (2.3.13)–(2.3.16) imply that∣∣(gi− fn) (g− f)

∣∣k,B→ 0, for any

seminorm | · |k,B of L(F,G); therefore, (gi − fn)→ (g − f), as desired.


Remarks 2.3.15. 1) With similar arguments, we prove the continuity of

comp: LI(F)×L(F,G) −→ L(F,G) : (f, g) 7→ f g

For the sake of completeness, we sketch the proof: Taking seminorms, asequence (fn) and a net (gi) as in the main proof, we consider again twocases:

i) f = g = 0. Then, for an arbitrary seminorm | · |k,B of L(F,G) (againB ⊆ F bounded containing 0),

∣∣(fn gi)(x)∣∣k

=∣∣fn(gi(x))

∣∣k≤∣∣fn∣∣k·∣∣gi(x)

∣∣k, x ∈ B.

Consequently,

sup∣∣(fn gi)(x)

∣∣k, x ∈ B

≤∣∣fn∣∣k· sup

∣∣gi(x)∣∣k, x ∈ B

or, equivalently,∣∣fn gi

∣∣k,B≤ |f |n ·

∣∣gi∣∣k,B

. Since fn and gi converge to zero,the same is true for their composition.

ii) fn → f and gn → g, with f , g not necessarily zero. Then

∣∣(fn gi)− (f g)∣∣k,B

= sup∣∣fn(gi(x))− f(g(x))

∣∣k, x ∈ B

≤ sup∣∣(fn − f)

(gi(x)− g(x)

)∣∣k, x ∈ B

+ sup∣∣(fn − f)(g(x))

∣∣k, x ∈ B

+ sup∣∣f(gi(x)− g(x)

)∣∣k, x ∈ B

Since (fn − f)→ 0 and (gi − g)→ 0, it follows that

sup∣∣(fn − f)(gi(x)− g(x))

∣∣kx ∈ B

=∣∣(fn − f) (gi − g)

∣∣n,B−→ 0.

On the other hand, since g is continuous linear, g(B) is also a boundedset containing the zero element, thus providing a seminorm of

∣∣ ·∣∣k,g(B)

of

L(F,G), for which

sup∣∣(fn − f)(g(x))

∣∣kx ∈ B

=∣∣fn − f

∣∣k,g(B)

−→ 0

because (fn − f)→ 0 in LI(F) ⊆ L(F). Finally,

sup∣∣f(gi(x)− g(x))

∣∣k, x ∈ B

≤ |f |k · sup∣∣gi(x)− g(x)

∣∣k, x ∈ B

= |f |k ·∣∣gi − g

∣∣k,B−→ 0.


As a result,∣∣(fn gi)− (f g)

∣∣k,B→ 0, for any seminorm | · |k,B of L(F,G);

therefore, fn gi → f g.

2) We note that, in virtue of (2.3.4), LI(F) = ε(H(F)).

To prove the continuity of the evaluation map, we consider the Frechetspaces E and F, with topologies derived from the families of seminormspnn∈N and qnn∈N, respectively. As usual, L(E,F) is a locally convexspace with topology based on the seminorms given by

∣∣f∣∣n,B

= sup qn(f(x)), x ∈ B ,

where n ∈ N, and B ⊆ E is any bounded set containing 0.

Proposition 2.3.16. The evaluation map

ev : L(E,F)× E −→ F : (f, a) 7→ f(a)

is continuous.

Proof. Let (fi) be an arbitrary net in L(E,F) converging to a continuouslinear map f , and let (an) be sequence in E converging to a ∈ E. Then, forany seminorm ql of F,

ql(fi(an)− f(a)

)≤ ql

(fi(an)− f(an)

)+ ql

(f(an)− f(a)

).

Since f ∈ L(E,F), there exists a seminorm pk of E and a positive constantM such that ql(f(u)) ≤M · pk(u), for every u ∈ E. Then,

ql(f(an)− f(a)

)= ql

(f(an − a)

)≤M · pk(an − a) −→

n0 .

On the other hand, if we consider the bounded set

B = an − a |n ∈ N ∪ 0, a

and the corresponding seminorm | · |l,B of L(E,F), we have that∣∣fi − f

∣∣l,B

= supql((fi − f)(x)

), x ∈ B

−→i

0,

as well as

ql(fi(an)− f(an)

)≤ ql

(fi(an)− fi(a)− f(an) + f(a)) + ql(fi(a)− f(a)

)

= ql((fi − f)(an − a)

)+ ql

((fi − f)(a)

)

≤∣∣fi − f

∣∣l,B

+∣∣fi − f

∣∣l,B.

This completes the proof.

The following result is an obvious consequence of Propositions 2.3.14,2.3.16 and Remark 2.2.4(2).

Corollary 2.3.17. The preceding maps comp and ev are smooth.


2.4 Differential equations in Frechet spaces

One of the main problems in the study of infinite dimensional non-Banachlocally convex topological vector spaces is the lack of a general solvabilitytheory for differential equations (even the linear ones), analogous to the the-ory established in the Banach framework. This serious drawback directlyimpinges on many problems and applications of differential geometry. Forexample, Frechet modelled differential manifolds and fibre bundles with ap-plications (especially in theoretical physics) constantly increasing, appear tohave very poor geometrical structures because of the aforementioned weak-ness.

The question of solvability has been approached by a number of authorsin recent years by suggesting various ways of studying concrete classes ofdifferential equations in infinite-dimensional spaces, particularly in Frechetspaces (see, for instance, [Ham82], [Lem86], [Lob92] and [Pap80]). In thissection, a new approach, based on the representation of Frechet spaces byprojective limits of Banach spaces, is proposed.

Our method exploits the compatibility of algebraic, topological and dif-ferential structures of Frechet spaces with projective limits. We note thatthis approach not only addresses the solvability question of the equations athand, but also provides a detailed description of the solutions, a fact not al-ways achieved in other approaches. Clearly, this is essentially advantageousin many applications.

In what follows, F will denote a Frechet space whose topology is gener-ated by the countable family of seminorms (pi)i∈N and is realized as the limitof a projective system of Banach spaces Ei; ρjii,j∈N. In this framework,the notion of Lipschitz (continuity) condition is generalized as follows.

Definition 2.4.1. A map φ : F→ F will be called k-Lipschitz, where k isa positive real number, if

pi(φ(x2)− φ(x1)) ≤ k · pi(x2 − x1)

for every x1, x2 ∈ F and i ∈ N.

In the case where φ is the projective limit of a projective system of mapsφi : Ei → Eii∈N, we check whether φ satisfies Definition 2.4.1 or not bythe behavior of its components. More precisely:

Proposition 2.4.2. A map φ = lim←−

φi is k-Lipschitz if and only if each φi

is k-Lipschitz in Ei.

2.4. Differential equations in Frechet spaces 103

Proof. Let xi = ρi (x) , yi = ρi (y) be two arbitrarily chosen points in Ei.(Here, deviating from our usual notation, we set

(xi)i∈N

for typographical

reasons which will be clear soon.) The relations between φ and (φi) lead to

∥∥φi (xi)− φi (yi)∥∥Ei = pi

(φ(x)− φ(y)

),

(recall that ‖ · ‖Ei stands for the norm of Ei). Therefore, a k-Lipschitz con-dition for φ in the generalized sense of Definition 2.4.1, implies the classicalanalog for each φi in Ei, i.e.

∥∥φi (xi)− φi (yi)

∥∥Ei ≤ k · pi(x− y) = k · ‖xi − yi‖Ei .

Conversely, the k-Lipschitz conditions for all φi’s (with the same constantk, of course) yield

pi(φ(x)− φ(y)) ≤ k · ‖xi − yi‖Ei = k · pi(x− y),

by which we conclude the proof.

The previous generalization of the Lipschitz continuity allows one toapproach a wide class of differential equations in F by using the techniquesof projective limits, as the next result illustrates.

Theorem 2.4.3. Let φ= lim←−

φi : R× F→ F be a projective limit k-Lipschitzmap. If, for an initial point (t0, x0) ∈ R× F, there exists a constant τ ∈ R

such that

(* ) M := suppi(φ(t, x0)); i ∈ N, t ∈ [t0 − τ, t0 + τ ]

< +∞,

then the differential equation

(2.4.1) x′ = φ(t, x)

admits a unique solution defined on the interval I = [t0−a, t0+a], satisfying

the initial condition x(t0) = x0. Here a = infτ, 1

M1+k

.

Proof. We define the system of ordinary differential equations in the Banachspaces Ei

(2.4.2) x′i = φi(t, xi).


According to Proposition 2.4.2, each φi is k-Lipschitz and, for every i ∈ N

and x ∈ Ei with ‖xi − ρi(x0)‖i ≤ 1, we see that

∥∥φi(t, x)∥∥Ei ≤

∥∥φi(t, ρi(x0))∥∥Ei +

∥∥φi(t, x)− φi(t, ρi(x0))∥∥Ei

≤∥∥ρi(φ(t, x0))

∥∥Ei + k ·

∥∥x− ρi(x0)∥∥Ei

= pi(φ(t, x0)) + k ≤ M1 + k.

Therefore (see [Car67(a), Corollary 1.7.2]), each Equation 2.4.1 admits aunique solution xi on the interval I = [t0 − a, t0 + a].

On the other hand, the previous solutions form a projective system. Thisis deduced from

(ρji xj

)′(t) = ρji(x′j(t)) = ρji

(φj(t, xj(t))

)= φi(t, (ρji xj)(t)),

which means that ρji xj is a solution of (2.4.2) with the same initial con-dition (t0, ρ

i(x0)). As a result, the C1-map x = lim←−

xi can be defined on Iand determines the desired solution of (2.4.1), since

x′(t) =(x′i(t)

)=(φi(t, xi(t))

)=

=(φi(t, ρi(x(t)))

)=(ρi(φ(t, x(t)))

)= φ(t, x(t)).

This is the unique solution satisfying the initial condition (t0, x0). Indeed,if y : J ⊆ R → F is another solution of (2.4.1) with y(t0) = x0, then ycan be realized as the limit of the projective system ρi yi : J → Eii∈N.Employing analogous computations as before, we check that ρi y coincideswith xi (for each i ∈ N) as a solution of (2.4.2) with the same initial condition(t0, ρ

i(x0)). Thusy = lim←−

(ρi y) = lim←−

xi = x,

as claimed.

Since a Lipschitz condition is fulfilled in the case of linear differentialequations, taking into account (2.3.12), we prove the following:

Theorem 2.4.4. The n-order linear differential equation

x(n) = A0 · x+A1 · x′ + · · ·+An−1 · x

(n−1) +B,

where Ai : [0, 1] → L(F) and B : [0, 1] → F are continuous maps, admitsa unique solution for a given initial condition, provided that each factor Aidecomposes to

Ai = ε A∗i ,

where A∗i : [0, 1]→H(F) is continuous and, as in (2.3.4),

ε : H(F) −→ L(F) :(f i)i∈N7→ lim←− f

i.

2.4. Differential equations in Frechet spaces 105

Proof. The assumptions imply that each factor Ai can be thought of as aprojective limit of continuous maps; namely,

Ai(t) = ε(A∗i (t)) = lim←−j

(Aji (t)

),

where Aji : [0, 1] → L(Ei) (j ∈ N) are the components of A∗i . In this way,the n-order differential equation in study is equivalent to

X ′ = Φ(t,X),

where X =(x, x′, ..., x(n−1)

)and Φ(t,X) = A(t) ·X +B, with

A : [0, 1] −→ H(Fn) : t 7−→

0 1F 0 . . . 00 0 1F . . . 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 0 0 . . . 1F

A0(t) A1(t) A2(t) ... An−1(t)

.

Since Ai = ε A∗i , the operator A can be realized as the projective limit ofthe continuous operators Aj given by

Aj : [0, 1] −→ Hj(Fn) : t 7−→

0 1Ej 0 . . . 00 0 1Ej . . . 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 0 0 ... 1Ej

Aj0(t) Aj1(t) Aj2(t) ... Ajn−1(t)

.

The same is true for the map B : [0, 1] → F, since B = lim←−(ρj B). As aresult, Φ itself is the projective limit of the maps

Φj(t,X) = Aj(t) ·X +Bj,

which are linear with respect to the second factor, hence Lipschitz. Applyingnow Theorem 2.4.3, we conclude the proof.

Remark 2.4.5. It is worth noticing at this point that the above approachgeneralizes the works of R. S. Hamilton [Ham82] and N. Papaghiuc [Pap80]on linear differential equations.

In particular, R. S. Hamilton has studied ordinary differential equationsof first order x′ = ϕ(t, x), where the function ϕ decomposes into

ϕ = g f, where g : B→ F, f : F→ B,


B denoting a Banach space. Maps of this type are called smooth-Banachfunctions. However, it is proved (see [Gal97(a)]) that

a smooth-Banach linear operator can always be realized as a projectivelimit, falling thus within the framework of Theorem 2.4.4

On the other hand, the linear differential equations studied by N. Pa-paghiuc are restricted to the Frechet space LI(F) ≡ H(F) (see Proposi-tion 2.3.13, thus the result of [Pap80] is again a special case of Theorem 2.4.4.

The projective limit approach, although not answering the general prob-lem of solvability of differential equations in Frechet spaces, neverthelessapplies to every Frechet space and not to particular classes as is the casein, e.g., [Lob92]. In fact, the latter is dealing with operators admittingconverging exponential series.

Chapter 3

Frechet manifolds

The study of manifolds modelled on infinite-dimensional spaces is receiv-ing an increasing interest in recent decades due to interactions and appli-cations extending beyond the borders of (classical) differential geometryand associated problems in mathematical analysis and theoretical physics.For instance, fibrations and foliations, jet fields, connections, sprays, La-grangians and Finsler structures ([EE67], [AA96], [AIM93], [DRP95], [GP05]and [Sau87]) are objects naturally listed in this framework. In particular,manifolds modelled on non-Banach locally convex spaces have been studiedfrom different points of view as in [Omo70], [Omo74], [Omo78], [Omo97],[KM97], [AM99], [Nee06], [Vero74] and [Vero79]. However, several ques-tions remain open as a result of the internal problems of the space models.

To be more precise, in conjunction with the problems encountered inthe framework of Frechet spaces or more general topological vector spacesand discussed at the end of §,2.1, two issues still remain critical in thestudy of infinite-dimensional manifolds: The lack of a general solvabilitytheory for differential equations and the pathological structure of the generallinear groups involved in this framework. Both issues seriously affect thestudy of topological, differential and geometrical aspects in the non-Banachframework.

To give a concrete example, let us take the tangent bundle TM of asmooth manifold M modelled on a Frechet space F. Even the existence of avector bundle structure on TM cannot be ensured, since the general lineargroup GL(F), serving as the structural group in the finite-dimensional case,does not admit a reasonable Lie group structure here.

On the other hand, the study of many geometric features on M , suchas connections, parallel translations, holonomy groups, etc., reduce to ap-

107

108 Chapter 3. Frechet manifolds

propriate differential equations in the models. The problems regarding thesolvability of the latter raise questions about whether the corresponding re-sults, already obtained for finite-dimensional or Banach manifolds, can bestill transferred to the non-Banach case.

In this chapter we intend to propose a new way for addressing the abovementioned problems, by taking advantage of the realization of Frechet spacesas projective limits of Banach spaces, already discussed in §,2.3. Exploitingthe fact that projective limits are compatible with algebraic structures onthe models, as well as with the differentiability tools adopted in Chapter 2,we are able to study a wide sub-category of infinite-dimensional non-Banachmanifolds, namely those modelled on Frechet spaces that can be viewed asprojective limits of Banach manifolds.

3.1 Smooth structures on Frechet manifolds

In Chapter 2, we have exploited the compatibility of projective limits withalgebraic and topological data in order to reduce basic problems in Frechetspaces to their Banach counterparts. However, the same reduction is notalways successful when dealing with differential geometric objects, even atthe lower level of the smooth structure of a manifold. For example, withoutappropriate restrictions, the domain of a projective limit of local chartsmay collapse to a singleton. The following definition provides the optimumconditions ensuring the smooth interaction between projective limits andmanifolds.

Definition 3.1.1. Let M i;µjii,j∈N be a projective system of smooth man-ifolds modelled on the Banach spaces Eii∈N, respectively (with smoothconnecting morphisms). A system of corresponding charts (U i, φi)i∈N willbe called a (projective) limit chart if and only if the limits lim

←−U i, lim←−

φi

can be defined, and the sets lim←−

U i, lim←−

φi(lim←−

U i) are open in lim←−

M i, lim←−

Ei,respectively.

Limit charts will determine a smooth structure on the projective limitof Banach manifolds. Before proving this, we need the following:

Definition 3.1.2. With the notations of Definition 3.1.1, the space M =lim←−M

i;µjii,j∈N is called a projective limit of Banach manifolds, orplb-manifold for short, provided that:

(1) The models Eii∈N form a projective system with connecting mor-phisms ρji : Ej → Ei; j ≥ i and limit the Frechet space F = lim←−Ei.

3.1. Smooth structures on Frechet manifolds 109

(2) M is covered by a family (Uα, φa)α∈I of limit charts, where Uα =lim←−

U iα and φa = lim←−

φiα, the limits taken with respect to i ∈ N.

If there is no danger of confusion, we simply write M = lim←−Mi.

Proposition 3.1.3. A plb-manifold M = lim←−Mi, as before, is a smooth

manifold modelled on the Frechet space F.

Proof. The desired differential structure on M is naturally determined bythe limit charts

(lim←−

U iα, lim←−φiα). They obviously cover M . Their (smooth)

compatibility is ensured by the compatibility of the corresponding factorsand the fact that projective limit of differentiable maps remain also differ-entiable. More precisely, if

(Uα = lim←−U

iα, φα = lim←−φ

iα

)and

(Uβ = lim←−U

iβ, φβ = lim←−φ

iβ

)

are two limit charts with Uα ∩Uβ 6= ∅, then the transition functions are thediffeomorphisms

φβ φ−1α = lim←−

(φiβ (φiα)−1

): φα(Uα ∩ Uβ) −→ φβ(Uα ∩ Uβ).

Remarks 3.1.4. 1) We recall that the smooth structure of Proposition 3.1.3is based on the differentiability of J. A. Leslie ([Les67], [Les68]) discussed in§,2.2. Other methods of differentiability (for instance those of M. C. Abbati-A. Mania [AM99], A. Kriegl-P. Michor ([KM97]), M.E. Verona [Vero74],[Vero79]) can also be applied, because the projective limits are compatiblewith the algebraic and topological structures involved in the definitions.However, the respective manifold structures are not in general equivalent tothat of the aforementioned proposition.

2) Let us elaborate on the requirement concerning the existence of limitcharts covering the projective limit M = lim←−M

i: The assumption that

M i;µjii,j∈N is a projective system of smooth manifolds already means,by definition, that the connecting morphisms µji : M j → M i (j ≥ i) arecompatible with the smooth structures of the manifolds M i (i ∈ N); thatis, they are smooth in the ordinary sense. Therefore, for each j ≥ i, thereexist charts (U i, φi) and (U j , φj) of M i and M j, respectively, such thatµji(U j) ⊆ U i and the local representation

(3.1.1) φi µji (φj)−1 : Ej ⊇ φj(U j) −→ φi(U i) ⊆ Ei

is smooth. This fact does not necessarily lead to the construction of corre-sponding smooth charts on M , a gap that led a number of authors (see, e.g.,


[AM99], [Vero74], [Vero79]) to adopt rather algebraic approaches. However,our definition of limit charts ensures that (3.1.1) not only is a smooth mapbut also coincides with the connecting morphisms of the models, i.e.

(3.1.2) φi µji (φj)−1 = ρji j ≥ i.

In other words, the charts under consideration connect the projective sys-tems of the manifolds and the corresponding models involved.

As an application of the preceding constructions, we prove the followingimmediate result.

Proposition 3.1.5. If M = lim←−M i;µjii,j∈N is a plb-manifold, then the

canonical projections µi : M →M i (i ∈ N) are smooth.

Proof. Let (U = lim←−

xi, φ = lim←−

φi) be any chart at an arbitrary point in

x ∈ M . The local representation of µji with respect to the charts (U, φ) ofM and (xi, φi) of M i, coincides with the corresponding canonical projectionon the model spaces; that is,

φi µi φ−1 = ρi,

thus proving the smoothness of µi at x.

The previous proposition, in conjunction with Remark 3.1.4(2), ensuresthe categorical consistency of the derived structures.

Examples 3.1.6.

1. Every Banach manifold M can be trivially considered as a plb-manifoldcoinciding with the limit of the single element projective system M, idM.

2. Every Frechet space F, being always a projective limit of Banach spaceslim←−Ei, fulfils also the assumptions of the Definition 3.1.2 with respect to the

total chart (F, idF) =(lim←−

Ei, lim←−

idEi

).

3. The group C0(R,R+) of all continuous positive real-valued curves is aplb-manifold modelled on the Frechet space C0(R,R) via the isomorphismg = lim←− g

n, where

gn : C0(R,R+) −→ C0 ([−n.n],R+) : f 7→ f∣∣[−n,n]

; n ∈ N,

and corresponding limit charts(C0([−n, n],R+), φn

)with φn(f) = log f .

3.1. Smooth structures on Frechet manifolds 111

4. The group of all smooth maps C∞(M,G) from a compact manifold Mto a finite dimensional Lie group G is a Frechet plb-manifold since

C∞(M,G) ∼=⋂

n∈N

Cn(M,G)

with corresponding charts(Cn(M,V ), φn

)n∈N

, where V is an open subsetof G over which the exponential map expG of G is a diffeomorphism and

φn(f) = exp−1G f.

5. H. Omori in [Omo70] introduced the notion of inverse limit manifoldsas the intersection of a countable number of Banach manifolds forming anested sequence

M1 ⊇M2 ⊇ · · · ⊇Mn ⊇Mn+1 ⊇ · · ·

on which projective limits of charts can be defined. The space M = ∩n∈NMn

cannot be always endowed with the structure of a smooth manifold, since thedomains of the charts employed may collapse to single point sets. However,if the intersection of the domains are open sets, then Omori’s manifolds arespecial cases of plb-manifolds with connecting morphisms being the naturalembeddings.

6. The space of infinite jets J∞(E) of a Banach vector bundle (E, π,B) isa plb manifold modelled on the Frechet space

E× L(B,E)× L2s(B,E)× L3s(B,E)× · · · ,

where Lks(B,E) is the space of continuous symmetric k-linear maps betweenthe model spaces B and E of the base B and the total space E, respectively.In particular, J∞(E) is isomorphic with the projective limit of the finitedimensional jets:

J∞(E) ∼= lim←−

Jn(E).

The charts of J∞(E) are the pairs (J∞(U), φU ), where

φU : J∞(U) −→ U × L(B,E)× L2s(B,E)× · · ·

is given byφU (j∞x ξ) =

(x, ξ(x), dξ(x), d2ξ(x), . . .

),

if j∞x ξ denotes the infinite jet (: equivalent class) of a section ξ of E, and Uis an open subset of B. Further details will be given in § 6.3.

7. M .E. Verona in [Vero74] and [Vero79] studied projective limits of mani-folds adopting, however, a rather topological approach.


Smooth maps between plb-manifolds are defined in the usual way. How-ever, there is a particular category of maps which deserves special attention.

Definition 3.1.7. Let M i;µjii,j∈N and N i; νjii,j∈N be two plb-mani-folds and let f i : M i → N ii∈N be a projective system of smooth maps.Then, the limit

lim←−

f i : (xi)i∈N 7−→ (f i(xi))i∈N

is called a projective limit of smooth maps (pls-map, in short).

The two commutative diagrams of Definition 2.3.4, relating a projectivesystem of maps and its limit with the connecting morphisms and the canon-ical projections of the projective systems of the domains and ranges of themaps, translated in the present situation read as follows:

νji f j = f i µji,(3.1.3)

νi f = f i µi,(3.1.4)

for all i, j ∈ N and j ≥ i.

In the approaches of [AM99], [Pap80], [Vero74], [Vero79], projective lim-its of smooth maps between projective limits of smooth manifolds are taken,by definition, as the smooth maps between projective limits of smooth man-ifolds. This is a rather algebraic way of defining differentiability. In ourframework, the differentiation defined in Section § 2.2) and the definition ofplb-manifolds allow one to prove that pls-maps are rendered smooth in thestandard way, i.e. they have differentiable local representations with respectto limit charts. We clarify this matter in the next proposition.

Proposition 3.1.8. Every pls-map

f = lim←−

f i : M = lim←−

M i −→ N = lim←−

N i

is a smooth map between plb-manifolds.

Proof. Let x = (xi)i∈N be an arbitrary point in M . We consider the limitcharts (U, φ) =

(lim←−

xi, lim←−

φi)

and(V, ψ) = (lim

←−V i, lim←−

ψi)

of M and Ncontaining x and f(x), respectively. Without loss of generality, we mayassume that f(U) ⊆ V , otherwise we restrict ourselves to an open subset ofU ∩ f−1(V ). Now, the local representation ψ f φ−1 : φ(U)→ ψ(V ) of f ,with respect to the chosen charts, is given by

ψ f φ−1 = lim←−

ψi lim←−

f i lim←−

(φi)−1 = lim←−

(ψi f i φi)−1.

Therefore, f is smooth at x if and only if its local representation is smoothat φ(x); equivalently (in virtue of Proposition 2.3.11) if and only if eachψi fi φ

−1i : φi(Ui)→ ψi(Vi) is smooth at xi, for all i ∈ N.

3.2. The tangent bundle of a plb-manifold 113

Remarks 3.1.9. 1) It is worth noting here that the previous result, al-though ensuring the smoothness of pls-maps, does not restrict the set ofsmooth maps between plb-manifolds only to pls-maps, as is the case of[AM99]), [Pap80], [Vero74] and [Vero79] mentioned earlier [see also Re-mark 3.1.4(1)]. Therefore, smooth maps that are not necessarily projectivelimits are also included in our framework.

2) In the special case of smooth curves C∞(R,M), where M is a plb-manifold, the projective limit approach proves to be very convenient sinceevery curve α : R→ M can be naturally realized as the inverse limit of thecorresponding projections to the factors, i.e.

α = lim←−

(µi α),

where µi : M → M i (i ∈ N) are the canonical projections of the projectivelimit to the factors. We conclude that:

A curve on a plb-manifold is smooth if and only if it is a projective limitof smooth curves. Consequently,

(3.1.5) C∞(R,M) ≡ lim←−C∞(R,M i)

within a natural isomorphism.

3) The preceding identification of smooth curves might be used to find away connecting the differentiability adopted here with that of [KM97]. Thisis an open question.

3.2 The tangent bundle of a plb-manifold

Using the tools of the previous section, we proceed to the study of thetangent spaces and bundles of projective limit manifolds. The identification(3.1.4) allows one to follow the classical pattern.

Definition 3.2.1. Let M = M i;µjii,j∈N be a plb manifold and x =(xi)i∈N ∈M . Then, the tangent space of M at x is the quotient space

TxM = C∞x (R,M)/ ∼x

with respect to the equivalence relation

(3.2.1) α ∼x β ⇔ (φ α)′(0) = (φ β)′(0)

between smooth curves in M with α(0) = β(0) = x, and for any chart (U, φ)of M with x ∈ U .


We note that (see also Proposition 2.3.11)

(3.2.2)

(φ α)′(t) = (D(φ α)(0)) (1)

=(D(lim←−

φi α)(0))

(1)

=(D(lim←−

φi lim←−

αi))(0))

(1)

=(D(lim←−(φi µi α))(0)

)(1)

= lim←−

(D(φi µi α)(0)

)(1)

As usual, (3.2.1) does not depend on the choice of the chart at x. Hence,it is sufficient to check this condition for a limit chart.

The equivalence class of a curve α passing through x is denoted by[α, x], instead of the customary [(α, x)]. This is a minor deviation aimingto simplify expressions involving successive parentheses, as will be often thecase below.

Obviously,

TM =⋃

x∈M

TxM

is the tangent bundle of M . Its projection is the map

π : TM −→M : [α, x] 7→ x.

We shall show that the tangent bundle of a plb-manifold remains inthe same category of manifolds, thus we obtain yet another example of aplb-manifold. First we prove the following:

Proposition 3.2.2. Let M = lim←−

M i be a plb-manifold. Then the tangent

space TxM at an arbitrary point x = (xi)i∈N ∈ M is in bijective correspon-dence with lim

←−TxiM

i. Therefore, we obtain the identification

(3.2.3) TxM ≡ lim←−TxiMi

Proof. The differentials Txjµji : TxjM

j −→ TxiMi (j ≥ i) of the connect-

ing morphisms of M determine the connecting morphisms of the projectivesystem TxjM

j, Txjµji, since

Txkµkj Txjµ

ji = Txk(µkj µji) = Txkµki, k ≥ j ≥ i.

Then we relate lim←−

TxiMi with the tangent space TxM by the map

(3.2.4) Rx : TxM −→ lim←−TxiMi : [α, x] 7→

([µi α, µi(x)]i

)i∈N

,


where the bracket [ ]i stands for the equivalence class of curves in M i.Rx is an injection: Assume that Rx([α, x]) = Rx([β, x]). If (U, φ) =

(lim←−

xi, lim←−

φi) is any limit chart of M containing x, then, in virtue of (3.2.2),we see that

(φ α)′(0) = (φ β)′(0)

⇔ lim←−

(D(φi µi α)(0)

)(1) = lim

←−

(D(φi µi β)(0)

)(1)

⇔(D(φi µi α)(0)

)(1) =

(D(φi µi β)(0)

)(1)

⇔ [µi α, xi]i = [µi β, xi]i

⇔ [µi α, µi(x)]i = [µi β, µi(x)]i

⇔ Rx([α, x]) = Rx([β, x])

from which we obtain the injectivity of Rx.Rx is a surjection: Take any element

([αi, xi]i

)i∈N

in lim←−TxiMi. Using

again the limit chart (U,ϕ) =(lim←−x

i, lim←−ϕi), we define the elements of

F = lim←−Ei

u = (φi(xi))i∈N and v =((φi αi)′(0)

)i∈N

,

(recall that Ei is the model of M i, for each i ∈ N), and the smooth curves

h : R −→ F : t 7→ u+ t · v,

α : J −→M : t 7→ φ−1(h(t)),

where J is an open interval such that α(J) ⊂ U . We shall show thatRx([α, x]) := ([µi α, µi(x)]i)i∈N = ([αi, xi]i)i∈N.

To this end we recall that, by the definition of the projective limit ofmaps, the following diagram is commutative :

M ⊇ lim←−

U iφ- lim←−

φi(U i) ⊆ F

M ⊇ U i

µi

?

φi- φi(U i) ⊆ Fi

ρi

?

Therefore,

(µi α)(0) = (µi φ−1)(h(0))

= (φi)−1(ρi((φi(xi))i∈N

))

= (φi)−1(φi(xi)) = xi

= µi(x) = (µi α)(0).


Similarly, since ρi is continuous linear for all i ∈ N,

(φi µi α)′(0) = (ρi φ α)′(0) = (ρi h)′(0)

=(D(ρi h)(0)

)(1) =

(Dρi(h(0)) Dh(0)

)(1)

= ρi(h′(0)) = ρi((

(φi αi)′(0))i∈N

)

= (φi αi)′(0).

As a result, the curves µi α and αi are equivalent at xi ∈M i (i ∈ N), thusproving the desired surjectivity of Rx.

Corollary 3.2.3. By means of Rx, the structure of a Frechet space onlim←−TxiM

i is transferred to TxM . Therefore, TxM is isomorphic to theFrechet model F, and (3.2.3) is completed as follows:

(3.2.5) TxM ≡ lim←−

TxiMi ∼= lim←−

Ei = F,

for every x = (xi)i∈N

Of course, the isomorphism TxM ∼= F can be established as in the caseof finite-dimensional or Banach manifolds. However, (3.2.5) shows that thetangent space of a plb-manifold is a projective limit itself and provides acomplete picture of the way the tangent spaces of a plb-manifold are relatedwith the tangent spaces of the factors and their models.

The differential of a smooth map between plb-manifolds can be definedas in the Banach case.

Definition 3.2.4. Let M = lim←−Mi;µjii,j∈N and N = lim←−N

i; νjii,j∈N betwo plb-manifolds, and let f : M → N be a differentiable map at the pointx = (xi)i∈N ∈M . Then the differential of f at x is the map

Txf : TxM −→ Tf(x)N : [α, x] 7→ [f α, f(x)].

Accordingly, the (total) differential of f is the map Tf : TM → TN ,given by Tf |TxM = Txf.

Working as in the Banach framework, one easily verifies that Txf is alinear map. In particular, if f is a pls-map, then its differential can berealized as the projective limit of the differentials of the factor maps. As amatter of fact, we prove the following:


Proposition 3.2.5. Let f := lim←−

f i : lim←−

M i → lim←−

N i be a pls-map as in

Proposition 3.1.5. Then the differential of f at any point x = (xi)i∈N ∈ Mcoincides—up to isomorphism—with the projective limit of the differentialsof the f i’s, i.e.

(3.2.6) Txf ≡ lim←−

Txifi.

Proof. Differentiating (3.1.2) at every xi ∈ TxjMj, we obtain

(3.2.7) Tfj(xj)νji Txjf

j = Tµji(xj)fi Txjµ

ji.

Taking into account that the differentials of the connecting morphisms µjiand νji are themselves connecting morphisms for the projective systemsof the tangent spaces (see the beginning of the proof of Proposition 3.2.2),we immediately see that lim

←−Txif

i exists.

For the proof of (3.2.6) it suffices to verify that the diagram

TxMTxf

- Tf(x)N

lim←−TxiMi

RMx

?

lim←−Txifi- lim←−Tf i(xi)N

i

RNf(x)

?

is commutative, where the vertical maps are the corresponding isomorphisms(3.2.4). Indeed, for an arbitrary [α, x] ∈ TxM , we have that

(RNf(x) Txf

)([α, x]) =

([νi f α, νi(f(x))]

)i∈N

,

whereas

(lim←−

Txifi RMx

)([α, x]) =

([f i µi α, f i(µi(x))]

)i∈N

.

The right-hand sides of the above equalities coincide in virtue of (3.1.4);hence, the diagram is commutative, as claimed.

Corollary 3.2.6. The canonical projections qi : lim←−

TxiMi → TxiM

i satisfy

Txµi = qi Rx, i ∈ N, where Txµ

i : TxM → TxiMi are the differentials of

the canonical projections of M = lim←−

M i. Therefore, in virtue of (3.2.4),

Txµi = qi up to isomorphism.


Proof. For every [α, x] ∈ TxM ,

(qi Rx

)([α, x]) = qi

(([µi α, µi(x)]i

)i∈N

)

= [µi α, µi(x)]i = Txµi([α, x]).

Schematically, the diagram

TxMRx

- lim←−

TxiMi

TxiMi

qi

?

Txµi

-

is commutative. This completes the proof, since Rx is a (linear) isomor-phism.

Proposition 3.2.7. If M = lim←−Mi is a plb-manifold, then so is lim←−TM

i.

Proof. Since M is a plb-manifold there exist local charts(

lim←−Ui, lim←−φ

i)

covering M . Each chart (U i, φi) of M i induces the corresponding chart(π−1i (U i),Φi

)of TM i, where πi : TM

i →M i is the projection and

Φi : π−1i (U i) −→ φi(U i)× Ei

is given by Φi(xi) =(xi, φi(xi)

), xi ∈ TxiM

i. Recall that φi : TxiMi → Ei

is the linear isomorphism with φi(xi) := (φi α)′(0) if xi = [α, xi].

It is an easy exercise to verify that:

i)TM j;Tµji : TM j → TM i

i,j∈N

is a projective system, thus the limit

lim←−TMi exists.

ii) The pairs(π−1i (U i),Φi

), for all i ∈ N, form a projective system of

charts as in Definition 3.1.1.

We shall show that lim←−

π−1i (U i) and lim←−

Φi(π−1i (U i)

)are open subsets of

TM and F× F, respectively. Indeed, by

lim←−π−1i (U i) = lim←−π

−1i (lim←−U

i) =(lim←−πi

)−1(lim←−U

i)

we see that lim←−π−1i (U i) ⊂ TM is open, because lim←−U

i is open in M and

lim←−πi : lim←−TMi → lim←−M

i = M is a continuous map.


On the other hand,

lim←−

Φi(π−1i (U i)

)= lim←−

(φi(U i)× Ei

)=

= lim←−φi(U i)× lim←−Ei = lim←−φ

i(U i)× F.

Since φi(U i) ⊂ Ei is open, then lim←−φi(U i) is open in F, consequently the set

lim←−Φi(π−1i (U i)

)is an open subset of F× F.

The previous assertions show that the pairs(π−1i (U i),Φi

), for all i ∈

N, determine a projective limit chart (see Definition 3.1.1). Moreover, all(lim←−π

−1i (U i), lim←−Φi

)cover lim←−TM

i, thus, in virtue of Definition 3.1.2, thelatter becomes a plb-manifold (see also Proposition 3.1.3).

The tangent bundle TM of a plb-manifold M = lim←−M i;µjii,j∈N is en-

dowed with a differential structure in the standard way of finite-dimensionalor Banach manifolds: As in the previous proof, for every chart (U, φ) ofM (a limit chart or not), we define the chart (π−1(U),Φ) of TM , whereπ : TM →M is the projection of the bundle and

Φ: π−1(U) −→ φ(U)× F : [α, x] 7→(φ(x), (φ α)′(0)

).

By routine checking, we see that the previous charts determine the structureof a Frechet manifold on TM . However, the realization of the tangent spacesof TM as projective limits of the corresponding tangent spaces of the factors(see Proposition 3.2.2) allows us to show that TM is diffeomorphic withthe projective limit of the corresponding tangent bundles of the factors.Therefore, as alluded to before Proposition 3.2.2, the tangent bundle of aplb-manifold remains—up to a diffeomorphism—within the category of plb-manifolds.

Theorem 3.2.8. Within a diffeomorphism, the tangent bundle TM of aplb-manifold M = lim

←−M i coincides with lim

←−TM i.

Proof. We consider the map R : TM → lim←−TMi, given by [see (3.2.4)]

R∣∣TxM

= Rx : TxM −→ lim←−

TxiMi.

The map R is well-defined because

lim←−

TM i =.⋃

x∈M

lim←−

TxiMi ≡

⊔

x∈M

lim←−

TxiMi (disjoint union).

This is a consequence of the fact that the projective limits, involved in bothsides of the equality, have the same connecting morphisms.


Obviously, R is a bijection by the analogous properties of the Rx’s, es-tablished in the proof of Proposition 3.2.2.

To show that R is a diffeomorphism, it suffices to examine its local be-haviour. To this aim let u ∈ TxM be an arbitrary tangent vector. For a limitchart

(U = lim←−x

i, φ = lim←−φi)

of M at x, we construct the chart (π−1(U),Φ)of TM at u (see the comments preceding the present statement), and thechart

(lim←−π

−1i (xi), lim←−Φi

)of lim←−TM

i (see the proof of Proposition 3.2.7).

We see immediately that R((π−1(U)

)= lim←−π

−1i (xi), thus the local rep-

resentation, say, Rφ of R, with respect to the last two charts, is given byRφ = lim←−Φi R Φ−1 as in the following diagram:

π−1(U)R

- lim←−π−1i (xi)

φ(U)× F

Φ

? Rφ- lim←−

φi(xi)× F

lim←−

Φi

?

But now φ(U) = lim←−φi(xi), and

(lim←−

Φi R)([α, x]) = lim←−

Φi(Rx([α, x]))

=(Φi([µi α, µi(x)]i)

)i∈N

=(φi(xi), (φi µi α)′(0)

)i∈N

=(lim←−

φi)(x), (lim←−

(φi α)′)(0))

= (φ(x), (φ α)′(0)) = Φ([α, x]);

in other words, Rφ = idφ(U)×F. This shows that R is a diffeomorphism at uand concludes the proof.

Corollary 3.2.9. Let f = lim←−

f i be a pls-map between the plb-manifolds

M = lim←−

M i and N = lim←−

N i. Then the total differential Tf : TM → TNis also a pls-map such that

(3.2.8) Tf = lim←−Tfi

within a diffeomorphism.

Proof. The total differentiation of (3.1.3) yields [see also (3.2.7)]

Tνji Tf j = Tf i Tµji;

3.3. Vector fields 121

hence, in virtue of Definition 2.3.4, lim←−

Tf i exists. by

Now, denoting by RM : TM → lim←−TMi and RN : TN → lim←−TN

i theisomorphism of Theorem 3.2.8, for M and N respectively, and taking intoaccount the commutative diagram in the proof of Proposition 2.3.6, we havethat

RN Tf∣∣TxM

= RNf(x) Txf = lim←−

Txifi RMx = lim

←−Tf i RM

∣∣TxM

,

for every x ∈M . Therefore,

(3.2.9) RN Tf = lim←−Tfi RM ,

whence the result.

For the sake of completeness, the diffeomorphisms of Theorem 3.2.8 andCorollary 3.2.9 can be written in the respective forms

T(lim←−

M i)∼= lim←−

TM i,

T(lim←− f

)∼= lim←−Tf

i.

Consequently, up to isomorphism, lim←−

commutes with the tangent functorT in the category of plb-manifolds.

3.3 Vector fields

Having established the structure of a Frechet manifold on plb-manifolds andtheir tangent bundles, we can now define vector fields. We are interested inprojective limits of vector fields, in consistency with our framework.

Definition 3.3.1. Let M = M i;µji be a plb-manifold. A projectivesystem of vector fields is a (countable) family of smooth vector fields onthe factors

ξi : M i −→ TM i | i ∈ N

satisfying the natural condition

(3.3.1) Tµji ξj = ξi µji; j ≥ i,

which means that ξj and ξi are µji-related vector fields.

The previous definition, along with Proposition 3.1.8, implies that

lim←−

ξi : M = lim←−

M i −→ lim←−

TM i

exists and is a pls-map, thus it is smooth. Moreover,

(3.3.2)(lim←−πi

)(lim←− ξi

)= lim←− (πi ξ) = lim←− idM = idM .


Proposition 3.3.2. With the previous notations, lim←−

ξi identifies with a

smooth vector field of M = lim←−

M i.

Proof. Let R = RM : TM → lim←−TMi be the diffeomorphism of Theo-

rem 3.2.8. We set ξ := R−1 lim←− ξi. Clearly, ξ is a smooth map. By

routine computations, we see that

(3.3.3) π =(lim←−πi

)R.

Therefore, (3.3.3) and (3.3.2) imply that π ξ = idM ; hence ξ is a smoothvector field on M .

Henceforth, in virtue of the preceding proposition, projective limit ofvector fields ξ = lim

←−ξi will be thought of as a smooth vector field of M .

We now proceed to the study of the integral curves of a vector fieldξ = lim←− ξ

i as before. As in the classical case [see § 1.1.11 and (1.1.15)], weare led to solving a local equation of the form

(3.3.4) β′(t) = ξφ(β(t)), t ∈ Jβ

where now ξφ : φ(U) → F, with F a Frechet space. Therefore, we are con-fronted with the problems already discussed in §,2.4. The existence anduniqueness of the solutions depend on appropriate conditions.

The main result here is the following:

Theorem 3.3.3. Let M = lim←−Mi;µjii,j∈N be a plb-manifold modelled on

a Frechet space F = lim←−Ei. Assume that every M i is a Hausdorff space, and

ξ = lim←−

ξi is a vector field on M . If there exists a limit chart(lim←−

xi, lim←−

ϕi)

at a given point x0 =(xi0)∈ M , such that condition (* ) of Theorem 2.4.3

is satisfied, then there is a unique integral curve α of ξ with initial conditionα(0) = x0.

Proof. Equation (3.3.4) is equivalent to the system of countable equations

(3.3.5)(βi)′

(t) = ξiφi(βi(t)); i ∈ N,

with βi(0) = φi(xi0), where βi = µi β. By the assumptions and Theo-rem 2.4.3, there exists a unique solution (βi), with all the curves βi defined

on the same interval, say J . Clearly, each αi =(φi)−1 βi is an integral

curve of ξi with initial condition xi0. Moreover, following the argument ofthe same Theorem 2.4.3, we obtain the limit β = lim←−β

i which is a solution

of (3.3.4) with β(0) = φ(x0); hence, α = φ−1 β is an integrable curve of

3.4. Frechet-Lie groups 123

ξ with α(0) = x0. The uniqueness of α is obvious: If γ is any other inte-gral curve of ξ with γ(0) = x0, then the first part of the proof implies thatαi = γi; hence, α = γ. The proof is now complete.

Remarks 3.3.4. 1) Regarding the above statement, it should be notedthat the assumption that condition (* ) of Theorem 2.4.3 is satisfied is notan additional requirement, imposed by the differential structure of the man-ifolds under consideration. It is a necessary condition in order to resolve theinvolved equations in the context of Frechet spaces.

2) The same condition guarantees the existence of solutions of (3.3.4)whose domains do not collapse to a single point. Indeed, without it, thefactor solutions could have different domains, thus the projective limit ofthe latter could be a single point.

3.4 Frechet-Lie groups

Here we extend the mechanism of plb-manifolds to the case of Lie groupsmodelled on Frechet spaces. Plb-groups defined below admit always anexponential map. Note that projective limits of Lie groups have been usedin the study of the group of diffeomorphisms of a compact manifold (see[Les67], [Omo70]).

Let Gi; gjii,j∈N be a projective system of groups. By definition, theconnecting morphisms of the system are group homomorphisms. The limitG = lim

←−Gi is also a group whose multiplication and inversion are given,

respectively, by

(xi) · (yi) := (xi · yi),((xi)

)−1:=((xi)−1

),

for every x = (xi) and y = (yi) in G, with i ∈ N. We denote by

gi : G = lim←−Gi −→ Gi

the canonical projection of G to Gi, which are group (homo)morphisms.If Lx : G → G is the left translation of G by x = (xi) ∈ G and, analo-

gously, Ljy : Gj → Gj that of Gj by y ∈ Gi, it is immediate that

Lx = lim←−

Lixi(3.4.1)

gi Lx = Lixi gi,(3.4.2)

gji Ljy = Ligji(y) gji.(3.4.3)


In the remainder of this chapter we are dealing with projective systemsof Banach-Lie groups; that is, projective systems of groups satisfying theconditions of Definition 3.1.2. The connecting morphisms gji : Gj → Gi aremorphisms of Banach-Lie groups.

Proposition 3.4.1. Let Gi; gjii,j∈N be a projective system of Banach-Lie groups, where Gi is modelled on the Banach space Gi (i ∈ N). Thenthe plb-manifold G = lim←−G

i is a Lie group modelled on the Frechet space

G = lim←−Gi.

Proof. By Proposition 3.1.3, G is a Frechet manifold. On the other hand, thecomment at the beginning of the present section implies that the operationsof multiplication and inversion of G coincide with the projective limits oftheir counterparts on the factor Banach-Lie groups. As a result, in virtueof Proposition 3.1.8, the former operations are smooth maps, and G is a Liegroup.

The group G lb-group is called a plb-group. Note that the connectingmorphisms gji : Gj → Gi (j ≥ i) are, by definition, Lie group morphisms,and so are the canonical projections gi : G → Gi, in virtue of Proposi-tion 3.1.5.

Examples 3.4.2.

1. Every Banach-Lie group G is trivially a Frechet-Lie group by settingG = lim

←−Gi, with Gi = G for all i ∈ N.

2. Let F = lim←−

Ei be a Frechet space represented by a projective limit of

a countable family of Banach spaces (see Theorem 2.3.8). If Gi := (Ei,+),then F = lim

←−Gi is an (abelian) Frechet-Lie group.

3. The groups C0(R,R+) and C∞(M,G) (where M is a compact manifoldand G a finite dimensional Lie group), defined in Examples 3.1.6, are alsoFrechet-Lie groups.

Proposition 3.4.3. Let G = lim←−

Gi be a Frechet-Lie group as in the previ-

ous statement. If ξ = lim←−

ξi is a smooth vector field on G, then:

i) ξj and ξi are gji-related, for every j ≥ i.ii) ξ and ξi are gi-related.

Moreover, a vector field on G is left invariant if and only if it is a projectivelimit of invariant vector fields on the factors, i.e.

ξ ∈ L(G) ⇔ ξ = lim←− ξi : ξi ∈ L(Gi) ∀ i ∈ N.


Proof. Property i) is a result of the analog of (3.3.1), whereas ii) is a con-sequence of the second diagram of Definition 2.3.4 adapted to the presentcase.

Take now any ξ ∈ L(G). Then ξ corresponds to a unique v ∈ TeG ≡lim←−TeiG

i such that ξe = v ≡(vi). Clearly, e ≡ (ei) is the identity element

of G and ei the identity of Gi. Also, in a reverse way, each vi determines aunique ξi ∈ L(Gi), with ξi

ei= vi, for every i ∈ N. We shall prove that the

collection(ξi)

is a projective system of vector fields. To this end it sufficesto prove the analog of (3.3.1), namely

(3.4.4) Tgji ξj = ξi gji, j ≥ i.

Before proving this, we recall that Tejgji are the connecting morphisms

and Tegi the canonical projections of lim

←−TeiG

i, thus Tejgji(vj) = vi and

Tegi(v) = vi (i, j ∈ N; j ≥ i). Therefore, for every x ∈ Gj , (3.4.3) implies

that

Txgji(ξi(x)) = Txg

ji(TejL

jx(ξj

ej))

= Tej (gji Ljx)(vj)

= Tej(Ligji(x) g

ji)(vj) = Tgji(ej)L

igji(x)

(Tejg

ji(vj))

= TeiLigji(x)(v

i) = TeiLigji(x)(ξ

iei) = ξ(gji(x)),

which proves (3.4.4). Consequently, lim←− ξi exists.

To prove that ξ = lim←−

ξi, we need to show that

(3.4.5) Tgi ξ = ξi gi,

according to Proposition 2.3.5. Indeed, for every x ∈ G, working as in theproof of (3.4.4) and applying (3.4.2), we find:

Txgi(ξ(x)) = Txg

i(TeLx(ξe)

)= Te(g

i Lx)(ξe)

= Te(Lixi g

i)(v) = Tgi(e)L

ixi

(Teg

i(v))

= Tgi(e)Lixi(v

i) = Tgi(e)Lixi(ξ

iei)

= ξi(xi) = ξi(gi(x)),

thus we obtain (3.4.5). In conclusion, we have proved that every ξ ∈ L(G)is the projective limit of ξi ∈ L(Gi)i∈N.

Conversely, if ξ is a vector field of G such that ξ = lim←− ξi, where ξi ∈

L(Gi), for every i ∈ N, then necessarily ξ ∈ L(G). This is so, for if x ∈ G,


then

TeLx(ξ(e)) = TeLx(lim←− ξ

i(e))

= TeLx

((ξi(ei)

)i∈N

)

= lim←−

TeiLixi

((ξi(ei)

)i∈N

)=(TeiL

ixi(ξi(ei)

))i∈N

=(ξi(xi)

)i∈N

=(lim←−

ξi)

(e) = ξ(x),

which completes the proof.

Theorem 3.4.4. For G = lim←−Gi as before, its Lie algebra L(G) can be also

realized as a projective limit, i.e.

L(G) ≡ lim←−L(Gi),

by an isomorphism of Lie algebras, where L(Gi) is the Lie algebra of Gi

(i ∈ N). Hence, L(G) is a Frechet-Lie algebra.

Proof. From the identification (1.2.3), equality (3.2.3) and the proof ofProposition 3.4.3, we obtain the linear isomorphisms

(3.4.6) L(G) ≡ TeG ≡ lim←−

TeiGi ≡ lim←−L(Gi),

thus L(G) is a Frechet space.

It remains to show that L(G) and lim←−L(Gi) are isomorphic Lie algebras.

To this end we first show that g := TeG and lim←− gi := lim←−TeiGi are isomor-

phic Lie algebras. Hence we need to define an appropriate bracket on lim←− gi.

As one may guess, the latter will be the projective limit of the brackets [ , ]i

of the factor algebras gi. To ensure the existence of such a limit of brackets,it suffices to prove that the following the diagram is commutative:

gj × gj[ , ]j

- gj

gi × gi

Tejgji × Tejg

ji

?

[ , ]i- gi

Tejgji

?

Indeed, assume that uj, vj are any elements of gj, and ξj , ηj the corre-sponding (left invariant) vector fields in L(Gj). We denote by ξi, ηi ∈ L(Gi)the respective fields of the Tejg

ji(uj) and Tejgji(vj) in gi. As in the proof of


Proposition 3.4.3, we obtain equality (3.4.4), thus ξj and ξi are gji-related,and similarly for ηj , ηi. Therefore,

[Tejg

ji(uj), Tejgji(vj)

]i=[ξi, ηi

]i (ei)

=([ξi, ηi

]i gji

)(ej) =

=(Tgji

[ξj, ηj

]j)(ej) = Tejg

ji([uj , vj

]j).

This proves the desired commutativity and, consequently, the existence oflim←−

[ , ]j .

The next step is to show that Tegi : g→ gi is a morphism of Lie algebras

with respect to the corresponding brackets [ ] and [ ]i: For arbitrary vectorsu, v ∈ g, we consider the corresponding fields ξ, η ∈ L(G). Also, we denoteby ξi, ηi ∈ L(Gi) the fields corresponding to Teg

i(u), Tegi(u) ∈ gi. Then,

working as in the proof of the commutativity of the previous diagram, wesee that [ξ, η] and [ξi, ηi]i are gi-related. Therefore,

[Teg

i(u), Tegi(u)

]=[ξi, ηi

]i(e) =

([ξi, ηi

]i gi)

(e) =

= Teg([ξi, ηi

](e))

= Teg([ξie, η

ie

])= Teg([u, v]);

that is, Tegi is a morphism of Lie algebras. Taking now the limits, we have

thatlim←−

Tegi : TeG −→ lim

←−TeiG

i

is also a morphism of Lie algebras, which gives an isomorphism after theidentification TeG ≡ lim

←−TeiG

i. As a matter of fact,

[ , ] ≡ lim←− [ , ]i.

Finally, (3.4.6) extends the previous Lie algebra isomorphism to one betweenL(G) and lim←−L(Gi).

We come now to the question of the existence of an exponential map.Naturally, this is related to the solution of appropriate differential equationswhich determine the integral curves of the left invariant vector fields ofthe Lie groups at hand. As we have already mentioned in §,2.4, we lacka general solvability theory of differential equations in Frechet or, moregenerally, non-Banach locally convex spaces. This deficiency has led manyauthors to propose various approaches to define a kind of exponential map(in this respect see the instructive notes by K. H. Neeb [Nee06]). Here,following the main methodology of this work, we prove the existence of anexponential map in the classical sense.


Theorem 3.4.5. Let G = lim←−

Gi be a Frechet-Lie group as in Proposition3.4.1. Then G admits an exponential map expG : TeG→ G satisfying

expG ≡ lim←−

expGi ,

if expGi (i ∈ N) are the exponential maps of the factor groups.

Before the proof we need the following auxiliary result:

Lemma 3.4.6. Let G = lim←−Gi be a Frechet-Lie group and ξ ∈ L(G), thus

(by Proposition 3.4.3) ξ = lim←− ξi, with ξi ∈ L(Gi). Let also an arbitrary

point x0 ≡(xi0)∈ G. If, for every i ∈ N, αi : R→ Gi is the integral curve of

ξi, with initial condition αi(0) = xi0, then α := lim←−

αi exists and α : R → Gis the integral curve of ξ, with initial condition α(0) = x0.

Proof. For every j ≥ i,

(gji αj)(0) = gji(xj0) = xi0.

On the other hand, the velocity vector of gji αj : R → Gi at any t ∈ R

yields [see also equality (1.1.11) and the ensuing comments]:

(gji αj

).(t) = Te(g

ji αj)( ddt

∣∣∣t

)

= Tegji((αj).(t)

)= Teg

ji(ξj(αj(t)

)

or, in virtue of (3.4.4),

=(Teg

ji ξj)

(αj(t)) =(ξi gji

)(αj(t))

= ξi((gji αj)(t)

).

Therefore, gji αj is an integral curve of ξi with initial condition xi0; hence,gji αj = αi, which means that

(αi)i∈N

is a projective system. We set

α := lim←−

αi.

We claim that α is the integral curve of ξ with initial condition α(0) = x0.Indeed,

α(0) =(lim←−α

i)

(0) =(αi(0)

)i∈N

=(xi0)i∈N

= x0.

Since both α(t) and ξ(α(t)) are elements of Tα(t)G ≡ lim←−Tαi(t)Gi, equality

α(t) = ξ(α(t)) holds if and only if the previous vectors have the same pro-jections (components). Here, by Corollary 3.2.6, the canonical projections


are Tα(t)gi : Tα(t)G→ Tαi(t)G

i. Therefore,

α(t) = ξ(α(t))

⇔ Tα(t)gi(α(t)) = Tα(t)g

i(ξ(α(t)))

⇔ Tα(t)gi(Tα(t)

( ddt

∣∣∣t

))= Tα(t)g

i(lim←−

ξi(α(t)))

⇔ Tt(gi α)

( ddt

∣∣∣t

)= Tα(t)g

i((ξi(αi(t))

)i∈N

)

⇔(gi α

)·

(t) =(ξi αi

)(t)

⇔(αi)·

(t) =(ξi αi

)(t),

which is true because αi is an integral curve of ξ.

We are now in a position to give the

Proof of Theorem 3.4.5. Let an arbitrary vector v ≡ (vi) ∈ TeG ≡ lim←−

T ieGi

[recall that e = (ei) is the identity of G and ei the one of Gi (i ∈ N)]. Ifξ ∈ L(G) is the left invariant vector field of G corresponding to v, thenξ =

(ξi)i∈N

, where ξi ∈ L(Gi) (see Proposition 3.4.3). By the precedinglemma, we obtain

expG(v) = α(1) =(lim←−

αi)

(1) =(αi(1)

)i∈N

=(expGi(vi)

)i∈N

=(lim←− expGi

)(v),

thus concluding the proof.

Remark 3.4.7. It should be noted that expG is not necessarily a localdiffeomorphism at 0 ∈ TeG although each expiG is, for every i ∈ N. Infact, for each i ∈ N, there are open neighborhoods U i of 0 ∈ TeiG

i, andN i of ei ∈ Gi, such that expiG : U i → N i is a diffeomorphism. ThenexpG |lim←−

U i : lim←−Ui → lim←−N

i is defined, but lim←−Ui is not always an open

neighborhood of 0 ∈ TeG. However, the existence of an exponential map,which is a local diffeomorphism at the identity of the group, characterizescommutative Frechet-Lie groups in the sense of the next result.

Theorem 3.4.8 ([Gal96]). Let G be a commutative Frechet-Lie group, andassume that there is a smooth map f : TeG→ G satisfying the properties:

i) f(u+ v) = f(u) · f(v), for every u, v ∈ TeG;

ii) There are open neighborhoods V0 ⊆ TeG, Ne ⊆ G of 0 and e, respec-tively, such that f : V0 → Ne is a diffeomorphism.

Then G is a projective limit of Banach-Lie groups.


3.5 Equations with Maurer-Cartan differential

We briefly discuss the analog of equation (1.2.7) in the context of Frechetmanifolds and Frechet-Lie groups.

Let B be Banach space and G = lim←−

Gi a Frechet-Lie group with Lie

algebra g = lim←−

gi. As in § 1.4.4 [constructions (c) and (e)], we consider thethe linear map bundle L(B, g). Anticipating the general theory of projectivelimit vector bundles studied in Chapter 5, we prove:

Proposition 3.5.1. The following properties hold:i)

(L(B, gi); pji)i,j∈N

is a projective system of Banach vector bundles,

where pji(f) := Tejgji f .

ii) lim←−L(B, gi) ≡ L(B, g), and the canonical projections are

pi : L(B, g) −→ L(B, gi) : f 7→ pi(f) = Tegi f.

Regarding now differential forms with values in g, we have:

Proposition 3.5.2. If θ ∈ Λ1(B, g), there are θi ∈ Λ1(B, gi), such thatθ = lim←− θ

i. Moreover, for every vector fields X,Y ∈ X (B), we have:

(i) θ(X) = lim←−

(θi(X)

),

(ii) Y (θ(X)) = lim←−

(Y (θi(X))

),

(iii) dθ(X,Y ) = lim←−

(dθi(X,Y )

),

(iv) [θ, θ](X,Y ) = lim←−

([θi, θi]i(X,Y )

).

within appropriate isomorphisms.

Proof. For every i ∈ N, we define the map

θi : B → L(B, gi) : x 7→ Tegi θx,

where gi : G→ Gi (i ∈ N) are the canonical projections of G = lim←−

Gi. It is

clear that θi ∈ Λ1(B, gi), for every i ∈ N.Easy computations show that the diagrams (for all i, j ∈ N, j ≥ i)

Bθj

- L(B, gj) Bθ

- L(B, g)

L(B, gi)

pji

?

θi-

L(B, gi)

pi

?

θi-

3.5. Equations with Maurer-Cartan differential 131

are commutative. The left of them implies that the limit lim←−

θi exists, while

the right one and Proposition 2.3.5 imply that θ = lim←−

θi.To prove (i), we check that

(Tejg

ji θj(X))

(x) = Tejgji(θjx(Xx)

)= θi(X)(x), x ∈ B.

This implies the existence of the lim←−

(θi(X)

). Similarly, Tegiθ(X) = θi(X),

thus (again by Proposition 2.3.5) θ(X) = lim←−

(θi(X)

).

For the proof of (ii) we note that

(3.5.1)(Tejg

ji Y (θj(X))

(x) =(Tejg

ji Tx(θj(X)))

(Yx), x ∈ B.

Also, from (i) and Proposition 3.2.2,

Tx(θ(X)) = Tx(lim←−

θi(X))

= lim←−

Tx(θi(X)

),

thus Tx(θi(X)

)= Tejg

ji(Txθ

j(X)). As a result, (3.5.1) transforms into

(Tejg

ji Y (θj(X))

(x) = Tx(θi(X)

)(Yx) = Y (θi(X))(x);

that is, TejgjiY (θj(X) = Y (θi(X)), for all i, j ∈ N with j ≥ i. This ensures

the existence of lim←−

Y (θi(X)). On the other hand, by similar arguments,

Tegi Y (θ(X) = Y (θi(X)), for every i ∈ N. Hence, we obtain (ii).Property (iii) is a direct consequence of (i):

dθ(X,Y ) = X(θ(Y ))− Y (θ(X))− θ([X,Y ])

= lim←−

X(θi(Y ))− lim←−

Y (θi(X)) − lim←−

(θi([X,Y ])

)

= lim←−

(dθi(X,Y )

).

For (iv) we apply (i) along with Theorem 3.4.4 as follows:

[θ, θ](X,Y ) = [θ(X), θ(Y )] =[lim←−

θ(X), lim←−

θ(Y )]

= lim←−[θi(X), θi(Y )]i = lim←−

([θi, θi]i(X,Y )

).

For simplicity, in what follows, we denote by D : C∞(B,G) → Λ1(B, g)the operator induced by the right Maurer-Cartan differential (see § 1.2.6).We intend to study the analog of (1.2.7) in the present setting, i.e. equation

(3.5.2) Dx = θ,

where θ ∈ Λ1(B, g).


Lemma 3.5.3. A differential form θ ∈ Λ1(B, g) is integrable; that is, dθ =12 [θ, θ], if and only if every θi (i ∈ N) is integrable.

Proof. Immediate consequence of Proposition3.5.2.

Theorem 3.5.4. Let an arbitrary (x0, g0) ∈ B×G. Equation (3.5.2) has aunique solution f : U → G (U open neighborhood of x0) with f(x0) = g0 ifand only if θ is integrable.

Proof. We first prove the statement for B simply connected. Assumingthat θ is integrable, Lemma 3.5.3 implies that each equation (in the Banachframework)

(3.5.3) Dxi = θi; i ∈ N,

has a unique (global) solution f i : B → Gi with initial condition f i(x0) =gi(g0) := gi0. Because the connecting morphisms gji : Gj → Gi are grouphomomorphisms (see the beginning of § 3.4, we easily check that

D(gji f j)(x)

)= Tejg

ji Df j(x) = Tejgji θjx = θix. x ∈ B.

Since (gji f j)(x0) = gi(g0), it follows that gji f j is also a solution of(3.5.3) with the same initial condition, thus gji f j = f i, for every j ≥ i.As a result f := lim←− f

i : B → G exists and determines a smooth map inthe sense of our framework (see also Proposition 3.1.8). On the other hand,if Rg (g ∈ G) denotes the right translation of G by g, and Rigi (gi ∈ Gi),

denotes the right translation of Gi by gi, then the analog of (3.4.1) implies

Df(x) = Tf(x)Rf(x)−1 Txf

= lim←−

(Tf i(x)R

if i(x)−1 Txf

i)

= lim←−

Df i(x) = lim←−

θi(x) = θ(x),

for every x ∈ B. Hence, f is a solution of (3.5.2) satisfying f(x0) =(f i(x0)

)i

=(gi(g0)

)i

= g0.

We shall show that f is the unique solution of (3.5.2) with the giveninitial condition. Indeed, assume that ϕ : B → G is another solution suchthat ϕ(x0) = g0. Then, ϕ = lim

←−ϕi, where ϕi = gi ϕ : B → Gi. We check

that Dϕi = θi as follows: Since Dϕ = θ, we have that

Tegi Tϕ(x)Rϕ(x)−1 Txϕ = Teg

i θx,

3.6. Differential forms 133

or, by Corollary 3.2.6 applied to the case of G,

Tϕi(x)Riϕi(x)−1 Txϕ

i = θix; x ∈ B,

thus Dϕi = θi. In addition, ϕi(x0) = gi(g0) holds for all i ∈ N. Conse-quently, by the uniqueness of the solutions of (3.5.3), ϕi = f i, for all i ∈ N;hence, ϕ = f as claimed.

Conversely, assume that (3.5.2) has a unique solution f with f(x0) = g0.Setting f i := gif (i ∈ N), we have that f = lim

←−f i. Using similar arguments

as before, we check that Df i = θi, i ∈ N. Therefore, θi is integrable for alli ∈ N, which by Lemma 3.5.3 implies the integrability of θ.

We consider now the case where B is not necessarily simply connected.Denoting by (B, π, B) the universal covering space of B, we set θ := π∗θ.

It is immediate that θ = lim←−

θi. If θ is integrable, so are the forms θi and

θi, for every i ∈ N. Thus each lifted equation Dzi = θi [see (1.2.8)] has

a unique (global) solution f i : B → Gi with f i(x0) = gi(g0), where x0 isan arbitrarily chosen point of B with p(x0) = x0. By the structure of theuniversal covering, there is an open neighborhood U of x0 and a connectedcomponent V of π−1(U) such that π : V → U is a diffeomorphism. Setting,

f i := f i π−1|U , i ∈ N, we check that Df i = θi and f i(x0) = gi(g0). Asin the first case of the proof, f := lim

←−f i : U → G exists and is the unique

solution of (3.5.2) with f(x0) = g0.For the converse we proceed as in its counterpart in the first case.

In conjunction with the terminology of fundamental solution (of the liftedequation on the universal covering) and the monodromy homomorphism ofthe original equation, induced in the end of § 1.2.6, we obtain the followingbyproduct of the proof of Theorem 3.5.4:

Corollary 3.5.5. Let θ ∈ Λ1(B, g) be an integrable form such that θ =lim←− θ

i, with θi ∈ Λ1(B, gi), i ∈ N. Then:

(i) Fθ = lim←−

Fθi ,

(ii) θ# = lim←−

θi#.

3.6 Differential forms

Since differential forms are smooth sections of map bundles, arbitrary formson Frechet manifolds, with values in Frechet spaces, are not necessarilyrepresented as projective limits of forms on Banach manifolds. Nevertheless,


as we explain below (Proposition 3.6.3), the converse is partially true: ‘point-wise’ projective limits of ordinary vector valued differential forms on Banachmanifolds yield differential forms on Frechet manifolds. On the other hand,connection forms on limit principal bundles (treated in § 4.2) provide animportant example of differential forms that can always be represented—ina point-wise fashion—as projective limits of (connection) forms on Banachprincipal bundles.

In view of concrete applications in Chapter 4, throughout this sectionwe consider:

• A Frechet manifold M = lim←−

M i, derived from the projective system of

Banach manifolds M i;µjii,j∈N, as in § 3.1. We recall that M is modelledon F = lim

←−Ei, where Ei is the model of M i (i ∈ N), and Ei; ρjii,j∈N is the

projective system generating F.

• A Frechet-Lie group G = lim←−

Gi, where Gi; gjii,j∈N is a projectivesystem of Banach-Lie groups. We already know (see Theorem 3.4.4) thatthe Lie algebras gi ≡ L(Gi) of Gi (i ∈ N) determine a projective system suchthat g ≡ L(G) = lim←− gi. The connecting morphisms of the latter system are

the Lie algebra morphisms gji ≡ Tejgj : gj → gi, (j ≥ i), induced by gji

after the identification (1.2.3).

Then we define the set

L(TM, g) :=⋃

x∈M

L(TxM, g)

and the natural projection

L : L(TM, g) −→M : f 7→ L(f) := x, if f ∈ L(TxM, g).

Proposition 3.6.1. L(TM, g) is a smooth manifold modelled on the locallyconvex space L(F, g).

Proof. Let (U, φ) be a chart of M and (π−1(U),Φ) the induced chart of thetangent bundle (TM,M, π). We define the chart

(L−1(U), Φ

)of L(TM, g),

where the mapΦ : L−1(U) −→ φ(U)× L(F, g)

is given by

Φ(f) =(φ(L(f)), f φ

−1L(f)

)=(φ(x), f φ

−1x

),

if f ∈ L(TxM, g). Recall that φ−1x : TxM → F is the linear isomorphism

induced by φ.


Let (V, ψ) be another chart of M , compatible with (U,Φ), U ∩ V 6= ∅,and let (L−1(V ), Ψ) the corresponding chart of L(TM, g). To verify thecompatibility of

(L−1(U), Φ

)and

(L−1(V ), Ψ

), we evaluate

Ψ Φ−1 : φ(U ∩ V )×L(F, g) −→ ψ(U ∩ V )× L(F, g)

at any point (a, g) of the domain:

(Ψ Φ−1

)(a, g) = Ψ

(φ−1(a), g φφ−1(a)

)

=((ψ φ−1)(a), g φφ−1(a) ψ

−1φ−1(a)

)

=((ψ φ−1)(a), g D(φ ψ−1)

((ψ φ−1)(a)

)),[see § 1.1.4]

or, setting F := φ ψ−1,

(Ψ Φ−1

)(a, g) =

(F−1(a), g DF (F−1(a))

).

Hence, for the desired compatibility it suffices to show that

(3.6.1) χ : φ(U ∩ V )× L(F, g) −→ L(F, g) : (a, g) 7→ g DF(F−1(a)

)

is a smooth map. To this end, we exploit the structure of plb-manifolds. Inthis regard, we can choose charts of M such that (U, φ) =

(lim←−

xi, lim←−

φi).

Then φx = lim←−φixi , for every x = (xi) ∈ M , Φ = lim←− Φi, where (xi, Φi) is

the analogous chart of the Banach bundle L(TM i, gi) [see § 1.4.4(c)], andF = lim←−F

i, with

F i := φi (ψi)−1 : Ei ⊇ ψi(xi ∩ V i) −→ φi(xi ∩ V i) ⊆ Ei.

Consequently, since F is a diffeomorphism, so is the map

φ(U ∩ V ) ∋ a 7−→ DF (F−1(a)) ∈ Lis(F) ⊂ L(F)

in virtue of Proposition 2.3.12. The commutative diagram below clarifiesthe latter argument.

F ⊇ φ(U ∩ V ) ∋ aF−1

- F−1(a) ∈ ψ(U ∩ V ) ⊆ F

DF(F−1(a)

)∈ L(F)

Prop. 2.3.12

?-


Now, because DF(F−1(a)

)is a projective limit of corresponding differ-

entials, it follows in particular that DF(F−1(a)

)∈ LI(F) [see (2.3.11) and

Proposition 2.3.13]. Therefore, applying the composition map

comp: L(F, g)×LI(F) −→ L(F, g) : (g, h) 7→ g h,

which is continuous bilinear, hence smooth [see Proposition 2.3.14 and Re-mark 2.2.4(2)], we conclude that (3.6.1) is smooth, by which we prove thecompatibility of the aforementioned charts and the existence of a smoothstructure on L(TM, g).

Corollary 3.6.2. The triplet (L(TM, g),M,L) is a locally trivial fibrationof fibre type L(F, g)

As usual, the smooth sections of L(TM, g) are the g-valued 1-forms onM and their set is denoted by Λ1(M, g).

In preparation of the connection forms, treated in § 4.2, we prove thefollowing.

Proposition 3.6.3. Let θi ∈ Λ1(M i, gi), i ∈ N, be differential forms suchthat the projective limit lim←− θ

i(xi) exists, for every x = (xi) ∈ M . Then

the map θ : M → L(TM, g), given by θ(x) ≡ θx := lim←− θi(xi), is a g-valued

1-form on M .

Proof. In virtue of Proposition 3.2.2, L θ = idM ; thus we need only toshow the smoothness of θ. For this purpose, we modify the pattern of theproof of Proposition 2.3.12. More precisely, using the limit charts (in theproof) of Proposition 3.6.1, we consider the composite map Φ θ|U , shownalso in the diagram:

Uθ- L−1(U) = L(TM, g)|U

L(F, g)

Φ

?-

Hence, for every x = (xi) ∈ U = lim←−Ui

(3.6.2)

(Φ θ

)(x) = Φ(θx) = Φ

(lim←− θ

i(xi))

=(φ(x),

(lim←−

θi(xi)) φ−1x

)

=(φ(x), lim

←−

(θi(xi) φ

−1x

)).


The first component of the right-hand side of (3.6.2) is smooth. For thesmoothness of the second one we define the maps [see also (2.3.5)]

Ri : U i −→ L(Ei, gi) : xi 7→ θi(xi) (φixi)−1

,(3.6.3)

Qi : U i −→ Hi(F, g) : xi 7→(R1(µi1(xi)

), R2

(µi2(xi)

), . . . , Ri(µi)

).(3.6.4)

The map (3.6.3) is smooth because θi ∈ Λ1(M i, gi), thus θi(xi) (φixi)−1

coincides with the second component of Φi θi. Also, each Qi takes valuesin Hi(F, g) since the diagram

EjRj(µij(xi))

- gj

Ei

ρjk

?

Rk(µik(xi))- gk

gjk

?

commutes, for every i, j, k ∈ N with (caution !) i ≥ j ≥ k. Indeed, theexistence of lim

←−θi(xi) implies that

gij θj(µij(xi)) = θk(µik(xi)) Tµij(xi)µjk;

therefore,

gjk Rj(µij(xi)

)= gjk θj

(µij(xi)

)(φjµij(xi)

)−1

= θk(µik(xi)

) Tµij (xi)µ

jk (φjµij(xi)

)−1

= θk(µik(xi)

)(φkµik(xi)

)−1 ρjk

= Rk(µik(xi)

) ρjk.

On the other hand, if

hji : Hj(F, g)→Hi(F, g) : (f1, f2, . . . , f j) 7→ (f1, f2, ..., f i); j ≥ i,

it follows that

(hji Qj)(xj) = hji(R1(µj1(xj)

), R2

(µj2(xj)

), . . . , Rj(xj)

)

=(R1(µj1(xj)

), R2

(µj2(xj)

), . . . , Ri

(µji(xj)

))

=(R1(µi1(µji(xj)

)), R2

(µi2(µji(xj)

)), . . . , Ri

(µji(xj)

))

=(Qi µji

)(xj);


hence, the map

Q = lim←−Qi : U = lim←−U

i −→ H(F, g) = lim←−Hi(F, g)

exists and is smooth.

Finally, applying the continuous linear embedding

ε : H(F, g) −→ L(F, g) : (f i) 7→ lim←− fi,

along with the identifications of Example 2.3.3(2), we see that

(ε Q)((xi)

)=(ε lim←−

Qi) (

(xi))

=

= ε(R1(x1),

(R1(x1), R2(x2)

),(R1(x1), R2(x2), R3(x3)

), . . .

)

≡ ε(R1(x1), R2(x2), R3(x3), . . .

)

= lim←−Ri(xi) = lim←−

(θi(xi)

(φi−1

xi

).

This means that the last limit is a smooth map (since both Q and ε aresmooth), and so are (3.6.2) and θ.

Remark 3.6.4. The previous result clarifies the terminology applied in theintroduction of the present section: θ ∈ Λ1(M, g) is determined point-wiseby the limits of (θi)i∈N, i.e. θ(x) = lim

←−θi(xi), x = (xi).

To define differential forms of higher degree, we consider the space of con-tinuous k-linear maps Lk(F, g) = L(F, . . . ,F; g) (k factors), equipped with anappropriate topology (e.g. the bornologification, [KM97]), so that the com-position map is continuous, and the space of continuous k-alternating (skew-symmetric) mapsAk(F, g) is a closed subspace of Lk(F, g), thus a locally con-vex space itself. Accordingly, we construct the triplet (Ak(TM, g),M,Lk),where

Ak(TM, g) :=⋃

x∈M

Ak(TxM, g),

Lk : Ak(TM, g) −→M : f 7→ Lk(f) := x, if f ∈ Ak(TxM, g).

The analogs of Proposition 3.6.1 and Corollary 3.6.2 are stated in thefollowing.

Proposition 3.6.5. (Ak(TM, g),M,Lk) is a locally trivial fibration of fibretype Ak(F, g).


Proof. We sketch its main steps since it is a simple extension of that ofProposition 3.6.1. From the charts (U, φ) and (π−1(U),Φ) of M and TM ,respectively, we define the chart

(L−1k (U), Φk

)of Ak(TM, g), where

Φk : L−1k (U) −→ φ(U)×Ak(F, g)

is given by

Φk(f) :=(φ(x), f

(φ−1x × · · · × φ

−1x

))

if f ∈ A(TxM, g).For the compatibility of two charts

(L−1k (U), Φk

)and

(L−1k (V ), Ψk

), with

U ∩ V 6= ∅, we check that

Ψk Φ−1k : φ(U ∩ V )×Ak(F, g) −→ ψ(U ∩ V )×Ak(F, g)

has the form

(Ψk Φ−1k

)(a, h) =

(F−1(a), g

(DF (F−1(a)) × · · · ×DF (F−1(a))

),

for every (a, h) ∈ φ(U ∩ V )×Ak(F, g), where F := φ ψ−1. Its smoothnessis shown as in Proposition 3.6.1, taking into account that the involved com-position map comp: Ak(F, g)× LI(F) × · · · × LI(F) → Ak(F, g) is given bycomp (h; f1, . . . , fk) = h (f1 × · · · × fk).

We close by adding that the exterior differentials

d : Λk(M, g)→ Λk+1(M, g); k ≥ 0,

are defined in the usual way; that is, by differentiating the local principalparts of the sections representing the given differential forms. The ordinaryproperties hold true also in the present context.

Chapter 4

Projective systemsof principal bundles

In the previous chapter, taking advantage of the compatibility betweenFrechet structures and projective limits, we set up an appropriate back-ground for the study of certain manifolds and Lie groups modelled on Frechetspaces. The results obtained allowed us to handle a wide variety of man-ifolds that cannot be modelled on Banach spaces, addressing the difficultythat prevents the transfer of classical tools from the finite-dimensional andBanach cases.

The same background will be exploited in order to study more compli-cated geometric structures. More precisely, in this chapter, we shall focus onprincipal bundles and their geometric properties within the Frechet frame-work. It will be shown that the projective limit approach gives a way outof a number of significant difficulties emerging from the peculiarities of thespace models. For example, in the case of an arbitrary Frechet principal bun-dle, connections may not have parallel displacements, the standard proof ofCartan’s (second) structural equation fails as based on the existence of 1-parameter subgroups of transformations, and so on. All the above issues canbe addressed, up to a point, by using, whenever possible, the mechanism ofprojective limits.

141

142 Chapter 4. Projective systems of principal bundles

4.1 Projective systems and Frechet principal bun-

dles

For the definition of projective limits of principal bundles, we follow themethod adopted in the previous sections: All the spaces and local chartsinvolved should form projective systems.

Definition 4.1.1. A countable family

ℓi = (P i, Gi, B, πi); F ji =

(pji, gji, idB

)i,j∈N

of Banach principal bundles and corresponding morphisms will be called aprojective system of principal bundles if the following conditions arefulfilled:(PLPB. 1) P i; pjii,j∈N is a projective system of Banach smooth manifoldsconverging to the plb-manifold P = lim

←−P i

(PLPB. 2) Gi; gjii,j∈N is a projective system of Banach Lie groups con-verging to the plb-group G = lim

←−Gi.

(PLPB. 3) For every j ≥ i, each

F ji = (pji, gji, idB) : ℓj = (P j , Gj , B,Pij) −→ ℓi = (P i, Gi, B, πi)

is a principal bundle morphism.

(PLPB. 4) There exists a trivializing cover (Uα,Φα)α∈I of P , whoseelements are obtained as projective limits of corresponding trivializations ofthe factor bundles P i; that is,

(4.1.1) (Uα,Φα) =

(lim←−i∈N

U iα, lim←−i∈N

Φiα

), α ∈ I.

As a matter of notation, if B is the model of B and the group G ismodelled on the Frechet space G = lim

←−Gi, ζjii;j∈N, then P is modelled on

P = lim←−Pi, ρji = idB×ζ

jii,j∈, where Pi = B×Gi.

It is important to note that all the factor bundles ℓi (i ∈ N) in the pre-vious definition have the same base space. This assumption simplifies manytechnical details of subsequent computations, without any loss of generality.Alternatively, one could consider base spaces Bi (i ∈ N), assuming, at thesame time, that they converge to a plb-manifold. However, this would leadto undue complications without any reasonably significant gain. Thus, re-garding (PLPB 4), in many cases it will be sufficient to take U iα = Uα forall i ∈ N.

4.1. Projective systems and Frechet principal bundles 143

The conditions of Definition 4.1.1 ensure the existence of the projectivelimit of the system ℓi;F

jii,j∈N, namely

ℓ = lim←−

ℓi =(lim←−

P i, lim←−

Gi, B, lim←−

πi),

called a projective limit of (Banach) principal bundles (or plb-princi-pal bundle, for short). We shall denote by pi : P = lim←−P

i → P i the canon-ical projection of the limit to the i-th factor bundle. The base manifold Bis assumed to be a Hausdorff space and admits smooth partitions of unity.The first assumption implies the uniqueness of solutions of differential equa-tions (with given initial conditions), while the second ensures the existenceof connections on the factor bundles.

Theorem 4.1.2. Every plb-principal bundle is a Frechet principal bundle.

Proof. In virtue of Propositions 3.1.3 and 3.4.1, in conjunction with Def-inition 4.1.1, the total space P = lim

←−P i is a Frechet manifold, and the

structure group G = lim←−

Gi is a Frechet-Lie group.

Denoting by πi : P i → B the projections of the factor bundles ℓi (i ∈ N),and by δi : P i ×Gi → P i the actions of Gi respectively on the total spaces,we see that (PLPB. 3) implies the equalities

πi pji = πj ,(4.1.2)

pji δj = δi (pji × gji); j ≥ i.(4.1.3)

As a result, the pls-maps

π := lim←−

πi : P → B, and δ := lim←−

δi : P ×G→ P

exist and are smooth in virtue of Proposition 3.1.8. Moreover, the compat-ibility of the projective limits with any algebraic structure ensures that δ isa smooth action. Indeed, for every u = (ui) ∈ P and every g1 = (gi1), g2 =(gi2) ∈ G, we check that

δ(u, e) = lim←−

δi((ui), (ei)

)=(δi(ui, ei)

)i∈N

= (ui) = u,

as well as

δ(δ(u, g1), g2) = lim←− δi(lim←− δ

i((ui), (gi1)

), (gi2)

)

= lim←−

δi((δi(ui, gi1), (gi2)

)i∈N

)

=(δi(δi(ui, gi1), (gi2)

))i∈N

=(δi(ui, gi1g

i2

))i∈N

= lim←− δi((ui), (gi1g

i2))

= δ(u, g1g2).


Customarily, we simplify the notations by writing u ·g instead of δ(u, g),and analogously for the actions δi. This convention will be systematicallyapplied throughout this chapter.

The trivializing cover for the desired principal bundle structure on P isprovided by the charts (Uα,Φα) given by (4.1.1), because, for every u =(ui) ∈ π−1(U) and g = (gi) ∈ G,

(pr1 Φα

)(u) = pr1

(Φα(u)

)= pr1

(Φα((ui))

)

=(pr1(Φ

iα(ui))

)i∈N

=(πi(ui)

)i∈N

= π(u),

while

Φα(u · g) = lim←−

Φiα

((ui) · (gi)

)=(Φiα(ui · gi)

)i∈N

=(πi(ui · gi),Φi

α,2(ui · gi)

)i∈N

=(πi(ui),Φi

α,2(ui) · (gi))i∈N

=(π(u),Φα,2(u) · g)

)=(π(u),Φα,2(u)

)· g

= Φα(u) · g,

where Φα,2 : π−1(Uα)→ G is the projection of Φα to the second factor.

The previous bundle structure allows to transfer a number of classicalresults to the Frechet framework.

Proposition 4.1.3. Let ℓ =(P = lim←−P

i, G = lim←−Gi, B, π = lim←−π

i)be

a plb-principal bundle. Then, for every b ∈ B, the fibre π−1(b) is a plb-manifold such that

π−1(b) = lim←−

((πi)−1(b)

).

Proof. By (PLPB. 4) of Definition 4.1.1, we can find a trivialization, say,(U,Φ) = (lim←−U

i, lim←−Φi) of P , with b ∈ U (for simplicity we omit the index

α), where (U i,Φi) are trivializations of the bundles P i (i ∈ N). We alreadyknow (see § 1.6.1) that, for every i ∈ N, the fibre (πi)−1(b) is a Banachmanifold diffeomorphic with the structure group Gi via the map

(4.1.4) Φib := pr2 Φ

i∣∣(πi)−1(b)

: (πi)−1(b)≃−−→ Gi.

The fact that F ji = (pji, idB , gji) is a principal bundle morphism implies

that πi pji = πj and

pji((πj)−1(b)

)⊆ (πi)−1(b), j ≥ i.

On the other hand, the restrictions

pji∣∣(πj)−1(b)

: (πj)−1(b) −→ (πi)−1(b)


are smooth, since each (πi)−1(b) is a canonical submanifold of the corre-sponding total space P i (i ∈ N). Therefore, we obtain the projective systemof smooth manifolds

(πi)−1(b); pjii,j∈N

.

We shall show that the conditions of Definition 3.1.1 are satisfied, therefore,lim←−

((πi)−1(b)) = π−1(b) is a plb-manifold.

Indeed, the models of (πi)−1(b), identified with Gii∈N, form a pro-jective system with connecting morphisms identified with the connectingmorphisms ζji : Gj → Gi (j ≥ i). Also, the fact that (U,Φ) is the projectivelimit of (U i,Φi) implies that

(4.1.5) (idB ×gji) Φj = Φi pji,

which, together with (4.1.4), gives for every u = (ui) ∈ lim←−

((πi)−1(b)

):

(4.1.6) gji(Φjb(u

j))

= Φib(p

ji(uj)) = Φib(u

i).

Therefore,(Φib(u

i))∈ G = lim

←−Gi and we can find a limit chart

(V = lim

←−V i,

Ψ = lim←−

Ψi)

of G so that(Φib(u

i))∈ V . Setting now

W i = (Φib)−1(V i) and χi = Ψi Φi

b,

we obtain a chart (W i, χi) of (πi)−1(b). Then, by (4.1.6),

pji(W j) = pji((Φj

b)−1(V j)

)= (Φi

b)−1(gji(V j)

)= (Φi

b)−1(V i) = W i;

in other words, the domains of the charts (W i, χi), i ∈ N, define a projectivesystem. The same holds true for the corresponding maps, since

ζji χj = ζji Ψj Φjb = Ψi gji Φj

b = Ψi Φib p

ji = χi pji,

for all j ≥ i. In addition, the projective limits of the domains and the imagesof the same maps are open sets, because

lim←−

χi(W i) = lim←−

(Ψi(V i)),

lim←−Wi = lim←−

((Φi

b)−1(V i)

)=(

lim←−Φib

)−1(lim←−V

i).

Summarizing, we have seen that a limit chart, in the sense of Definition 3.1.1,can be defined at every point of lim←−((πi)−1(b)); hence, the latter is a plb-manifold as claimed.


Remark 4.1.4. It is useful to mention here another approach to the sub-manifold structure on the fibres π−1(b) of P : In the presence of the trivial-izations Φ = lim

←−Φi, each Φb := pr2 Φ|π−1(b) : π−1(b)→ G (b ∈ B) coincides

with lim←−

Φib = lim←−

(pr2 Φ

i|(πi)−1(b)

), a fact ensuring that the Frechet mani-

fold structure, determined as in the Banach case (see the list of properties ofa Banach principal bundle in § 1.6.1), coincides with the structure obtainedby the projective limit approach given here.

Proposition 4.1.5. Let(P = lim

←−P i, G = lim

←−Gi, B, π = lim

←−πi)be a plb-

principal bundle. The canonical projections pi : P → P i and gi : G→ Gi ofthe total space and the structure group, respectively, determine the principalbundle morphism

F i = (pi, gi, idB) : (P,G,B, π) −→ (P i, Gi, B, πi).

Proof. Since P = lim←−Pi is a plb-manifold and G = lim←−G

i a plb-group, it

follows that pi : P → P i is a smooth map and gi : G → Gi a Lie groupmorphism, for every index i ∈ N (see Propositions 3.1.5 and 3.4.1). On theother hand, π = lim

←−πi implies that

πi pi = idB π, i ∈ N.

Moreover, pi is equivariant with respect to G and Gi:

pi(u · a) = pi((ui), (ai)

)= pi

((ui · ai)

)= ui · ai = pi(u) · gi(a),

for every (u, a) ∈ P ×G.

Concerning the local sections of a plb-principal bundle we obtain:

Proposition 4.1.6. Every local section s : U ⊆ B → P of a plb-principalbundle

(P = lim

←−P i, G = lim

←−Gi, B, π = lim

←−πi)coincides with a projective

limit of local sections si : U → P i on the factor bundles; that is, s = lim←− si.

Proof. The desired components of s are obtained by projecting it via thecanonical projections of P to the factor bundles P i. Indeed, each

(4.1.7) si := pi s : U ⊆ B → P i

is a smooth map (as a composite of smooth maps). It is also a section ofthe Banach bundle (P i, Gi, B, πi) since

πi si = πi pi s = π s = idU .


Finally, the equalities

pji sj = pji pj s = pi s = si; j ≥ i,

ensure that lim←−

si exists and coincides with s, in virtue of the definition of

the components (si) and Proposition 2.3.5.

In Definition 4.1.1 we required the existence of limit trivializations inthe structure of a projective limit principal bundle. However, by precedingproposition, and the close relationship between the local sections and thetrivializations of a principal bundle, every trivialization is a limit. Moreprecisely:

Theorem 4.1.7. Every trivialization of a plb-principal bundle(P = lim

←−P i,

G = lim←−Gi, B, π = lim←−π

i)can be realized as the projective limit of trivial-

izations of the factor bundles.

Proof. Let (U,Φ) be an arbitrary trivialization of P . Then the map

s : U −→ π−1(U) : x 7→ Φ−1(x, e),

where e denotes the unit of G, is a smooth (local) section of P . As in theprevious proposition, s = lim←− s

i, with si given by (4.1.7). It is a typicalresult of the theory of (Banach) principal bundles [see § 1.6.3 and equality(1.6.5)] that each si determines the local trivialization of P i

Φi : (πi)−1(U) −→ U ×Gi : u 7→(πi(u), ki(u)

),

where ki : (πi)−1(U)→ Gi is the smooth map, uniquely determined by

(4.1.8) u = si(πi(u)) · ki(u), u ∈ (πi)−1(U).

[Observe that ki = ki (si πi, id(πi)−1(U)

), where ki is the analog of 1.6.6.]

We see that the family ki : (πi)−1(U) → Gii∈N is a projective system ofsmooth maps, since, for every j ≥ i and u ∈ (πi)−1(U), (4.1.8) leads to

pji(u) = pji(sj(πj(u))) · gji(kj(u)) = si(πi(pji(u))) · gji(kj(u));

hence, again by (4.1.8),

ki(pji(u)) = gji(kj(u)), u ∈ (πi)−1(U);

equivalently,

(4.1.9) ki pji = gji kj.


As a result, the pls-map lim←−

ki : U −→ G is defined.Analogously, the section s defined by (U,Φ) in the beginning of the proof,

determines the map k : π−1(U)→ G, given by

(4.1.10) u = s(π(u)) · k(u), u ∈ π−1(U).

Now, for every u ∈ π−1(U), (4.1.10) yields:

pi(u) = pi(s(π(u))) · gi(k(u)) = si(πi(pi(u))) · gi(k(u)),

from which follows that ki(pi(u)) = gi(k(u)); that is,

(4.1.11) ki pi = gi k, i ∈ N.

Therefore, in virtue of Proposition 2.3.5, k = lim←−

ki.The above equalities lead to the desired relationship between the trivi-

alizations (U,Φ) and (U,Φi). First observe that (4.1.9) yields

(4.1.12)

(idU ×gji) Φj = (idU ×g

ji) (πj , kj) = (πj , gji kj)

= (πi pji, ki pji) = (πi, ki) pji

= Φi pji,

thus lim←−

Φi exists. Moreover, in virtue of (4.1.11),

(4.1.13)

(idU ×gi) Φ = (idU ×g

i) (π, k) = (π, gi k)

= (πi pi, ki pi) = (πi, ki) pi

= Φi pi.

This implies lim←−

Φi = Φ, by which we complete the proof.

In the same vein, the transition functions of a limit bundle remain alsoin the category of projective limits.

Proposition 4.1.8. The transition functions of a plb-principal bundle(P =

lim←−Pi, G = lim←−G

i, B, π = lim←−πi)are the pls-maps

tαβ = lim←− t

iαβ

α,β∈I

,

wheretiαβα,β∈I

are the transition functions of P i, for every i ∈ N.

Proof. According to Definition 4.1.1, there is a limit trivialization

(Uα,Φα) = (lim←−Uiα, lim←−Φi

α)α∈I

of P . Let

tαβ : Uαβ −→ G : x 7→(

Φα,x Φ−1β,x

)(e); α, β ∈ I,


be the corresponding transition functions, where [see also the analog of(4.1.4)]

Φα,x := pr2 Φα|π−1(x) : π−1(x)≃−−→ G.

Analogously, the transition functions of the factor bundles (P i, Gi, B, πi),defined by the trivializations (U iα,Φ

iα)α∈I (i ∈ N), are

tiαβ : U iαβ −→ Gi : x 7→(Φiα,x (Φi

β,x)−1)

(ei).

Recall that e and ei denote the unit elements of G and Gi, respectively.Since Φα = lim←−Φi

α, for every α ∈ I, we obtain the analogs of (4.1.12)and (4.1.13); namely,

(idB ×gji) Φj

α = Φiα p

ji,

(idB ×gi) Φα = Φi

α pi,

for every i, j ∈ N with j ≥ i.Consequently, by restriction to the fibres, we obtain:

(gji tjαβ)(x) = gji(Φjα,x

((Φj

β,x)−1(ej)))

= Φiα,x

(pji((Φj

β,x)−1(ej)

))

= Φiα,x

((Φi

β,x)−1(gji(ej))

)=(Φiα,x (Φi

β,x)−1)(ei)

= tiαβ(x),

for every α, β ∈ I, j ≥ i and every x ∈ Uαβ . This proves that the limit

lim←−

tiαβ : Uαβ −→ G

exists. Moreover, by the same token,

(gi tαβ)(x) = gi(Φα,x

((Φβ,x)

−1(e)))

= Φiα,x

(pi((Φβ,x)−1(e)

))

= Φiα,x

((Φi

β,x)−1(gi(e)))

=(Φiα,x

(Φiβ,x)−1)(ei)

= tiαβ(x),

which implies that tαβ = lim←− tiαβ , for all α, β ∈ I.

The previous proposition, in conjunction with the bijective correspon-dence between principal bundles and cocycles (see § 1.6.3), can be exploitedto obtain the following fundamental result in our framework.

Theorem 4.1.9. Every Frechet principal bundle ℓ = (P,G,B, π) with aBanach base and structure group a plb-group coincides, up to isomorphism,with a projective limit of Banach principal bundles.


Proof. Let Gi; gjii,j∈N be the projective system of Banach-Lie groupssuch that G = lim

←−Gi, the latter being modelled on the Frechet space

G = lim←−Gi, ρji. Also, take B as the model of B. If (Uα,Φα)α∈I is

a trivializing cover of P and tαβ the corresponding transition functions,then, for every i ∈ N, the smooth maps

tiαβ := gi tαβ : Uαβ −→ Gi; i ∈ N,

(: gi : G→ Gi the canonical projections of G) determine a projective system,because

gji tjαβ = gji gj tαβ = gi tαβ = tiαβ.

Thus, by the definition of the factors, tαβ = lim←− tiαβ, for all α, β ∈ I.

On the other hand, for every i ∈ N,(tiαβ)α,β∈I

is a (Gi-valued) smooth1-cocycle, since, for every x ∈ Uαβγ , we have:

tiαβ(x) · tiβγ(x) = gi(tαβ(x)) · gi(tβγ(x)) =

= gi(tαβ(x) · tβγ(x)) = gi(tαγ(x)),

for all α, β, γ ∈ I. Therefore, a unique Banach principal bundle ℓi =(P i, Gi, B, πi) can be defined with transition functions

(tiαβ)α,β∈I

. Specifi-

cally, following the construction described in the last part of § 1.6.3, P i =Si/∼i , where

Si =⋃

α∈I

(α × Uα ×G

i)

and the equivalence relation is defined by

(α, x, g) ∼i (β, x′, g′) ⇔ x = x′ and g′ = tiβα(x) · g.

The natural projection of P i is

πi : P i −→ B : [(α, x, g)]i 7→ x,

where [(α, x, s)]i denotes the equivalence class of (α, x, s), while the actionof Gi on the (right of) P i is given by

[(α, x, g)]i · g′ = [(α, x, g · g′)]i.

Finally, a (local) trivializing cover (Uα,Φiα) of P i is obtained by taking

Φiα : (πi)−1(Uα) −→ Uα ×G

i : [(γ, y, h)]i 7→ (y, tiαγ(y) · h).


The bundles ℓi (i ∈ N) are interconnected by the maps

pji : P j −→ P i : [(α, x, g)]j 7→ [(α, x, gji(g))]i.

They are well-defined, since [(α, x, g)]j = [(β, x, g′)]j yields g′ = tjβα(x) · g.

Moreover, the fact that the connecting morphisms gji : Gj → Gi (j ≥ i) aregroup morphisms and the transition functions tiαβ form a projective systemsimplies that

gji(g′) = gji(tjβα(x) · g) = tiβα(x) · gji(g),

which shows that [(α, x, gji(g))]i = [(α, x, gji(g′))]i.

On the other hand, the maps pji (j ≥ i) first commute with the naturalprojections of the bundles:

(πi pji) ([(α, x, g)]j ) = πi([(α, x, gji(g))]i

)= x = πj([(α, x, g)]j ),

and, secondly, they respect the bundle actions:

pji([(α, x, g)]j · g

′)

= pji([(α, x, g · g′)]j

)

=[(α, x, gji(g · g′)

)]i

= [(α, x, gji(g) · gji(g′))]i

= [(α, x, gji(g))]i · gji(g′) = pji([(α, x, g)]j ) · g

ji(g′),

where [(a, x, g)]j and g′ are arbitrarily chosen elements of P j and Gj respec-tively. In addition, pji are smooth maps since they are locally projected, viathe trivializing maps, to idUα ×g

ji, as it follows from

(Φiα p

ji)

([(α, x, g)]j ) = Φiα

([(α, x, gji(g))]i

)

=(x, tiαβ(x) · gji(g)

)=(x, gji(tjαβ(x) · g)

)

= (idUα ×gji)(Φjα([(α, x, g)]j )

).

The last two arguments prove that the triplets

F ji = (pji, gji, idB) : ℓj −→ ℓi; j ≥ i,

are principal bundle morphisms. Moreover, the family (ℓi, F ji)i,j∈N is aprojective system, because

(pik pji)([(α, x, g)]j ) = [(α, x, gik(gji(g)))]k

= [(α, x, gjk(g))]k = pjk([(α, x, g)]j ),


for every j ≥ i ≥ k. This system fulfils all the requirements of Defini-tion 4.1.1. As a matter of fact, taking into account that the trivializations

(Uα, lim←−

Φiα

)α∈I

satisfy condition (PLPB. 4), the only thing that remains to be checked isthat P i; pjii,j∈N is a projective system of Banach manifolds convergingto the plb-manifold lim←−P

i. The necessary plb-charts are obtained by the

previously defined trivializations (Uα,Φiα) of the principal bundles, the cor-

responding charts(Vα = lim←−V

iα,Ψα = lim←−Ψi

α

)of the plb-group G, and

the atlas (Uα, ϕα,B)α∈I of B. More precisely, for each i ∈ N, we definethe local chart with map

µiα := (ϕα ×Ψiα) Φi

α : (Φiα)−1(Uα × V

iα) −→ ϕα(Uα)×Ψi

α(V iα),

where ϕα(Uα), Ψiα(V i

α) and (Φiα)−1(Uα × V

iα) are open subsets of B, G and

π−1(Uα), respectively.These charts commute with the connecting morphisms of the involved

projective limits. Indeed, recalling that ζji : Gj → Gi are the connectingmorphisms of the projective system of the models of Gi,

(idB×ζji) µjα = (idB×ζ

ji) (ϕα ×Ψjα) Φj

α

=(ϕα × (ζji Ψj

α)) Φj

α =(ϕα × (Ψi

α gji))Φj

α

= (ϕα ×Ψiα) (idB×g

ji) Φjα = (ϕα ×Ψi

α) (Φiα p

ji) = µiα pji.

Therefore, the charts(lim←−

((Φi

α)−1(Uα × Viα)), lim←−

µiα)

can be defined. The fact that (Vα,Ψα)α∈I is a family of plb-charts on Gensures that the domains and ranges of the previous limit charts are opensets, because

lim←−

((Φi

α)−1(Uα × Viα))

=(lim←−

Φiα

)−1 (Uα × lim

←−V iα

),

lim←−

(µiα((Φi

α)−1(Uα × Viα)))

=

= lim←−

(ϕα(Uα)×Ψi

α(V iα))

= ϕα(Uα)× lim←−

(Ψiα(V i

α)).

Summarizing, the plb-(and therefore Frechet) principal bundle

lim←− ℓi = lim←−(P i, Gi, B, πi)

is now completely defined. Its identification with the initial bundle ℓ =(P,G,B, π) follows from the fact that both bundles have the same transitionfunctions

tαβ = lim←− t

iαβ

, as a consequence of their local structure.

4.2. Connections on limit principal bundles 153

Obviously, the family P i; pjii,j∈N, constructed in the previous proof, isan example of a projective system of principal bundles as in Definition 4.1.1.

4.2 Connections on limit principal bundles

This section focuses on the notion of connections on Frechet principal bun-dles in the context of the preceding § 4.1. Although the bundles in questionare projective limits of Banach principal bundles, a direct definition of con-nections as projective limits is not possible (see also the introductory dis-cussion in § 3.6). For this reason, we propose a generalized approach to pro-jective systems of connections resulting in their convergence to connectionson projective limit principal bundles, so as to satisfy the usual properties.This generalization provides a characterization of connections on such limitbundles.

The most convenient way to handle connections, in the present context,is by using global and local connection forms. The latter play a key rolein our approach and allow us to transfer to the Frechet framework manyimportant geometric features and results of finite-dimensional and Banachprincipal bundles. For instance, we mention the existence of parallel trans-lations (in spite of the problems arising in solving differential equations inFrechet spaces), holonomy groups, and flat connections. Of course, split-tings of appropriate exact sequences (in the sense of § 1.7.1) can also beused. However, they are quite cumbersome and there is no significant gainfrom their use.

Throughout this section, ℓ = (P,G,B, π) is a Frechet principal bundleover a Banach manifold, with structure group a Frechet-Lie group G repre-sented by a projective limit of Banach Lie groups, i.e. G = lim←−G

i; gjii,j∈N.Then, by Theorem 4.1.9, ℓ coincides, up to isomorphism, with a projectivelimit of Banach principal bundles,

ℓ ≡ lim←−

ℓi = lim←−

(P i, Gi, B, πi).

This means that the total space and the projection of the bundle are alsoprojective limits, i.e. P ≡ lim

←−P i, pji and π ≡ lim

←−πi.

Assume now that each bundle ℓi admits a connection whose correspond-ing connection form is denoted by ωi ∈ Λ1(P i, gi), i ∈ N (see § 1.7.2 and§ 3.6). Recall that gi is the Lie algebra of Gi. Then, an unconditional exis-tence of a projective limit for the family ωii∈N cannot be expected, sincethe manifolds L(TP i, gi)i,j∈N fail to form a projective system. A way


out of this problem is to require that the previous connections are properlyrelated.

Before giving a precise definition, we note that the value of a differentialform ωi ∈ Λ1(P i, gi) at a point ui ∈ P i, customarily denoted by ωiui , forconvenience will be written as ωi(ui). In this respect, it is understood thatωi(ui)(V ) denotes the value of the linear map ωiui = ωi(ui) at a vectorV ∈ TuiP

i.

Definition 4.2.1. By a projective system of connections on ℓ we meana countable family ωii∈N of connections (forms) on P ii∈N, respectively,such that, for every pair of indices (i, j) with j ≥ i, the connections ωj andωi are (pji, gji, idB)-related in the sense of § 1.7.5 [see also (1.7.19)]. Thenthe map

ω : P −→ L(TP, g) : u = (ui)i∈N 7→ ω(u) = lim←−(ωi(ui))

is said to be the projective limit of ωii∈N and will be simply denotedby ω = lim←−ω

i.

The preceding notation is used only for simplicity, in spite of the afore-mentioned remarks about the non existence of limit of differential forms.Thus, its real meaning is the aforementioned point-wise convergence;that is to say, ω(u) ≡ ωu = lim

←−ωi(ui), for every u = (ui) ∈ P .

From the above definition it becomes clear that related connections areof particular importance to our approach, because they ensure the existenceof the projective limit involved therein. Indeed, according to the explicitformula following the equivalent conditions (1.7.19), the requirement thatωj and ωi are (pji, gji, idB)-related (j ≥ i) implies that

Tejgji ωj(uj) = ωi(ui) Tujp

ji,

for every uj ∈ P j, ui = pji(uj). This is exactly the condition implying theexistence of lim

←−ωi(ui), according to Definition 2.3.4. Note that Tejg

ji = gji

in the notations of (1.7.19).

Our immediate goal is to prove that ω is a connection form in the usualsense (see § 1.7.2). To this end, we need a number preparatory lemmas.

Lemma 4.2.2. The map ω : P → L(TP, g), as in Definition 4.2.1, deter-mines a g-valued differential 1-form on P .

Proof. Direct consequence of Proposition 3.6.3.


Lemma 4.2.3. If X is a left invariant vector field of the structure groupG and X∗ the corresponding fundamental (Killing) vector field on P , thenω(X∗) = Xe ≡ X.

The reason for changing the customary notation X∗ of the fundamentalvector field (see § 1.3.2) to X∗ will be clear in the course of the proof.

Proof. By Theorem 3.4.4, every X ∈ L(G) ≡ g identifies with lim←−

Xi, where

Xi ∈ gi. We denote by Xi∗ the fundamental vector fields on P i, correspond-

ing to Xi (i ∈ N). If uj ∈ P j then, with respect to the action of Gj on (theright) of P j, we define the map

uj : Gj −→ P j : g 7→ uj · g.

Hence, for any choice of indices i, j with j ≥ i, we see that:

(Tpji Xj∗)(u

j) = Tujpji(Tej u

j(Xj(ej)

))= Tej

(pji uj

)(Xj(ej)

).

On the other hand, since (pij , gji, idB) : ℓj → ℓi is a principal bundle mor-phism,

(pji uj)(g) = pji(uj · g) = pji(uj) · gji(g) =(

(pji(uj) gji)

(g),

for every g ∈ Gi. Combining the preceding equalities, we obtain

(Tpji Xj∗)(u

j) = Tej(pji(uj) gji

)(Xj(ej)

)

= Tei pji(uj)

(Tejg

ji(Xj(ej)

)

= Tei pji(uj)(Xi(ei))

= (Xi∗ p

ji)(uj);

hence, Xi∗i∈N is a projective system and lim

←−Xi∗ exists.

Working similarly for the fields X∗ and Xi∗ on P and P i, respectively,

and taking into account that the canonical projections of P and G inducethe principal bundle morphisms (P i, gj , idB) : ℓ→ ℓi , we check that

Tpi X∗ = Xi∗ p

i, i ∈ N.

As a result, X∗ = lim←−

Xi∗, in virtue of Proposition 2.3.5. Therefore, for every

u = (ui) ∈ P , condition (ω. 2) of § 1.7.2 implies

ωu(X∗(u)) =(lim←−ω

i(ui)) (Xi∗(u

i)i∈N

=(ωi(ui)

(Xi∗(u

i)))i∈N

=(Xi(ei)

)i∈N

= Xe,

which concludes the proof.


The last requirement needed is the interplay of ω with the adjoint rep-resentation Ad of G.

Lemma 4.2.4. Let Rg : P → P : u 7→ u · g be the right translation of P byg ∈ G. Then, in analogy to (ω. 1) of § 1.7.2,

R∗gω = Ad(g−1)ω.

Proof. The adjoint representation of G = lim←−

Gi takes a projective limitform when applied to any element g ∈ G:

Ad(g−1) = lim←−

(Adi(gi)−1

),

as a consequence of Proposition 3.2.5. Similarly, Rg = lim←−

Rigi

. Then, for

every u = (ui) ∈ P , g = (gi) ∈ G and w = (wi) ∈ TuP ,

(R∗gω)u(w) = ωu·g(TuRg(w))

=(lim←−

ωi(ui · gi)) (

lim←−

TuiRigi(w

i))

=(ωi(ui · gi)

(TuiR

igi(w

i)))

i∈N

=((Ri∗giω

i)(ui)(wi)

)i∈N

=(

Adi((gi)−1

)(ωi(ui)(wi)

))i∈N

= lim←−

(Adi(gi)−1

) (lim←−

(ωi(ui)(wi)

))

= Ad(g−1)(ωu(w)).

The previous lemmata now prove the following main result.

Theorem 4.2.5. Let (ωi)i∈N be a projective system of connections on ℓ ≡lim←−

ℓi = lim←−

(P i, Gi, B, πi). Then the differential form ω = lim←−

ωi ∈ Λ1(P, g)is a connection on P .

Corollary 4.2.6. The connections ω and ωi are (pi, gi, idB)-related.

Proof. We recall that pi : P → P i and gi : G→ Gi are the canonical projec-tions. Thus, for every u = (ui) ∈ P and w = (wi) ∈ TuP , Corollary 3.2.6implies that

(4.2.1)((pi)∗ω

)u(w) = ωi(ui)

(Tup

i((wi)) = ωi(ui)(wi).

On the other hand,

ωu(w) =(

lim←−ωi(ui)

)((wi)) = lim←−

(ωi(ui)(wi)

)∈ lim←− gi = g,


or, applying gi := Teigi

(4.2.2) gi(ωu(w)) = ωi(ui)(wi).

From (4.2.4) and (4.2.5) it follows that (pi)∗ω = giω, which proves thestatement.

The horizontal and vertical subspaces of TP are naturally determinedby the corresponding spaces on the factors.

Proposition 4.2.7. Let ω = lim←−

ωi be a connection on the limit bundle

(P,G,B, π) = lim←−

(P i, Gi, B, πi). Then, for every u = (ui)i∈N ∈ P , the re-spective horizontal and vertical subspaces HuP and VuP of TuP , determinedby ω, coincide with the projective limits of their Banach counterparts, i.e.

HuP = lim←−

HuiPi, VuP = lim

←−VuiP

i.

Proof. In virtue of Proposition 3.2.2 and Corollary 3.2.6, the tangent spacesTuiP

ii∈N form a projective system with connecting morphisms and canoni-cal projections the differentials of the connecting morphisms and projections,respectively, of P = lim←−P

i. Since ωj and ωi are (pji, gji, idB)-related, for

every i, j ∈ N with j ≥ i, it follows that Tujpj(HujP

j) ⊆ HuiPi, thus

lim←−HuiPi is defined. In addition,

w = (wi) ∈ HuP ⇔ ω(u)(w) = 0

⇔ lim←−

(ωi(ui)

)((wi)) = 0

⇔ ωi(ui)(wi) = 0, i ∈ N

⇔ wi ∈ HuiP ⇔ w ∈ lim←−HuiPi,

which proves the statement for the horizontal spaces.

On the other hand,

VuP = TuP −HuP = lim←−TuiPi − lim←−HuiP

i

= lim←−

(TuiP

i −HuiPi)

= lim←−

VuiPi.

We turn now to the local connection forms of a limit connection. Thisis an advantageous approach within the framework of limit bundles, sincethey provide an equivalent way of studying limit connections, while theirdomains remain in the same fixed base. The first result in this context israther expected.


Proposition 4.2.8. Let ω = lim←−

ωi be a connection on (P,G,B, π) =

lim←−

(P i, Gi, B, πi). Let also sα : Uα → Pα∈I be the family of natural localsections of P over an open cover Uαα∈I of B. Then the local connectionforms ωαα∈I of ω are given by

(4.2.3) ωα = lim←−i∈N

ωiα; α ∈ I,

whereωiαα∈I

are the local connection forms of the factor connection ωi,for every i ∈ N.

Before the proof we notice that the limit (4.2.3) has the ordinary senseand not the point-wise sense of Definition 4.2.1 concerning connection forms.

Proof. By definition [see also (1.7.5)] and Proposition 4.1.6,

ωα = s∗αω : Uα −→ L(TUα, g),

ωiα = (siα)∗ωi : Uα −→ L(TUα, gi),

where siα := pi sα and pi : P = lim←−

P i → P i (i ∈ N) the canonical pro-jections. Because TUα is the same for all indices i ∈ N, as we explain indetail in § 6.2, the linear map bundle L(TUα, g) is a projective limit of vec-tor bundles. More precisely, L(TUα, g) = lim←−L(TUα, g

i), with connectingmorphisms

(4.2.4) λji : L(TUα, gj) −→ L(TUα, g

i) : f 7→ Tejgji f ; j ≥ i,

and canonical projections

(4.2.5) λi : L(TUα, g) −→ L(TUα, gi) : f 7→ Teg

i f ; i ∈ N,

both induced by the corresponding elements of the group G.Now, for every x ∈ Uα and every pair of indices (j, i) with j ≥ i, we see

that

(λji ωjα)(x) =(Tejg

ji ωjα)

(x)

= Tejgji ωj(sjα(x)) Txs

jα

= ωi(pji(pj(sα(x)))

) Tx(pji pj sα)

= ωi(pi(sα(x))

) Tx(pi sα)

= ωi(siα(x)) Txsiα = ωiα(x),

i.e λji ωjα = ωiα which means thatωiαi∈N

exists, for each α ∈ I. Anal-

ogous computations for the canonical projections show that λi ωα = ωiα.Therefore, we obtain (4.2.3).


The use of the local connection forms yields a much more importantresult characterizing all the connections of a limit principal bundle. Moreprecisely:

Theorem 4.2.9. Every connection on a projective limit principal bundle(P,G,B, π) = lim

←−(P i, Gi, B, πi) is the limit of a projective system of con-

nections.

The proof relies on a series of auxiliary results. To this end, we recall thefollowing facts needed below: Let Uαα∈I be an open cover of the base spaceB, over which we define the local trivializations of the bundles involved. Wedenote by sα : Uα → Pα∈I the family of natural local sections of the bundleP . By projecting to each factor bundle, we obtain the local sections

(4.2.6)siα := pi sα : Uα → P i

α∈I

,

Similarly, the local connection forms ωα = s∗αωα∈I of ω, induce the localforms [see also (4.2.5)]

(4.2.7) ωiα := λi ωα : Uα → L(TUα, gi), i ∈ N.

on the limit factors. With these notation, we obtain:

Lemma 4.2.10. For each i ∈ N, the following compatibility condition holdsfor every α, β ∈ I:

ωiβ = Adi((giαβ)−1

)ωiα +

(giαβ)−1

dgiαβ .

More explicitly, for every x ∈ Uαβ and v ∈ TxB,

ωiβ,x(v) = Adi((giαβ(x))−1

).ωiα,x(v) + Tx

(Li(gi

αβ(x))−1 g

iαβ

)(v),

where Adi denotes the adjoint representation of Gi, and Ligi : Gi → Gi is

the left translation of Gi by any gi ∈ Gi.

Proof. As in the ordinary case, the local connection forms of ω satisfy thecompatibility condition

ωβ,x(v) = Ad((gαβ(x))−1

)(ωα,x(v)) + Tx

(L(gαβ(x))−1 gαβ)(v),

for every x ∈ Uαβ, v ∈ TxB and α, β ∈ I (Lg denotes the left translation ofG). On the other hand, both the adjoint representation of G and the left


translations can be realized as projective limits of their Banach counterparts[see (3.4.1) and Proposition 3.2.5], i.e.

Ad(g) = lim←−

(Adi(gi)

); Lg = lim

←−

(Ligi), ∀ g = (gi) ∈ G.

Therefore, in virtue of (4.2.5),

ωiβ,x(v) =(λi(ωβ,x)

)(v) =

(Teg

i ωβ,x)(v)

= Tegi(Ad(gαβ(x)−1).ωα,x(v)

)

+ Tegi(Tx(L(gαβ(x))−1 gαβ

)(v))

= (Tegi lim←−

(Adi

(giαβ(x))−1

)).ωα,x(v)

+ Tx

(gi L(gαβ(x))−1 gαβ

)(v)

=(Adi

((giαβ(x))−1

) Teg

i).ωα,x(v))

+ Tx

(Li(gi

αβ(x))−1 g

i gαβ)

(v)

= Adi(giαβ(x))−1

).λi(ωα,x))(v)

+ Tx

(Li(gi

αβ(x))−1 g

iαβ

)(v)

= Adi((giαβ(x))−1

).ωiα,x(v)

+ Tx

(Li(gi

αβ(x))−1 g

iαβ

)(v).

A direct consequence of the the preceding is:

Corollary 4.2.11. An arbitrary connection ω ≡ ωα on P = lim←−

P i in-

duces on each factor bundle P i a connection ωi, whose local connection formsωiα are given by equality (4.2.7).

As the reader may guess, the previous connections ωii∈N will convergeto ω. But, to be able to verify Definition 4.2.1, we need also the following:

Lemma 4.2.12. For every i, j ∈ N with j ≥ i, the connections ωj and ωi

of Corollary 4.2.11 are (pji, gji, idB)-related.

Proof. By Proposition 1.7.1, it suffices to prove the analog of (1.7.20)

(4.2.8) gjiωjα = Ad(h−1α )ωiα + h−1α dhα,

for every α, β ∈ I, where the smooth maps hα : Uα → Gi are determined bythe equalities pji(sjα(x)) = siα(x) ·hα(x), for every x ∈ Uα. Since, by (4.2.6),

pji sjα = pji pj sα = pi sα = siα,


it follows that hα(x) = ei, for all x ∈ Uα. Therefore, (4.2.8) reduces to

(4.2.8′) gjiωjα = ωiα.

This is true, because, for every x as before, (4.2.7) and (4.2.5) yield:

gji ωjα,x = Tejgji λj(ωα,x)

= Tejgji Teg

j ωα,x[e = (ei)]

= Te(gji gj) ωα,x = Teig

i(ωα,x)

= λi(ωα,x) = ωiα,x,

which proves (4.2.8′).

We are now in a position to give the

Proof of Theorem 4.2.9. Let ω be an arbitrary connection on P = lim←−

P i.

By Corollary 4.2.11, we obtain the projective system of connections ωii∈N.Lemma 4.2.12 implies that this is a projective system of connections in thesense of Definition 4.2.1; therefore, the latter system determines a limitconnection, say, ω = lim

←−ωi. Then ω = ω, since both have the same local

connection forms. Indeed, for every x ∈ Uα and v ∈ TxB, we obtain:

ωα,x(v) = (s∗αω)x(v) = ω(sα(x))(Txsα(v)

)

=(lim←−ω

i(siα(x)

)) (lim←−Txs

iα(v)

)

= lim←−

(ωi(siα(x)

)(Txs

iα(v)

))

= lim←−

(((siα)∗ωi

)x(v))

= lim←−

(ωiα(x)(v)

)= ωα,x(v).

This concludes the proof of the theorem.In virtue of Theorem 4.2.9, it is reasonable to call the connections on a

plb-principal bundle plb-connections.

Remark 4.2.13. Let (Q,G,B, πQ) be an arbitrary Frechet principal bundlewith structure group G = lim

←−Gi. Assume that θ is a connection on Q. Then,

by Theorem 4.1.9, there is an isomorphism (f, idG, idB) of (Q,G,B, πQ)onto a plb-bundle

(lim←−

P i, lim←−

Gi, B, lim←−

πi). Obviously, the isomorphism f

determines a unique connection ω on lim←−

P i such that θ = f∗ω. On the other

hand, Theorem 4.2.9 implies that ω = lim←−

ωi, where ωi is a connection on

P i, for every i ∈ N. Therefore, θ is (f, idG, idB)-related with lim←−

ωi. Roughlyspeaking, one may say that (Q, θ) coincides—up to isomorphism—with theprojective limit of pairs (P i, ωi).


4.3 Parallel translations and holonomy groups

The aim of this section is to study the geometric objects in the title. Theproblems in solving differential equations on the Frechet models make impos-sible the direct application, to our framework, of the classical pattern thatso successfully ensures the existence of parallel translations (along curves ofthe base space), as well as the holonomy groups, for finite-dimensional orBanach bundles. The key approach giving a way out is based again on theuse of projective limits, by means of which we realize the previous notions,without solving the corresponding differential equations. However, some in-teresting deviations from the classical case emerge in the study of holonomygroups, which seem to drop out of the projective limit category, as explicitlyshown in Theorem 4.3.5 below.

Throughout this section we consider a plb-principal bundle

ℓ ≡ (P,G,B, π) = lim←−

(P i, Gi, B, πi) ≡ lim←−

ℓi,

endowed with a connection ω = lim←−

ωi, as in Definition 4.2.1.Referring to §1.1.9 [see also (1.9.1)], we denote by α : I → P the hori-

zontal lift of a smooth curve curve α : I → B. We recall that.

α(t) ∈ Hα(t)P or, equivalently, ω(α(t))( .

α(t))

= 0.

Proposition 4.3.1. Let α : [0, 1] → B be a smooth curve with α(0) = b,and take any u = (ui) ∈ π−1(b). Then there exists a unique horizontal liftα : [0, 1] → P of α, such that α(0) = u. In particular, α coincides with theprojective limit of horizontal lifts of α on the factor bundles ℓi.

Proof. The idea here is to exploit the fact that the Banach bundle analogholds true. Thus, the horizontal lift αj : [0, 1] → P j of α on P j , withinitial condition αj(0) = uj , exists. Now, composing αj with the connectingmorphism pji : P j → P i (j ≥ i), we see that:

i) pji αj projects to α:

πi (pji αj) = πj αj = α,

ii) pji αj is horizontal with respect to ωi. Indeed, setting ∂t = ddt

∣∣t

andapplying (1.1.11),

ωi((pji αj)(t)

)( .

˜(pji αj)(t))

= ωi(pji(αj(t))

)(Tt(p

ji αj)(∂t))

= ωi(pji(αj(t))

)(Tαj(t)p

ji( .

αj(t)))

4.3. Parallel translations and holonomy groups 163

or, since ωj and ωj are (pji, gji, idB)-related,

ωi((pji αj)(t)

)( .

˜(pji αj)(t))

= Tejgji(ωj(αj(t))

)( .

αj(t))

= Tejgji(0) = 0,

iii) pji αj and the horizontal lift αi : [0, 1] → P i of α on P i have thesame initial condition:

(pji αj

)(0) = pji(uj) = ui.

Therefore, pji αj = αi, (j ≥ i), implying that the smooth curve

α := lim←−

αi : [0, 1] −→ P

can be defined. This is a horizontal curve with respect to ω = lim←−

ωi, because

ω(α(t)

)( .α(t)

)= lim←−

ωi(αi(t))( .

αi(t))

= 0.

It is also a lift of α,

(π α)(t) =(πi(pi(α(t))

))i∈N

=(πi(αi(t)

))i∈N

(α(t))i∈N = α(t),

satisfying the initial condition

α(0) =(αi(0)

)i∈N

=(ui)i∈N

) = u.

As a result, α := lim←− αi is the desired horizontal lift of α. The uniqueness of

α is checked using similar arguments, since any other curve, with the sameproperties, coincides with α on every factor bundle P i.

The previous result allows us to define the parallel translation of fibresalong any smooth curve in the base of a limit (Frechet) principal bundle.

Proposition 4.3.2. Let ℓ ≡ lim←−

ℓi be a plb-principal bundle as in the be-ginning of this section and α : I = [0, 1] → B a smooth curve such thatα(0) = b0 and α(1) = b1. If αu : I → P denotes the horizontal lift of α withαu(0) = u, then the following assertions hold:

i) The parallel translation or displacement along α

τα : π−1(b0) −→ π−1(b1) : u 7→ αu(1),

is defined.ii) τα = lim←− τ iα, where τ iα are the corresponding parallel translations on

the factor bundles.


Proof. The first assertion is an immediate consequence of Proposition 4.3.1.The second one is essentially based on the following fact (see [Vas82], equal-ity (1.7.20) and the notations of § 1.7.5):

If two connections ω and ω′ defined on the (Banach) principal bundles(P,G,B, π) and (P ′, G′, B′, π′), respectively, are (f, ϕ, h)-related, then

f τα = τ ′α′ f∣∣π−1(α(0))

,

where τα is the parallel displacement along α : I → B, and τ ′α′ thatalong α′ = h α.

Therefore, since ωj and ωi are (pji, gji, idB)-related, for every j, i ∈ N withj ≥ i, it follows that

pji τ jα = τ iα pji∣∣(πj)−1(α(0))

,

which implies that the projective limit

lim←− τ iα : lim←−

((πi)−1(b0)

)= π−1(b0) −→ π−1(b1) = lim←−

((πi)−1(b1)

)

is defined. The latter coincides with the parallel translation along α. Indeed,for every i ∈ N and u = (ui) ∈ π−1(b0), composition with the canonicalprojection pi : P → P i yields:

(pi τα)(u) = pi(αu(1)

)= pi

(lim←− α

iui(1)

)=

pi((αiui(1)

)i∈N

)= αiui(1) = τ iα(ui) = (τ iα p

i)(u).

These equalities, along with Proposition 2.3.5, prove the last claim andconclude the proof.

Corollary 4.3.3. With the notations of Proposition 4.3.2, the followingassertions are true:

i) For every g ∈ G and u ∈ P , τα(u · g) = τα(u) · g; consequently, τα isa G-equivariant diffeomorphism.

ii) τα−1 = τ−1α , if α−1 is the inverse (or reverse) curve of α.iii) τβ∗α = τβ τα, if β ∗α is the (appropriate) composition of α followed

by β.

Proof. An easy extension of the analogous results on ordinary (Banach orfinite-dimensional) bundles in the vein of Proposition 4.3.2.

Remark 4.3.4. Obviously, all the previous results, concerning horizontallifts and parallel displacements, are valid for piecewise smooth curves, underthe obvious modifications.

4.3. Parallel translations and holonomy groups 165

The fact that every connection on a plb-principal bundle is the projec-tive limit of connections (Theorem 4.2.9) and the results of this section onthe parallel displacements lead to interesting and some unexpected results,concerning the corresponding holonomy groups. Before exhibiting them, werecall a few facts from § 1.9, adapted to the present context.

Let u = (ui) ≡ (ui)i∈N ∈ P and b = π(u) =(πi(ui)

)i∈N∈ B. If Cb is the

loop group at b and C0b ⊂ Cb the group of 0-homotopic loops, we define the

group homomorphism ku : τα |α ∈ Cb → G by

τα(u) = u · ku(τα), u ∈ π−1(b).

Then Φu := ku(τα) |α ∈ Cb is the holonomy group of ω with referencepoint u, while Φ0

u =ku(τα) |α ∈ C0

b

is the respective restricted holonomy

group.

Theorem 4.3.5. The holonomy groups of ω are related with the correspond-ing ones of the component connections as follows:

i) Φu ⊆ lim←−

Φiui⊆ Φu, Φ0

u ⊆ lim←−

(Φiui

)0 ⊆ Φ0u, where Φu (resp. Φ0

u) is

the closure of Φu (resp. Φ0u) in the topology of G.

ii) gi(Φu) = Φiui, gi(Φ0

u) = (Φiui)

0.

iii) If Φu is open or closed in G, then Φu = lim←−Φiui. As result, in both

cases, Φu is a Frechet topological group.

iv) If the canonical projections pji, gji of the projective systems of bundlesand groups involved are the natural embeddings, then the holonomy groupsare Banach-Lie groups.

Proof. i) Since (pji, gji, idB) is a principal bundle morphism, we have invirtue of Proposition 4.3.2,

r ∈ Φjuj⇒ τ jα(uj) = uj · r ⇒ pji(τ jα(uj)) = pji(uj) · gji(r)

⇒ τ iα(ui) = ui · gji(r) ⇒ gji(r) ∈ Φiui ;

in other words, gji(Φjuj

) ⊆ Φiui

, for every j ≥ i, thus we obtain the projectivesystem Φi

ui ; gjii,j∈N yielding the limit lim

←−Φiui . By similar arguments we

obtain the limit of restricted holonomy groups lim←−(Φiui)

0.

Now, if r = (ri) is an arbitrary element of Φu, there is a piecewisesmooth curve α in B, such that α(0) = α(1) = b and τα(u) = u · r. Thenpi(τα(u)) = pi(u) · gi(r), equivalently τ iα(ui) = ui · ri; hence, ri ∈ Φi

ui andr = (ri) ∈ lim←−Φi

ui . As a result,

(4.3.1) Φu ⊆ lim←−Φiui .


To show that lim←−

Φiui⊆ Φu, we proceed as follows: For any r = (ri) ∈

lim←−

Φiui

, there exist smooth curves βi : [0, 1]→ B (i ∈ N), such that

βi(0) = βi(1) = πi(ui) = b and τ iβi(ui) = ui · ri.

This is equivalent to saying that

(4.3.2) βii(1) = ui · ri,

where βii denotes the horizontal lift of βi to the bundle P i, with initial

condition βii(0) = ui. Denoting, analogously, by βji the horizontal lift of βito P j (j > i), with βji (0) = uj, Proposition 4.3.1 implies that the limit

γi = lim←−j∈N

(βji)

: [0, 1] −→ P

exists, and coincides with the horizontal lift of βi to P , with initial conditionγi(0) =

(βji (0)

)j∈N

= (uj)j∈N = u; that is,

γi = lim←−j∈N

(βji)

= (βi)u.

Moreover, for every index n ∈ N, there exists an element zn ∈ G such thatγn(1) = u · zn, since γn(1) and u belong to the same fibre π−1(b). Then,τγn(u) = u · zn and (zn)n∈N is a sequence of elements of Φu. We claimthat the previous sequence converges to r = (ri) ∈ lim

←−Φiui

. To this end, itsuffices to prove that

limn→+∞

(gi(zn)

)= ri, ∀ i ∈ N.

This is the case, since [see also (4.3.2)]

γn = lim←−j∈N

(βjn)⇒ pi(γn(1)) = βin(1) ⇒ pi(u · zn) = ui · ri

⇒ ui · gi(zn) = ui · ri ⇒ gi(zn) = ri.

Therefore, lim←−

Φiui ⊆ Φu, which, together with (4.3.1) proves the first as-

sertion concerning Φu. The inclusions referring to the restricted holonomygroup are proved in a similar way.

ii) From the proof of i) it is ensured that, for every ri ∈ Φiui , there is

at least one element r ∈ Φu such that gi(r) = ri; hence, Φiui ⊆ gi(Φu).

4.4. The curvature of a plb-connection 167

Conversely, in virtue of i), gi(Φu

)⊆ gi

(lim←−

Φiui

)= Φi

ui, thus we prove

assertion ii).

iii) If Φu is closed in G, then the result is an immediate consequence ofassertion i). On the other hand, if Φu is open, there is an open neighborhoodU of the unit e ∈ G such that U ⊂ Φu. Then, for an arbitrarily chosenr ∈ lim←−

Φiui , based on i), we check that there is a sequence (zn)n∈N in Φu

with limn→+∞

zn = r and

limn→+∞

(zn · r−1) = e

⇒ zn · r−1 ∈ U ⊂ Φu; n ≥ n0,

⇒ r ∈ Φu.

Therefore, Φu = lim←−

Φiui

in this case too. This completes the proof ofassertion iii).

iv) Under the assumptions of the assertion iv), we have for every u ∈ P :

P 1 ⊇ P 2 ⊇ . . . , and lim←−Pi =

⋂

i∈N

P i,

G1 ⊇ G2 ⊇ . . . , and lim←−Gi =

⋂

i∈N

Gi,

Φ1u ⊇ Φ2

u ⊇ . . . , and lim←−Φiu =

⋂Φiu.

Moreover, for every element s ∈ Φ1u, there exists a piecewise smooth curve

α : [0, 1]→ B such that α(0) = α(1) = b = π(u) and τ 1α(u) = u · s. However,

τα = lim←− τ iα, which, in the case of a nested sequence as above, means that

τ 1α|π−1(b) = τα. In this way, for every u ∈ π−1(b), one has τα(u) = u · s,

which implies that s ∈ Φu, thus Φ1u ⊆ Φu. Since already Φ1

u ⊆ Φu, weconclude that Φu = Φ1

u. This completes the proof.

Remark 4.3.6. Regarding the assertion i) of the previous theorem, we notethat Theorem 4.5.5, in the end of § 4.5, provides a (counter)example showingthat the equality Φu = lim

←−Φiui

is not necessarily true.

4.4 The curvature of a plb-connection

Let ω = lim←−ωi be a plb-connection (recall Theorem 4.2.9 and the termi-

nology induced after its proof) on the plb-principal bundle (P,G,B, π) =(lim←−P

i, lim←−Gi, B, lim←−π

i). The curvature of ω is defined by Ω = dω(h×h),


where h : TP → HP is the projection to the horizontal subbundle. The cur-vature of ωi is denoted by Ωi, for all i ∈ N.

To establish the point-wise convergence of (Ωi) to Ω, we first need thefollowing result, whose proof is quite lengthy and technical.

Lemma 4.4.1. With the previous notations, if(dωi)i∈N

are the exterior

differentials of(ωi)i∈N

, then, for every uj ∈ P j and j ≥ i, the followingdiagram is commutative:

TujPj × TujP

j dωj(uj)- gj

TuiPi × TuiP

i

Tujpji × Tujp

ji

?

dωi(ui)- gi

Tejgji

?

where ui = pji(uj).

Proof. Let x := πj(uj) ∈ B. We consider a chart (U, φ) of B containingx and a trivialization (U,Φ = lim

←−Φi) of P (we can take the same U for

all of them). We choose an arbitrary u = (ui) ∈ π−1(U) and a plb-chart(V, ψ) =

(lim←−V

i, lim←−ψi)

of G such that pr2(Φ(u)) ∈ V . Since uj and pj(u)

belong to the same fibre of P j , there exists a (unique) aj ∈ Gj such thatuj = pj(u) · aj , thus Φj

2(uj) = Φj

2(pj(u)) · aj, where Φj2 = pr2 Φ

j . Then the

pair(V j · aj , ψj ρj

(aj)−1

)is a chart of Gj containing Φj

2(uj). Here ρj

(aj )−1

denotes the right translation of Gj by (aj)−1.Also, the equality uj = pj(u) · aj implies that ui = pji(uj) = pji(pj(u) ·

aj) = pi(u) · gji(aj), thus the pair(V ·gji(aj), ψj ρj

gji(aj)−1

)is a chart of

Gi containing Φi2(ui). Therefore, we may construct the charts (W j , χj) and

(W i, χi) of P j and P i, respectively, with

W j := (Φj)−1(U × V j · aj

), W i := (Φi)−1

(U × V i · gji(aj)

),

χj :=(φ×

(ψj ρj

(aj)−1

))Φj , χi :=

(φ×

(ψi ρigji(aj)−1

))Φi,

as well as the associated charts(τ−1P j (W j), χj

),(τ−1P i (W i), χi

)of the tangent

bundles (TP j , P j , τpj) and (TP i, P i, τpi).

Denoting by ζji : Gj → Gi the connecting morphisms of the model G =lim←−

Gi of G, we check that

(4.4.1)(

idφ(U)×ζji) χj = χi pji, j ≥ i.


Indeed,(

idφ(U)×ζji) χj =

(idφ(U)×ζ

ji)(φ×

(ψj ρj

(aj)−1

)) Φj

=(φ×

(ζji ψj ρj

(aj )−1

)) Φj

=(φ×

(ψi gji ρj

(aj)−1

)) Φj

=(φ×

(ψi ρigji(aj)−1 g

ji))Φj

=(φ×

(ψi ρigji(aj)−1

)) (idU ×g

ji) Φj

=(φ×

(ψi ρigji(aj)−1

))Φi pji

= χi pji.

Analogously, for the above charts of the tangent bundles,

(4.4.2)(

idφ(U)×ζji)×(

idB ×ζji) χj = χi Tpji, j ≥ i.

Taking now two arbitrary tangent vectors X,Y ∈ TujPj, we define the

following vector fields:

ξj : W j −→ TP j : z 7→(χj)−1(

χj(z), χjuj

(X)),

ξi : W i −→ TP i : y 7→(χi)−1(

χi(y), χiui(Tujpji(X))

),

ηj : W j −→ TP j : z 7→(χj)−1(

χj(z), χjuj

(Y )),

ηi : W i −→ TP i : y 7→(χi)−1(

χi(y), χiui(Tujpji(Y ))

),

where, as in (1.1.4), χrur : TurPr → B×Gr is the linear isomorphism induced

by the chart (W r, χr). Then the definition of the tangent charts [see (1.1.7)]implies that

(4.4.3)ξj(uj) = X, ξi(ui) = Tujp

ji(X),

ηj(uj) = Y, ηi(ui) = Tujpji(Y ).

On the other hand, in virtue of (4.4.2), (4.4.1),

Tpji ξj = Tpji (χj)−1(χj , χj

uj(X)

)

=(χi)−1(

idφ(U)×ζji)×(

idB×ζji)(χj , χj

uj(X)

)

=(χi)−1((

idφ(U)×ζji) χj ,

(idB ×ζ

ji)(χjuj

(X)))

=(χi)−1(χi pji, χj

uj

(Tujp

ji(X)))

=(χi)−1(χi, χj

uj

(Tujp

ji(X))) pji

= ξi pji;


that is, ξj and ξi (j ≥ i) are pji-related. Similarly, ηj and ηi are pji-related,and so are [ξj , ηj ] and [ξi, ηi].

Now, going back to the desired commutativity of the statement, we ob-serve that, for arbitrary X,Y ∈ TujP

j as before,

(Tejg

ji dωj(uj))

(X,Y ) =(dωi(ui) (Tujp

ji × Tujpji))

(X,Y )

⇔ Tejgji(dωj(uj)(ξj(uj), ηj(uj))

)= dωi(ui)

(ξi(ui), ηi(ui)

)

⇔ Tejgji(dωj(ξj , ηj)(uj)

)= dωi(ξi, ηi)(ui)(∗)

⇔ Tejgji((ξj(ωj(ηj))− ηj(ωj(ξj))− ωj([ξj , ηj ]

)(uj)

)=

=(ξi(ωi(ηi))− ηi(ωi(ξi))− ωi([ξi, ηi]

)(ui)

⇔ Tejgji(Tujω

j(ηj)(ξj(uj)

))− Tejg

ji(Tujω

j(ξj)(ηj(uj)

))−

− Tejgji(ωj(ξj).[ξi, ηi](uj)

)=

= Tuiωi(ηi)

(ξi(ui)

)− Tui(ω

i(ξi)(ηi(ui)

)− ωi(ui).[ξi, ηi](ui).

Thus, applying (4.4.3), it suffices to verify the equalities

(4.4.4a) Tejgji(Tujω

j(ηj)(X))

= Tuiωi(ηi)

(Tujp

ji(X))

(4.4.4b) Tejgji(Tujω

j(ξj)(Y ))

= Tuiωi(ξi)

(Tujp

ji(Y ))

(4.4.4c) Tejgji(ωj(uj).[ξj , ηj ](uj)

)= ωi(ui).[ξi, ηi](ui)

The (pji, gji, idB)-relatedness of ωj and ωi implies

Tejgji(ωj(uj).[ξj , ηj ](uj)

)= ωi(ui)

(Tujp

ji[ξj , ηj ](uj))

= ωi(ui).[ξj , ηj ](pji(uj)) = ωi(ui).[ξi, ηi](ui),

thus proving (4.4.4c). Equality (4.4.4a) will be a consequence of the com-mutative diagram

TujPj Tujω

j(ηj)- Tωj(ηj )(Y )g

j

TuiPi

Tujpji

?

Tuiωi(ηi)

- Tωi(ηi)(Y i)gi

Tωj(ηj )(Y )(Tejgji)

?


where Y i := Tujgji(Y ), derived by differentiation of

(4.4.5) Tejgji ωj(ηj) = ωi(ηi) pji.

The preceding equality is satisfied, because, for every z ∈ P j ,

Tejgji.ωj(ηj)(z) = ωi(ηi).pji(z)

⇔ Tejgji(ωjz(η

j(z)))

= ωipji(z)(ηi(pji(z))

)

⇔ ωjpji(z)

(Tzp

ji(ηj(z)))

= ωipji(z)(ηi(pji(z))

)

the last equality being true since, as already proved, ηj and ηi are pji-relatedvector fields. Therefore, after the identifications

Tωj(ηj )(Y )gj ≡ gj , Tωi(ηi)(Y )g

i ≡ gi, Tωj(ηj)(Y )

(Tejg

ji)≡ T e

j

gji

and differentiation of (4.4.5), we obtain (4.4.4a). The proof of (4.4.4b) issimilar. This completes the proof of the lemma.

Lemma 4.4.2. Let dω be the exterior differential of ω = lim←−

ωi. Then, for

every u = (ui) ∈ P = lim←−Pi, the diagram

TuP × TuPdω(u)

- g

TuiPi × TuiP

i

Tupj × Tup

j

?

dωi(ui)- gi

Tegi

?

is commutative, where pi : P → P i and gi : G = lim←−Gi → Gi are the canon-

ical projections.

Proof. Following the main lines of the previous proof, we consider a chart(W,χ) =

(lim←−

W i, lim←−

χi)

of P at u, as well as the induced charts (τ−1P (W ), χ)

and ((τP i)−1(W i), χi)) of (TP,P, τP ) and (TP i, P i, τP i). For arbitraryX,Y ∈TuP , we define the vector fields

ξ : W −→ TP : z 7→ χ−1(χ(z), χu(X)

),

ξi : W i −→ TP i : y 7→(χi)−1(

χi(y), χiui(Tupi(X))

),

η : W −→ TP : z 7→ χ−1(χ(z), χu(Y )

),

ηi : W i −→ TP i : y 7→(χy)−1(

χi(y), χyui

(Tupi(Y ))

),


It is clear that

ξ(u) = X, ξi(ui) = Tupi(X), η(u) = Y, ηi(ui) = Tup

i(Y ).

Moreover, ξ, ξi and η, ηi are pi-related, and so are [ξ, η] and [ξi, ηi]. Thus,the desired commutativity of the diagram is equivalent to

Tegi(dω(u)(X,Y )

)= dωi(ui)

(Tup

i(X), Tupi(Y )

)

⇔ Tegi(dω(u)(ξ(u), η(u))

)= dωi(ui)

(ξi(ui), ηi(ui)

)

⇔ Tegi(dω(ξ, η)(u)

)= dωi(ξi, ηi)(ui)

The last equality is proved as its analog (∗) in the proof of Lemma 4.4.1

Corollary 4.4.3. With the notation of the preceding lemmata,

dω(u) = lim←− dωi(ui); u = (ui) ∈ P, i ∈ N.

Proof. Lemma 4.4.1 implies the existence of lim←−

dωi. The latter coincideswith dω(u) in virtue of Lemma 4.4.2 and Proposition 2.3.5.

We prove now the following main result of this section.

Theorem 4.4.4. Let ω = lim←−ωi be a connection on a plb-principal bundle

ℓ = lim←−

ℓi. If Ω is the curvature of ω and Ωi the curvature of ωi, for all

i ∈ N, then Ω(u) = lim←−

Ωi(ui), for every u = (ui) ∈ P .

Proof. First we check that, for every uj ∈ P j, the diagram

TujPj × TujP

j Ωj(uj)- gj

TuiPi × TuiP

i

Tujpji × Tujp

ji

?

Ωi(ui)- gi

Tejgji

?

is commutative, where ui = pji(uj). Indeed, for any X,Y ∈ TujPj, we

denote by Xh and Y h their horizontal components. Then the definition ofcurvature implies that

(Tejg

ji Ωjuj

)(X,Y ) = Tejg

ji(dωj(uj)(Xh, Y h)

)

= dωi(ui)(Tujp

ji(Xh), Tujpji(Y h)

)[Lemma 4.4.1]

= dωi(ui)(Tujp

ji(X)h, Tujpji(Y )h

)[see (1.7.19)]

= Ωi(ui)(Tujp

ji(X), Tujpji(Y )

)

=(Ωi(ui)

(Tujp

ji × Tujpji))

(X,Y ).


For convenience, we have used the same superscript h to denote the hori-zontal projections of all the connections.

The previous conclusion implies the existence of lim←−

Ωi. Thus, to provethe statement, it is sufficient to check also the commutativity of the diagram

TuP × TuPΩ(u)

- g

TuiPi × TuiP

i

Tupi × Tup

i

?

Ωi(ui)- gi

Tegi

?

for every i ∈ N. Indeed, for every X,Y ∈ TuP ,

(Tegi Ω(u))(X,Y ) = Teg

i(dω(u)(Xh, Y h)

)

= dωi(ui)(Tup

i(Xh), Tupi(Y h)

)[Lemma 4.4.2]

= dωi(ui)(Tup

i(X)h, Tupi(Y )h

)

= Ωi(ui)(Tujp

ji(X), Tujpji(Y )

)

=(Ωi(ui)

(Tup

i × Tupi))

(X,Y ).

In view of the preceding result, under the reservations following Defini-tion 4.2.1, we symbolically write Ω = lim

←−Ωi.

Corollary 4.4.5. Ω is horizontal and R∗gΩ = Ad(g−1Ω.

Proof. By definition, Ω is horizontal. The second property is proved usingthe same arguments as in the proof of Lemma 4.2.4.

As expected, Ω satisfies the structural equation. But this requires thefollowing:

Lemma 4.4.6. Given a connection ω = lim←−

ωi on P = lim←−

P i, it follows

that [ω, ω](u) = lim←−

[ωi, ωi]i(ui), for every u = (ui) ∈ P , where [ , ] and [ , ]i

are the brackets of g and gi, for all i ∈ N.

Proof. In analogy to previous lemmata, we need to prove the equalities

Tejgji [ωj, ωj ]j = [ωi, ωi]i

(Tujp

ji × Tujpji)

(4.4.6)

Teg [ω, ω] = [ωi, ωi]i (Tup

i × Tupi)

(4.4.7)


Indeed, for every X,Y ∈ TujPj, the definition of the bracket of forms

and the (pji, gji, idB)-relatedness of ωj, ωi (j ≥ i) yield:

(Tejg

ji [ωj , ωj ]j)

(X,Y ) = Tejgji([ωj(uj)(X), ωj(uj)(Y )]j

)

=[ωi(ui)

(Tujp

ji(X)), ωi(ui)

(Tujp

ji(Y )) ]i

= [ωi, ωi]i (Tujp

ji × Tujpji)

(X,Y ),

which leads to (4.4.6). The proof of (4.4.7) is similar, using the (pi, gi, idB)-relatedness of ω and ωi.

Proposition 4.4.7. With the previous assumptions, the curvature Ω =lim←−

Ωi satisfies Cartan’s (second) structure equation dΩ = dω + 12 [ω, ω].

Proof. In virtue of the structure equation for Banach principal bundles(1.8.1), Corollary 4.4.3 and Lemma 4.4.6, we find for every u = (ui) ∈ P :

Ω(u) = lim←−Ωi(ui) = lim←−

(dωi(ui) +

1

2[ωi, ωi]i(ui)

)=

= lim←−

(dωi(ui)

)+

1

2lim←−

([ωi, ωi]i(ui)

)= dω +

1

2[ω, ω](u).

Corollary 4.4.8. The curvature Ω = lim←−Ωi satisfies the Bianchi identitydΩ = [ω, ω]. Equivalently, DΩ = 0.

Proof. We proceed by applying the limit process to (1.8.4) and (1.8.4′). Notethat the horizontal projection h : TuP → HuP , u = (ui) ∈ P , coincides withlim←−

hi : TuiPi → HuiP

i by Proposition 4.2.7.

4.5 Flat plb-bundles

We shall discuss the relationship of flat connections with the holonomygroups and holonomy homomorphisms.

Throughout this section we fix a plb-bundle ℓ = (P,G,B, π) with con-nected base B. If ω = lim

←−ωi is a connection on a P with curvature (form)

Ω, then, according to the terminology of §§ 1.8.2 and 1.9, ω is called flat ifΩ = 0. In this case, the pair (P, ω) is called a flat bundle.

Proposition 4.5.1. The following assertions are equivalent:i) ω is flat.ii) Every ωi, i ∈ N, is flat.iii) The restricted holonomy group Φ0

u of ω is trivial, for any u ∈ P .

4.5. Flat plb-bundles 175

Proof. i)⇒ ii): The assumption implies that lim←−

Ωi(ui) = Ω(u) = 0, for

every u = (ui) ∈ P , thus

(4.5.1) Ωiui = Ωi(ui) = 0.

To prove ii), it suffices to show that Ωivi

(Xi, Y i) = 0, for arbitrary vi ∈ P i

and Xi, Y i ∈ HviPi. Indeed, if π(vi) = b ∈ B, we consider also an arbitrary

u = (ui) ∈ π−1(b). Since ui, vi ∈ (πi)−1(b), there is a (unique) a ∈ Gi suchthat vi = ui · a. Then, according to (1.7.4), TuiR

ia(HuiP

i) = HviPi; hence,

there are Xi, Y i ∈ HuiPi such that TuiR

ia(X

i) = Xi and TuiRia(Y

i) = Y i,where Ria is the right translation of P i by a ∈ Gi. Moreover, π pi = πyields Tuiπ

i Tupi = Tuπ, thus, restricting to the horizontal subspaces, we

obtain the commutative diagram

HuPTup

i- HuiP

i

TbB

Tuiπi

?

Tuπ-

Since Tuπ and Tuiπi are (top)linear isomorphisms, so is Tup

i. Therefore,there are X,Y ∈ TuP such that Tup

i(X) = Xi and Tupi(Y ) = Y i. The

previous considerations now yield:

Ωivi(X

i, Y i) = Ωiui·a

(TuiR

ia

(Tup

i(X)), TuiR

ia

(Tup

i(Y )))

=((Ria)

∗Ωi)ui

(Tup

i(X), Tupi(Y )

)

= Adi(a−1)(Ωiui

(Tup

i(X), Tupi(Y )

))

= Adi(a−1)(0) = 0.[by (4.5.1)]

ii)⇒ i): Obvious, in virtue of Theorem 4.4.4.i)⇒ iii): By ii) and the properties of the holonomy group in the Banach

context (see [Max72] and Theorem 1.9.1), we have that (Φiui

)0 = ei, for everyui ∈ P i. As a result, for any u = (ui) ∈ N, assertion i) of Theorem 4.3.5implies that

e ⊆ Φ0u ⊆ lim←−

(Φiui)

0 = lim←−

(ei) = e,

thus Φ0u = e.

iii)⇒ i): By the assumption and assertion ii) of Theorem 4.3.5,

(Φiui)

0 = gi(Φ0u

)= gi(e) = ei;

hence ωi is flat. This concludes the proof in virtue of i) ⇔ ii).


We turn now to the holonomy homomorphism of a flat connection ω =lim←−

ωi on a plb-bundle (P,G,B, π) with connected B. As we have sen in

§ 1.10, hω : π1(B) → G and hωi : π1(B) → Gi denote the correspondingholonomy homomorphisms of ω and ωi (i ∈ N).

Proposition 4.5.2. With the previous notations, hω = lim←−hωi .

Proof. In virtue of (1.10.1),

hω([α]) = ku(τα), hωi([α]) = kiui(τiα),

for any (fixed) u = (ui) ∈ P and every [α] ∈ π1(P ). Recall that ku is definedby τα(u) = u · ku(τα). The map kiui is defined analogously.

To prove the statement, first we check that

(4.5.2) gji hωj = hωi , j ≥ i.

To this end we observe that the existence of the limit τα = lim←− τ iα (see

Proposition 4.3.2) means that pji τ jα = τ iα pji, thus the bundle morphism

(pji, gji, idB) implies that

pji(τ jα(uj)

)=(uj · kj

uj(τ jα)

)= ui · gji

(kjuj

(τ jα))

⇒ τ iα = ui · gji(kjuj

(τ jα))

⇒ gji((kjuj

(τ jα))

= kiui(τiα)

⇒ (gji hωj )([α]) = hωi([α]),

for every [α] ∈ π1(B). This proves (4.5.2) ensuring, in turn, the existence oflim←−

hωi . By similar arguments, we see that gi hωj = hωi , for every i ∈ N.Therefore, Proposition 2.3.5 completes the proof.

Let G be a Frechet-Lie group. Following the construction describedin § 1.10, a homomorphism h : π1(B) → G determines a flat bundle (Q, θ)over B. In full terms, the bundle has the form ℓ = (Q,G,B, πQ), with totalspace Q = (B×G)/π1(B) and projection given by πQ([(x, s)]) = p(x). Moreprecisely, ℓ is associated, by h, to the principal bundle (B, π1(B), B, p) deter-mined by the universal covering B of B. On the other hand, if κ : B ×G→(B × G)/π1(B) is the canonical map, and ωo the canonical flat connectionon the trivial bundle ℓo = (B×G, B,pr1), then ωo is (κ, idG, p)-related withθ (actually this relationship determines θ). Recall that, if Ls is the lefttranslation of G by s ∈ G, ωo is given by

ωo(x, s).(X,Y ) = TsLs−1(Y ),


for every (x, s) ∈ B ×G and every (X,Y ) ∈ TxB × TsG.

In particular, if G = lim←−

Gi, we know (see Remark 4.2.13) that ℓ is—upto an isomorphism—a plb-principal bundle and θ is related with a projec-tive limit of connections. However, for such a G, we can define a concreteisomorphism relating (Q, θ) with a specific projective limit of Banach flatbundles. Before going into details, we define the group homomorphisms

(4.5.3) hi := gi h : π1(B) −→ Gi; i ∈ N,

where gi : G→ Gi, are the canonical projections. Thus h = lim←−

hi.

Theorem 4.5.3. Let B be a connected Banach manifold, G = lim←−

Gi a(Frechet) plb-group, and h : π1(B) ≡ π1(B,xo)→ G a Lie group morphism.If (Q, θ) is the principal bundle induced by h, then there is a principal bundleisomorphism (F, idG, idB) such that:

i) (Q,G,B, πQ) is (F, idG, idB)-isomorphic with(

lim←−Pi, G,B, lim←−π

i).

ii) θ is (F, idG, idB)-related with lim←−ωi,

where (P i, ωi), with P i ≡ (P i, Gi, B, πi), is the flat bundle induced by hi,for every i ∈ N.

Proof. We prove the properties of the statement in a series of steps.

Construction of lim←−

P i: Each hi induces a Banach principal bundle ℓi =

(P i, Gi, B, πi), equipped with a flat connection ωi. Now P i = (B×Gi)/π1(B)and πi([(x, s)] = p(x). The canonical flat connection ωio of ℓio = (B ×Gi, B,pr1) is (κi, idGi , p)-related with ωi, where κi : B×Gi → (B×Gi)/π1(B)is the canonical map. Analogously, ωio(x, a).(X,A) = TaL

ia−1(X,A), for ev-

ery (x, a) ∈ B × Gi and (X,A) ∈ TxB × TaGi, with Lia denoting the left

translation of Gi by a.

For every i, j ∈ N with j ≥ i, we define the map

(4.5.4) pji : P j −→ P i : [(x, s)]j 7→ [(x, gji(s))]i.

We recall that the connecting morphisms of G, gji : Gj → Gi, are grouphomomorphisms.


(a) pji is well-defined :

[(x, s)]j = [(y, t)]j

⇒ ∃ [γ] ∈ π1(B) : x, s) = (y, t) · [γ]

⇒ (x, s) =(y · [γ], hj([γ])−1 · t

)

⇒ (x, gji(s)) =(y · [γ],

(gji(hj([γ]))

)−1· gji(t)

)

⇒ (x, gji(s)) =(y · [γ], hi([γ])−1 · gji(t)

)

⇒ [(x, gji(s))]i = [(y, gji(t))]i.

(b) pji commutes with the bundle projections, i.e πi pji = πj:

(πi pji)([(x, s)]j

)= πi

([(x, gji(s))]j

)= p(x) = πj

([(x, s)]j

),

for every [(x, s)]j ∈ P j .

(c) pji is equivariant with respect to Gj and Gi:

pji([(x, s)]j · g

)= pji

([(x, s · g)]j

)=[(x, gji(s · g)

)]i

=[(x, gji(s) · gji(g)

)]i=[(x, gji(s)

)]i· gji(g)) = pji

([(x, s)]j

)· gji(g)

for every [(x, s)]j ∈ P j and g ∈ Gj .

(d) pji is smooth: This is checked locally as follows. Adapting the construc-tions of § 1.6.6(a) to the present context, we see that a local trivialization(U,Φ) of B, with Φ: B|U → U × π1(B) and corresponding natural sec-

tion σ : U → B, induces on each P i the local trivialization (U,Φi), with

Φi: P i|U → U ×Gi given by

Φi(

[(x, s)]i)

=(πi(x), hi([γ]) · s

)=(p(x), hi([γ]) · s

),

where [γ] ∈ π1(B) is determined by x = σ(p(x)) · [γ]. The inverse of the

trivializing map is given by(Φi)−1

(x, s) = [(σ(x), s)]i. Therefore, since

Φj(P j |U ) ⊆ P i|U , for every j ≥ i, we obtain the following diagram actually

yielding the local expression of pji, with respect to (U,Φj) and (U,Φ

i):

P j|Upji

- P i|U

U ×Gj

Φj

?

idU ×gji- U ×Gi

Φi

?


Indeed, for every (x, s) ∈ U ×Gj ,

(Φi pji

(Φj)−1)

(x, s) =(Φi pji

)([σ(x), s]j

)

= Φi(

[σ(x), gji(s)]i)

=(πi(σ(x)), hi([γ]) · gji(s)

)

=(p(σ(x)), hi([γ]) · gji(s)

)=(x, hi([γ]) · gji(s)

)

where [γ] ∈ Gi is now determined by σ(x) = σ(x) · [γ], thus [γ] = [exo ] (:the unit of π1(B)) and

(Φi pji

(Φj)−1)

(x, s) = (x, gji(s)) =(

idU ×gji)(x, s),

which proves the desired commutativity and, clearly, the smoothness of pji.

As a consequence of (a)–(d), F ji := (pji, gji, idB) : ℓj → ℓi is a princi-pal bundle morphism, and

ℓi;F ji

i,j∈N

is a projective system of Banachprincipal bundles, in virtue of Definition 4.1.1. Accordingly,

lim←− ℓi =

(lim←−P

i, G = lim←−Gi, B, lim←−π

i)

is a well-defined plb-bundle. We note that (4.1.1) in the aforementioneddefinition, concerning the local trivializations of the limit bundle, is now

satisfied by the local trivializations (U,Φi)i∈N of P i, derived each time by

the same pair (U,Φ) as the latter is running the set of all trivializations of B.

In fact, the existence of each lim←−

Φi

is ensured by the preceding commutativediagram.

Construction of F : We define the maps

F i : P −→ P i : [(x, s)] 7→ [(x, gi(s))]i,

for all i ∈ N. Working as in the case of pji, we easily check that F i is a well-defined, equivariant (with respect to G and Gi) map satisfying πi F i = πQ,for every i ∈ N. It is also smooth because locally identifies with idB ×g

i.Note that the charts of Q have a form analogous to that of P i mentionedearlier. Moreover, by the very definitions, pji F j = F i. Consequently, weobtain the smooth limit map F := lim

←−F i : P → lim

←−P i, commuting with the

projections πQ and lim←−

πi of ℓ and lim←−

ℓi, respectively, since

lim←−

πi F = lim←−

πi lim←−

F i = lim←−

(πi F i) = lim←−

πQ = πQ.

Hence, (F, idG, idB) is a G-B-(iso)morphism of (Q,G,B, πQ) onto lim←− ℓi.


Construction of lim←−

ωi: Let ωi denote the flat connection on ℓi, induced by

hi. Since the ωio is (κi, idGi , p)-related with ωi, it follows that

(4.5.5) ωio(x, s) = ωi([(x, s)]i) T(x,s)κi, (x, s) ∈ B ×Gi.

To show that ωj and ωi (j ≥ i) are (pji, gji, idB)-related, it suffices to checkthe commutativity of the diagram

TujPj ωj(uj)

- gj

TuiPi

Tujpji

?ωi(ui)

- gi

Tejgji

?

for every uj = [(x, s)]j ∈ P j, with ui = pji(uj) = [(x, gji(s))]i. Indeed, forany tangent vector W ∈ TujP

j, there are X ∈ TxB and Y ∈ TsGj such that

T(x,s)κj(X,Y ) = W , because κj is a submersion. Hence, in virtue of the

analog of (4.5.5) for ωjo and the definition of the canonical flat connectionωjo, we obtain:

(4.5.6)

(Tejg

ji ωj(uj))(W ) = Tejg

ji(ωj([(x, s)]j) T(x,s)κ

j .(X,Y ))

= Tejgji(ωjo(x, s).(X,Y )

)

= Tejgji(TsL

js−1(Y )

)

= Ts(gji Lj

s−1)(Y ).

Similarly,(ωi(ui) pji

)(W ) = ωi([(x, gji(s))]i)

(T[(x,s)]jp

ji(T(x,s)κj(X,Y )

)

= ωi([(x, gji(s))]i)(T(x,s)(p

ji κj)(X,Y ))

or, in virtue of pji κj = κi (idB ×gji),

(4.5.7)

(ωi(ui) pji

)(W ) =

(T(x,s)

(κi (idB ×g

ji))(X,Y )

)

= ωi([(x, gji(s))]i)(T(x,gji(s))κ

i(X,Tsg

ji(Y )))

= ωio(x, gji(s))(X,Tsg

ji(Y ))

= Ts(Ligji(s)−1 g

ji)(Y ).

Since, by (3.4.3), gjiLjs−1 = Li

gji(s)−1gji, equalities (4.5.6) and (4.5.7) prove

the commutativity of the preceding diagram, implying in turn that ωii∈N


is a projective system of connections as in Definition 4.2.1. Consequently,ω := lim

←−ωi is a connection on lim

←−P i. In virtue of Proposition 4.5.1 ω is flat.

Relatedness of θ and lim←−ωi: For any [(x, s)] ∈ Q and W ∈ T[(x,s)]Q, there

is a pair (X,Y ) ∈ TxB × TsG such that T(x,s)κ(X,Y ) = W . Then, the(κ, idG, p)-relatedness of θ and ωo implies

(4.5.8)θ[(x,s)](W ) = θκ(x,s)

(T(x,s)κ(X,Y )

)

= ωo(x, s).(X,Y ) = TsLs−1(Y ).

Analogously,

(4.5.9) (F ∗ω)[(x,s)](W ) = ωF ([(x,s)])

((T[(x,s)]F T(x,s)κ

)(X,Y )

).

On the other hand, taking into account the identifications of Proposi-tions 3.2.2 and 3.2.5, F = lim

←−F i implies that

T[(x,s)]F (T(x,s)κ(X,Y )) =(T[(x,s)]F

i(T(x,s)κ(X,Y )

))i∈N

=(T(x,s)(F

i κ)(X,Y ))i∈N

=(T(x,s)

(κi (idB ×g

i))(X,Y )

)i∈N

=(T(x,gi(s))

(X,Tsg

i(Y )))

i∈N.

Substituting the latter in (4.5.9), we find that

(F ∗ω)[(x,s)](W ) = (F ∗ω)[(x,s)]

((T(x,s)

(X,Tsg

i(Y )))

i∈N

)

=(ωiκi(x,gi(s))

(T(x,s)

(X,Tsg

i(Y ))))

i∈N

=((

(κi)∗ωi)(x,gi(s))

(X,Tsg

i(Y )))

i∈N

or, by the relatedness of ωio and ωi,

(4.5.10)(F ∗ω)[(x,s)](W ) =

(ωio(x, gi(s)).

(X,Tsg

i(Y )))

i∈N

=(Tgi(x)L

igi(s)−1(Tsg

i(Y )))i∈N

.


However,the aforementioned identifications, Corollary 3.2.6 and equality(3.4.1) imply that Y = (Tsg

i(Y ))i∈N; therefore,

TsLs−1(Y ) = TsLs−1(Y )((Tsg

i(Y ))i∈N)

=(

lim←−

Tgi(s)Ligi(s)−1

)((Tsg

i(Y ))i∈N)

[see (3.4.1)]

=(Tgi(s)L

igi(s)−1

(Tsg

i(Y )))

i∈N.

Consequently, in virtue of the preceding, equalities (4.5.8) and (4.5.10) yieldthe desired relatedness and complete the proof.

Using some of the tools applied in the previous study of flat bundles,we are in a position to give the counterexample mentioned in Remark 4.3.6.Before this, we need a few technicalities:

Set B := R2−(ν, 0) | ν ∈ N. ThenB is a connected manifold whose fun-damental group π1(B) is free with countable many generators (x1, x2, . . .).For every i ∈ N, we define the map ai : π1(B) → R, where ai(γ) equals thesum of exponents of the xi generator of γ (for simplicity we set γ = [γ]).Also, we define the maps

hi : π1(B) −→ Ri : γ 7→(a1(γ), a2(γ)), . . . , ai(γ)

),

for every i ∈ N.

With the previous notations we prove:

Lemma 4.5.4. The following assertions are true:

i) Each hi is a Lie group homomorphism.

ii) hi (π1(B)) = Zi.

iii) The homomorphism h := lim←−

hi : π1(B)→ R∞ is defined.

iv) h (π1(B)) 6= lim←−hi (π1(B)).

Proof. i) Considering, as usual, π1(B) with the discrete differential struc-ture, it suffices to show that every ai is a group homomorphism. Indeed,let

β = xεii1 · xε2i2· · · xεkik , γ = xδ1j1 · x

δ2j2· · · xδλjλ

be any elements of π1(B). If x1 appears in β at the positions ip1 , ip2 , . . . , ipxand in γ at the positions jq1 , jq2 , . . . , jqy , then for

β · γ = xεii1 · xε2i2· · · xεkik · x

δ1j1· xδ2ij2 · · · x

δλjλ


the same x1 appears at the ip1 , ip2 , . . . , ipx , jq1 , jq2 , . . . , jqy positions. There-fore,

a1(β · γ) = εp1 + εp2 + · · ·+ εpx + δq1 + δq2 + · · · δqy = a1(β) + a1(γ).

Similar arguments hold for every ai, i ≥ 2.

ii) Obviously, hi (π1(B)) ⊆ Zi since ai(γ) is an integer, for every γ ∈(π1(B)). Conversely, for an arbitrary element u = (k1, k2, . . . , ki) ∈ Zi, it

follows that u = hi(xk11 · x

k22 · · · x

kii

)∈ hi (π1(B)).

iii) It suffices to show that the diagram

p1(B)hj

- Rj

Ri

ρji

?

hi-

is commutative, for every j ≥ i, where ρji : Rj → Ri : (t1, t2, . . . , tj) 7→(t1, t2, . . . , ti) are the connecting morphisms of lim←−Ri = R∞. This is thecase, because

(ρji hj

)(γ) = ρji

(a1(γ), a2(γ), . . . , aj(γ)

)

=(a1(γ), a2(γ), . . . , ai(γ)

)

= hi(γ).

iv) We observe that lim←−hi (π1(B)) = lim←−Zi = Z∞. Therefore, if we

assume that h (π1(B)) = lim←−

hi (π1(B)), we would have that h (π1(B)) =Z∞, thus there would be some γ ∈ π1(B) such that h(γ) = (1, 1, · · · ) or,equivalently, ai(γ) = 1, for every i ∈ N. This means that all the generatorsxi (i ∈ N) would appear in the representation of γ, which is absurd. As aresult, we do have that h (π1(B)) 6= lim

←−hi (π1(B)).

Theorem 4.5.5. There exists a plb-connection ω = lim←−ωi whose holonomy

group does not coincide with the projective limit of the holonomy groups ofthe factor connections.

Proof. We consider the connected manifold B := R2 − (ν, 0) | ν ∈ N, theplb-group G := (R∞,+) = lim←−G

i, where Gi =(Ri,+

), as well as the Lie

group morphism h := lim←−hi : π1(B) → G defined in the previous lemma.


Then, as in Theorem 4.5.3 and the discussion before it, h determines a plb-principal bundle (Q, θ) which is (F, idG, idB)-isomorphic with the plb-bundle(lim←−

P i, lim←−

ωi), induced by the family (hi). Fixing the points q ∈ Q and

F (q) =: u = (ui), we obtain the corresponding holonomy groups of θ andω, Φq and Φu. It is an immediate consequence of (1.9.5) (see also [KN68,§ II.6]) that Φq = Φu. On the other hand, for each (P i, hi), induced byhi, we obtain the holonomy group Φi

ui . We also have that Φq = h (π1(B))and Φi

ui = hi (π1(B)). Therefore, if we assume that Φu = lim←−Φiui , then we

would have that

h (π1(B)) = Φq = Φu = lim←−Φiui = lim←−h

i (π1(B)) ,

which contradicts assertion iv) of Lemma 4.5.4. Hence, Φu 6= lim←−

Φiui , thus

verifying the statement for the (flat) connection ω = lim←−

ωi on the bundle(lim←−

P i, G,B, lim←−

πi).

Chapter 5

Projective systemsof vector bundles

Vector bundles of fibre type a Frechet space F are difficult to handle becauseof the pathology of the structure group GL(F). Therefore, the aim of thepresent chapter is to propose a method to address many relevant issues byusing the language of projective limits.

The structure of vector bundles induced by projective limits of Banachvector bundles is a bit more complicated than that of the principal bundles(studied in Chapter 4), and it is completely determined by the group H0(F)to be defined in Section 5.1 below, a group replacing the pathological GL(F).

Important examples are the infinite jets of sections of a Banach vec-tor bundle and spaces of linear and antisymmetric maps such as L(TB,F),Ak(TB,F), fully described in Chapter 6.

The important geometric notion of a (linear) connection in the presentcontext will be deferred to Chapter 7.

5.1 A particular Frechet group

Before delving into the structure of Frechet vector bundles in our framework,we introduce a particular type of a Frechet topological group which willessentially play the role of the structural group of the bundles under study.

As usual, F = lim←−

Ei is a Frechet space, projective limit of the Banach

spaces Ei, i ∈ N. Referring to (2.3.3), we set

(5.1.1) Hi(F) := Hi(F,F),

185

186 Chapter 5. Projective systems of vector bundles

and

(5.1.2) Hi0(F) := Hi(F)⋂ i∏

j=1

Lis(Ej),

where Lis(E) (also denoted by GL(E)) is the group of invertible elements ofL(E). Similarly, recalling (2.3.12), we set

(5.1.3) H0(F) := H(F)⋂ ∞∏

j=1

Lis(Ej),

Proposition 5.1.1. The following assertions are true:i) Every Hi0(F), i ∈ N, is a Banach-Lie group modelled on Hi(F), while

H0(F ) is a topological group with the relative topology of H(F).ii) The projective limit lim

←−Hi0(F) exists and coincides, up to an isomor-

phism of topological groups, with H0(F). Thus H0(F ) is a Frechet topologicalgroup.

Proof. Since each Lis(Ej) is open in L(Ej), (5.1.2) shows that Hi0(F) is anopen subset of Hi(F), i ∈ N. Clearly, Hi0(F) is a group with multiplication(the continuous bilinear) composition map:

(f1, . . . , f i

)·(g1, . . . , gi

):=(f1 g1, . . . , f i gi

).

Therefore, Hi0(F) is a Banach-Lie group modelled on Hi(F), for every i ∈ N.Assertion ii) is based on the proof of Theorem 2.3.10: Denoting again

by hji : Hj(F)→ Hi(F) (j ≥ i) the analogs of (2.3.6), we see that the maps

hji0 := hji∣∣Hj

0(F): Hj0(F) −→ Hi0(F)

are morphisms of topological groups satisfying the equalities hjk0 = hik0 hji0 ,

for every i, j, k ∈ N with j ≥ i ≥ k; hence, we obtain the projective system(Hi0(F);hji0

)whose limit lim

←−Hi0(F) is a topological group.

On the other hand, H0(F) is a topological group, with the obvious mul-tiplication (f i)i∈N ·(g

i)i∈N := (f i gi)i∈N, and topology the relative topologyas a subset of H(F). Also, we observe that the maps

hk0 := hk∣∣H0(F)

: H0(F) −→ Hk0(F) :(f i)i∈N7→(f1, . . . , fk

); k ∈ N,

are morphisms of topological groups satisfying hji0 hj0 = hi0. Thus we obtain

the morphism of topological groups

h0 := lim←−hi0 : H0(F) −→ lim←−H

i0(F).

5.2. Projective systems and Frechet vector bundles 187

Following the proof of Theorem 2.3.10, we check that h0 is a bijection, andh−10 is continuous as the restriction of the continuous map h−1 : lim

←−Hi(F)→

H(F) to lim←−Hi0(F) . Hence, H0(F) and lim

←−Hi0(F) can be identified as topo-

logical groups. Since lim←−Hi0(F) is a Frechet topological group, we conclude

the proof.

It should be noted that, although each Hi0(F) is a Banach-Lie group,H0(F) ≡ lim

←−Hi0(F) is not necessarily a Frechet-Lie group in the sense of

Proposition 3.4.1, since the projective system of the previous groups does notnecessarily satisfy the conditions of a plb-manifold in the sense of Definition3.1.2. More precisely, condition (2) of the aforementioned definition is inquestion because the projective limit of the open setsHi0(F) is not necessarilyopen.

5.2 Projective systems and Frechet vector bundles

Let (Ei, B, πi), i ∈ N, be Banach vector bundles (over the same base B)of fibre type Ei, respectively, and let f ji : Ej → Ei (j ≥ i), be vectorbundle morphisms over the identity. For every open U ⊆ B, we set EiU :=(πi)−1(U).

Definition 5.2.1. By a projective system of Banach vector bundlesEi; f jii,j∈N we mean a countable family (Ei, B, πi); f jii,j∈N satisfyingthe following additional conditions:

(PVB. 1) The Banach spaces Ei form a projective system with connectingmorphisms ρji.

(PVB. 2) For each x ∈ B, there exist local trivializations (U, τ i) of Ei (withthe same U), i ∈ N, such that x ∈ U and

τ i f ji = (idU ×ρji) τ j , j ≥ i.

Condition (PVB. 1) implies that F := lim←−Ei exists and has the structureof a Frechet space. The equality figuring in (PVB.2) is pictured in the nextcommutative diagram:

EjUf ji

- EiU

U × Ej

τ j

?

idU ×ρji- U × Ei

τ i

?


We note that Conditions (PVB. 1)–(PVB. 2), which enrich the set-theoretic definition of a projective system, are necessary to assure that theinduced projective limits remain within the category of vector bundles.

Projective systems of vector bundles, as in Definition 5.2.1, are also calledstrong (see, e.g., [Gal98]). However, we adhere to the simpler terminologyof our definition.

Proposition 5.2.2. If Ei; f jii,j∈N is a projective system of Banach vectorbundles, then E := lim←−E

i exists and is a plb-manifold; hence, by Proposition3.1.3, E is a Frechet manifold.

Proof. The existence of E follows from equality

(5.2.1) f jk = f ik f ji; j ≥ i ≥ k,

verified in the following way: Let u ∈ Ej, with πj(u) = x ∈ B, and let(U, τ i) be the trivializations satisfying condition (PVB. 2). Since u ∈ EjU ,f ji(u) ∈ EiU , and f ik(f ji(u)) ∈ EkU , we obtain

τk(f ik(f ji(u))

)=(

idU ×ρik)(τ i(f ji(u))

)

=((

idU ×ρik)(

idU ×ρji))

(τ j(u))

=((

idU ×ρjk) τ j

)(u) = τk

(f jk(u))

),

which leads to (5.2.1).If B is the (Banach space) model of the base B, each bundle Ei has

local charts with model B×Ei. Obviously,B× Ei; idB×ρ

ji

is a projectivesystem of Banach spaces inducing the Frechet space lim←−(B× Ei) = B× F.

The smooth structure of E is defined as follows: Let any u = (ui) ∈ E.Since πj(uj) = πi(f ji(uj)) = πi(ui), we set πi(ui) =: x, for every i ∈ N.If (U, τ i), i ∈ N, are the trivializations of (PV. 2) with x ∈ U , shrinkingU (if necessary), we may also consider the chart (U, φ) of B at x. Then,(EiU ,Φ

i) :=(EiU , (φ× idEi) τ i

)is a chart of Ei. It is clear that uj ∈ EjU ,

f ji(EjU ) ⊆ EiU , and the diagram

EjUf ji

- EiU

φ(U)× Ej

Φj

?

idφ(U)×ρji- φ(U) × Ei

Φi

?


is commutative, as a consequence of equalities

(idB×ρ

ji)Φj =

(idB×ρ

ji) (φ× idEj ) τ j

= (φ× ρji) τ j

= (φ× idEi) (idU ×ρji) τ j

= (φ× idEi) τ i f ji = Φi f ji.

Then,EiU , f

jii∈N

and Φii∈N are projective systems and their respective

limits lim←−EiU , lim←−Φi exist.

Similarly, the vb-morphisms f ji imply that πi f ji = πj, for all indicesi, j with j ≥ i, thus we define the continuous map

π := lim←−

πi : E = lim←−

Ei −→ B.

As a result,lim←−

EiU = lim←−

(πi)−1(U) = π−1(U),

which means that lim←−EiU is open in E. On the other hand,

lim←−

((Φ(EiU )

)= lim←−

(φ(U)× Ei

)= φ(U)× F;

that is, lim←−

(Φ(EiU )

)is an open subset of B × F. The last arguments prove

that the pairs(

lim←−EiU , lim←−Φ

)determine projective limit charts (see Defini-

tion 3.1.1) inducing the structure of a plb-manifold (Definition 3.1.2) on E.This concludes the proof.

The previous proposition shows that Ei; f jii,j∈N is a projective systemin the ordinary sense. However, the conditions of Definition 5.2.1 ensuresomething more: the structure of a Frechet manifold on lim

←−Ei.

Corollary 5.2.3. The map π := lim←−

πi : lim←−

Ei → B is smooth as a pls-map.

Proof. Immediate consequence of the equalities πi f ji = πj (j ≥ i) andProposition 3.1.8.

Definition 5.2.4. Let Ei; f jii,j∈N be a projective system of Banach vectorbundles. The triplet

(E := lim←−E

i, B, π := lim←−πi)

is called a plb-vectorbundle.

The term (plb-) vector bundle is justified by the following:

Theorem 5.2.5. A plb-vector bundle (E,B, π) is a Frechet vector bundle


Proof. We have already seen that E is a Frechet manifold and π a smoothmap.We need to show the Frechet analogs of (VB. 1)–(VB. 3) in § 1.4.1.

First observe that, for each x ∈ B,

(Eix := (πi)−1(x); f ji|

Ejx

)i,j∈N

is a projective system of Banach spaces, since f ji|Ej

x: Ejx → Eix are contin-

uous linear maps. Hence, lim←−Eix is defined and

Ex := π−1(x) =(lim←− (πi)−1

)(x) = lim←−

((πi)−1(x)

)= lim←−E

ix.

Indeed, if (U, τ i) are the local trivializations of (PVB 2), with x ∈ U ,then (see also the commutative diagram following Definition 5.2.1), for everyi, j ∈ N with j ≥ i, we obtain the diffeomorphism

τ := lim←−

τ i : π−1(U) = lim←−

((πi)−1(U)

)−→ lim←−

(U × Ei) = U × F.

Now, for every u = (ui) ∈ π−1(U),

(pr1 τ)(u) = pr1((τ i(ui))i∈N

)

= pr1((πi(ui),pr2(τ i(ui))

)i∈N

)

=(πi(ui)

)i∈N

= π(u);

that is, the following diagram is commutative:

π−1(U)τ

- U × F

U

pr1π-

On the other hand, if u = (ui) ∈ Ex (thus ui ∈ Eix, for every i ∈ N), similararguments show that

τ(u) =(τ ix(ui)

)i∈N

=(

lim←− τix

)(u);

hence,

τx = lim←− τix : Ex = lim←−E

ix −→ lim←−Ei = F


is an isomorphism of Frechet spaces. Therefore, for each x ∈ B, there is atrivialization (U, τ) of E satisfying the Frechet analog of (VB. 1).

For the proof of (the analog of) (VB. 2), we consider two trivializations(U, τ) and (V, σ), with U ∩ V 6= ∅, τ = lim

←−τ i, σ = lim

←−σi. Then, by the

previous arguments, τx σ−1x : F → F is an isomorphism of Frechet spaces,

for every x ∈ U ; hence, τx σ−1x ∈ Lis(F) ⊂ L(F).

To complete the proof it remains to show that the transition functions

(5.2.2) TUV : U ∩ V −→ L(F) : x 7→ TUV (x) := τx σ−1x ,

are smooth, for all overlapping trivializations of E.

We note that L(F) is not necessarily a Frechet space. Actually, it is aHausdorff locally convex topological vector space, whose topology is deter-mined by the uniform convergence on the bounded subsets of F. Hence,for the desired smoothness, we first consider the map [see also (2.3.12) and(2.3.3)]

(5.2.3) T ∗UV : U ∩ V : −→ H(F) : x 7→ T ∗UV (x) :=(τ ix (σix)−1

)i∈N

,

and we claim it is smooth. To this end, we define the maps

(5.2.4)(T ∗UV )k : U ∩ V −→ Hk(F), with

(T ∗UV )k(x) :=(τ1x (σ1x)−1, . . . , τ ix (σkx)−1

).

Since τkx (σkx)−1 = (T ∗UV )k (1 ≤ k ≤ i), where (T ∗UV )k is the transi-tion function of the Banach vector bundle (Ek, πk, B) over U ∩ V , then(T ∗UV )k : U ∩ V → L(Ek) is smooth, and so is (T ∗UV )i [recall from theproof of Theorem 2.3.10 that Hi(F) is a Banach space as a closed sub-space of

∏ik=1L(Ek)]. Using now the notations of Section 5.1, we routinely

verify that hji (T ∗UV )j = (T ∗UV )i, thus lim←−

(T ∗UV )i is defined. Since also

hi T ∗UV = (T ∗UV )i, Proposition 2.3.5 implies that T ∗UV = lim←−

(T ∗UV )i, fromwhich the smoothness of T ∗UV follows.

Next we see that

(5.2.5) TUV = ε T ∗UV ,

where ε : H(F) → L(F) is the map (f i)i∈N 7→ lim←−

f i. Because the latteris continuous linear, thus smooth in the sense of our differentiability (seeSection 2.2), we conclude that TUV is indeed a smooth map.


Remarks 5.2.6. 1) From equalities (5.2.3) and (5.1.3) we see that, in par-ticular, the maps T ∗UV take values in H0(F), i.e.,

T ∗UV : U ∩ V −→ H0(F) ⊂ H(F),

while, from (5.2.4) and (5.1.2),

(T ∗UV )i : U ∩ V −→ Hi0(F) ⊂ Hi(F).

Since H0(F) is only a topological group, T ∗UV is not smooth as an H0(F)-valued map. However, this does not affect the structure of (E,B, π), becauseonly the smoothness of T ∗UV as an H(F)-valued map matters. The signifi-cance of T ∗UV will be further illustrated in subsequent results.

2) Using the arguments of the last part of the preceding proof, we seethat, in fact,

T ∗UV = lim←−

(T ∗UV )i : U ∩ V −→ lim←−Hi0(F) = H0(F).

3) To remember that T ∗UV are smooth when they are considered asH(F)-valued maps, but not smooth as H0(F)-valued ones, we say that T ∗UV are generalized smooth maps in H0(F).

Definition 5.2.7. The local trivializations of (E,B, π) of the form (U, τ),with τ = lim τ i, are called plb-trivializations. Analogously, the mapsT ∗UV : U ∩ V → H0(F), derived from such trivializations are called plb-transition maps. They clearly satisfy equality

T ∗UV (x) = T ∗UW (x) T ∗WV (x); x ∈ U ∩ V ∩W,

thus T ∗UV is a cocycle with values in H0(F).

More precisely:

Definition 5.2.8. Let B be a Banach manifold, C = Uαα∈I an open coverof B, and let F = lim←−Ei be a Frechet space, where Ei; ρji is a projectivesystem of Banach spaces. An H0(F)-valued cocycle of B, with respect tothe cover C, is a family of smooth maps

T ∗αβ : Uαβ := Uα ∩ Uβ −→ H(F),

such that

T ∗αβ(x) ∈ H0(F); x ∈ Uαβ ,

T ∗αγ(x) = T ∗αβ(x) T ∗βγ(x); x ∈ Uαβγ .


Theorem 5.2.9. Let T ∗αβ be an H0(F)-valued cocycle over an open coverC = Uαα∈I of a Banach manifold B. Then there exists a unique—up toisomorphism—plb-vector bundle with plb-transition maps T ∗αβ.

Proof. For every k ∈ N and α, β ∈ I, we define the map

T kαβ := prk T∗αβ : Uαβ −→ L(Ek),

whereprk : H(F) −→ L(Ek) : (f i)i∈N 7→ fk.

T kαβ being the composite of smooth maps (in the sense of § 2.2), is also

smooth in the sense of Banach manifolds. Thus T kαβα,β∈I is a cocycle of

B over C with values in Lis(Ek) ⊂ L(Ek) and determines a Banach vectorbundle (Ek, B, πk), whose transition maps are precisely T kαβα,β∈I .

According to the construction expounded in § 1.4.2,

Ek =⋃

α,β∈I

(α × Uα × Ek

)/∼k

where(α, x, u) ∼k (β, y, v) ⇔ x = y, v = T kβα(u).

Denoting by [(α, x, u)]k the equivalence class of (α, x, u), we define the pro-jection πk : Ek → B by setting πk([(α, x, u)]k) := x, and the trivializations(Uα, τ

kα), with

τkα : (πk)−1(Uα) −→ Uα × Ek : τkα([(β, x, u)]k) := (x, T kαβ(x)(u).

For every i, j ∈ N with i ≤ j, we further define the map

f ji : Ej −→ Ei : [(α, x, u)]j 7→ [(α, x, ρji(u))]i.

The reader will have no difficulty to verify that f ji is well-defined and πi f ji = πj; thus, with respect to (Uα, τ

jα), we have that f ji

(EjUα

)⊆ EiUα

. Itis also easy to check that

(idUα ×ρji) τ jUα

= τ iUα f ji; j ≥ i,

from which follows that f ji is smooth, and its restriction to every fibre Ejxis precisely the continuous linear map

f ji∣∣Ej

x=(τ iα,x

)−1 ρji τ jα,x.


Moreover, by appropriate restrictions, we may consider the trivializations(over the same Uα ∈ C) (Uα, τ

jα) and (Uα, τ

iα) of Ej and Ei, respectively;

hence, the map

τ jiα : Uα ∋ x 7−→ τ iα,x fji∣∣Ej

x(τ jα,x

)−1

is constant; in fact, τ jiα (x) = ρji, for every x ∈ Uα, thus it is smooth. Thepreceding arguments prove that conditions (VBM. 1)–(VBM. 2) of § 1.4.3are fulfilled and, in turn, (f ji, idB) is a morphism of Banach vector bundles.

The previous constructions result in the projective system of Banachvector bundles Ei; f ji, consequently Proposition 5.2.2 determines the plb-vector bundle (E,B, π) with E = lim←−E

i and π = lim←−πi.

Assume now that T ′αβα,β∈I are the transition functions of E. By defi-nition,

T ′αβ(x) = τα,x (τβ,x)−1 =(τ iα,x (τ iβ,x)

−1)i∈N

.

But, for every u ∈ Ei,

(τ iα,x (τ iβ,x)

−1)(u) = τ iα,x([(β, x, u)]i) = T iαβ(x)(u),

therefore,

T ′αβ(x) =(T iαβ(x)

)i∈N

= T ∗αβ(x);

that is, the transition maps of E are exactly the given ones.

Finally, assume that there is another plb-vector bundle (E′, B, π) withtransition functions T ∗αβα,β∈I over C. Then, by Theorem 5.2.5, E andE′ are Frechet vector bundles with the same transition maps Tαβ = ε T ∗αβα,β∈I . Thus, applying to the Frechet framework the arguments of §1.4.3, relating cocycles and vb-isomorphisms, we conclude that the bundlesE and E′ are isomorphic.

Theorem 5.2.10. Let (E,B, π) be a Frechet vector bundle of fibre type F

and base a Banach manifold B. Then E is a plb-vector bundle if and onlyif the transition maps Tαβ : Uαβ → L(F) (α, β ∈ I) over an open cover C(determined by the local trivializations of the bundle) have a decompositionof the form Tαβ = ε T ∗αβ, where T

∗αβα,β∈I is a plb-cocycle of B over the

cover C.

Proof. If E is a plb-vector bundle, the desired decomposition is actuallyequality (5.2.5) obtained in the proof of Theorem 5.2.5.

5.3. Morphisms of plb-vector bundles 195

Conversely, assume that we have the decomposition of the statement. ByProposition 5.2.9, the cocycle T ∗αβαβ determines a plb-vector bundle, say,(E′, B, π′) whose plb-transition maps are T ∗αβαβ (see also Definition 5.2.7).Then, as proved in Theorem 5.2.5, E′ is a Frechet bundle with (ordinary)transition maps Tαβ = ε T ∗αβ . Hence, E ∼= E′. The identification induceson E the structure of a plb-vector bundle.

5.3 Morphisms of plb-vector bundles

Let S = Ei; f jii,j∈N and S = Ei; f jii,j∈N be two projective systems ofBanach vector bundles, over the Banach manifolds B and B, respectively,and corresponding fibre types Ei, Ei (see Definition 5.2.1).

Definition 5.3.1. A morphism of S into S is a family (gi, h)i∈N satis-fying the following conditions:

(PVBM. 1) Each (gi, h) is a vector bundle morphism between (Ei, πi, B)and (Ei, πi, B).

(PVBM. 2) The limit map lim←−

gi : lim←−

Ei → lim←−

Ei exists.

Of course, condition (PVBM. 2) is equivalent to the commutativity ofthe diagram (j ≥ i):

Ejgj

- Ej

Ei

f ji

?

gi- Ei

f ji

?

The next result, beside its interest per se, combines many technicalitiesmet so far.

Proposition 5.3.2. With the previous notations,(

lim←− gi, h)is a morphism

between the Frechet vector bundles (E,B, π) and (E, B, π), determined bythe systems S and S, respectively, in virtue of Theorem 5.2.5.

Proof. By Proposition 3.1.8, g is smooth. Also, in the proof of Theorem5.2.5, we found that Ex = lim←−E

ix, and similarly Eh(x) = lim←− E

ih(x), for every

x ∈ B. Condition (PVBM. 2) implies that

(5.3.1) f ji gjx = gix fji,


for every x ∈ B (with f ji and f ji now restricted to the respective fibres),thus lim

←−gix exists. But f i gx = gix f

i, where f i : E → Ei, f i : E → Ei arethe natural projections (also restricted to the fibres); hence (see Proposition2.3.5), gx = lim←− g

ix, which shows that the map gx : Ex → Eh(x) is continuous

linear, for every x ∈ B. The previous arguments prove the Frechet analogof (VB. 1) in § 1.4.3.

For the analog of (VB. 2), we proceed as follows: If x ∈ B is anarbitrary point, we choose a plb-trivialization (V, τ = lim←− τ

i) of E with

x ∈ V , and a plb-trivialization (U , τ = lim←− τi) of E with h(x) ∈ U . Tak-

ing U = V ∩ g−1(U) and appropriately restricting τ , we obtain the plb-trivialization (U, τ = lim

←−τ i) of E with x ∈ U (for simplicity we use the

same symbol for the trivializing maps), such that h(U) ⊆ U . We define themap

G : U 7−→ L(F, F) : x 7→ τh(x) gx τ−1x ,

where F = lim←−Ei and F = lim←− Ei are the fibre types of the bundles E andE, respectively. To achieve our goal, we need to show that G is smooth. Tothis end we consider the maps

Ri : U −→ L(Ei, Ei) : x 7→ τ ih(x) gix (τ ix)−1, i ∈ N.

They are smooth because every (gi, h) is a vb-morphism of Ei into Ei.Therefore, the map

Gi : U −→ L(E1, E1)× · · · × L(Ei, Ei),

with Gi := (R1, . . . , Ri), is also smooth. But Gi takes values in Hi(F, F).Indeed, if ρji : Ej → Ei and ρji : Ej → Ei are the connecting morphisms forj ≥ i, we check that

ρjk Rj(x) = ρjk τ jh(x) gjx (τ jx)−1

= τkh(x) fjk gjx (τ jx)−1[since E = lim←− Ei]

= τkh(x) gkx f

jk (τ jx)−1;[by (5.3.1)]

hence, by the fibre-wise restriction of the commutative diagram followingDefinition 5.3.1,

ρjk Rj(x) = τkh(x) gkx (τkx )−1 ρjk = Ri(x) ρjk.

The latter, in virtue of (2.3.5), proves the claim. Moreover, since Hi(F, F)is a closed subspace of

∏ik=1L(Ek, Ek), every Gi : U →Hi(F, F) is smooth.


Now, using the connecting morphisms hji : Hj(F, F) → Hi(F, F) [see(2.3.6)], we immediately check that hji Gj = Gi, for every j ≥ i, thuswe obtain the pls-map G∗ = lim

←−Gi : U → H(F, F) ≡ lim

←−Hi(F, F) [see also

equality (2.3.8)].We further verify that the diagram

UG∗

- H(F, F)

L(F, F)

ε

?

G-

is commutative [recall that ε is given by (2.3.4)]. This is the case because,for every x ∈ U ,

(ε G∗)(x) = ε((Gi(x)

)i∈N

)= ε

(((R1(x), . . . , Ri(x))

)i∈N

).

However, after the identification H(F, F) ≡ lim←−Hi(F, F), we have that

((R1(x), . . . , Ri(x))

)i∈N≡(Ri(x)

)i∈N

.

As a result,

(ε G∗)(x) = ε((Ri(x)

)i∈N

)= lim←−R

i(x)

= lim←−

(τ ih(x) g

ix (τ ix)−1

)

= τh(x) gx τ−1x = G(x),

from which, along with the smoothness of G∗ and ε, we conclude that G issmooth. This completes the proof.

Definition 5.3.3. A morphism(

lim←−

gi, h), as in Proposition 5.3.2, will be

called a plb-morphism. If all (gi, idB) are vb-isomorphisms, then (lim←− gi, h)

is called a plb-isomorphism.

With the cohomological classification of plb-vector bundles in mind, inthe remainder of this section we consider bundles over the same base andof the same fibre type. More precisely, let

(E = lim

←−Ei, B, π = lim

←−πi)

and(E = lim←− E

i, B, π = lim←− πi)

be plb-vector bundles of fibre type F = lim←−Ei.Then we prove:


Lemma 5.3.4. Every plb-bundle isomorphism (g = lim←−

gi, idB) of E ontoE corresponds bijectively to a family hα : Uα → H0(F)α∈I of generalizedsmooth maps over an open cover Uαα∈I of B such that

(5.3.2) T∗αβ(x) = hα(x) T ∗αβ(x) hβ(x)−1; x ∈ Uαβ,

if T ∗αβ and T∗αβ are the H0(F)-valued cocycles of E and E, respectively.

As usual, cocycles satisfying (5.3.2) are said to be cohomologous. Forthe definition of generalized smooth H0(F)-valued maps we refer to Re-mark 5.2.6(3).

Proof. Since each gi : Ei → Ei is an isomorphism between Banach bundles,there exists a family of smooth maps

hiα : Uα −→ GL(Ei); α ∈ I,

given by

hiα(x) = τ iα,x gix (τ iα,x

)−1,

where (Uα, τiα)α∈I , (Uα, τ

iα)α∈I are trivializations of Ei and Ei, respectively

(recall the discussion on the cohomological classification of Banach vectorbundles in § 1.4.3). The trivializations can be suitably chosen so that con-dition (PVB. 2) of Definition 5.2.1 be satisfied for both families of trivial-izations. If ρji : Ej → Ei are the connecting morphisms of the projectivesystem Eii∈N, then applying (PVB. 2) and (PVBM. 2) of Definition 5.3.1fibre-wise, we obtain

ρji hjα(x) = ρji τ jα,x gjx (τ jα,x

)−1

= τ iα,x fji gjx

(τ jα,x

)−1

= τ iα,x gix f

ji (τ jα,x

)−1

= τ iα,x gix (τ iα,x

)−1 ρji

= hiα(x) ρji,

for every i, j ∈ N with j ≥ i. As a result, the linear isomorphism

hα(x) := lim←−

hiα(x) : F = lim←−

Ei −→ lim←−

Ei = F

exists for each x ∈ Uα, thus we may define the generalized smooth map

(5.3.3) hα : Uα −→ H0(F) : x 7→(h1α(x),

(h1α(x), h2α(x)

), . . .

).


On the other hand, for every α, β ∈ I,

T∗αβ(x) hβ(x) T ∗βα(x) =

(Tiαβ(x) hiβ(x) T iβα(x)

)i∈N

=(τ iα,x

(τ iβ,x

)−1 hiβ(x) τ iβ,x

(τ iα,x

)−1)i∈N

=(τ iα,x g

ix (τ iα,x

)−1)i∈N

=(hiα(x)

)i∈N

= hα(x),

which is precisely (5.3.2).

Conversely, every family hα : Uα → H0(F)α∈I of generalized smoothmaps, defined by (5.3.3) and satisfying the compatibility condition (5.3.2),gives rise, for each i ∈ N, to a corresponding family of smooth maps (relativeto the bundle Ei)

hiα : Uα −→ GL(Ei); α ∈ I,

so that the equality

Tiαβ(x) hiβ(x) T iβα(x) = hiα(x); α ∈ I,

holds for every x ∈ Uα, where T iαβ and Tiαβ are the ordinary cocycles

of Ei and Ei, respectively. Therefore, following the discussion in § 1.4.3, wedefine the bundle isomorphism (over B) gi : Ei → Ei by setting

gix =(τ iα,x

)−1 hiα(x) τ iα,x, x ∈ B.

Then, working as in the first part of the proof, we have (for j ≥ i):

f ji gjx = f ji (τ jα,x

)−1 hjα(x) τ jα,x

=(τ iα,x

)−1 ρji hjα(x) τ jα,x

=(τ iα,x

)−1 hiα(x) τ iα,x f

ji

= gix fji,

for every x ∈ B, thus f ji gj = gi f ji. This equality ensures the existenceof the map g = lim←− g

i : E → E. Now, since both the conditions of Definition5.3.1 are fulfilled, Proposition 5.3.2 implies that (g, idB) is a vb-isomorphismof E onto E.


From the category of vector bundles (E,B, π) over the Banach mani-fold B, and of fibre type the Frechet space F, we single out those obtainedby projective systems of Banach bundles in the sense of Theorem 5.2.5.We denote their set by VBB(F). Considering the obvious equivalence rela-tion induced by the plb-isomorphisms, we obtain the corresponding quotientspace VBB(F)/∼ . Then we obtain the following cohomological classificationtheorem.

Theorem 5.3.5. If H0(F) denotes the sheaf of germs of H0(F)-valued gen-eralized smooth maps on B, then

VBB(F)/∼ = H1(B,H0(F))

within a bijection.

Here H1(B,H0(F)) is the first cohomology set of B with coefficients inH0(F) (see also the last part of § 1.4.3).

Proof. Let [E] be the equivalence class of a bundle E of the prescribed type.If T ∗αβ is the cocycle of E over a trivializing open cover U of B, then wedefine the map

V BB(F) ∋ [E] 7−→ [T ∗αβ] ∈ H1(B,H0(F)).

It is a matter of routine to verify that, in virtue of Lemma 5.3.4, this is awell-defined bijection.

5.4 The sections of plb-vector bundles

Let Ei; f jii,j∈N be a projective system of Banach vector bundles and(E,B, π) the induced plb-vector bundle (see Definitions 5.2.1, 5.2.4). Wedenote by Γ(Ei) and Γ(E) the C∞(B,R)-modules of smooth sections of Ei

and E, respectively.

Given a family of sections ξi ∈ Γ(Ei) | i ∈ N, an obvious consequenceof the definitions and the smoothness of the limits of smooth maps is thatlim←−

ξi ∈ Γ(E). The converse is also true;, namely, we have:

Lemma 5.4.1. Every section ξ ∈ Γ(E) has the form ξ = lim←−

ξi, where

ξi ∈ Γ(Ei), i ∈ N.

5.4. The sections of plb-vector bundles 201

Proof. For every i ∈ N, we set ξi := f i ξ (recall that f i : E = lim←−

Ei → Ei

is the canonical projection). Obviously, every ξi is smooth and satisfies

πi ξi = πi f i ξ = π ξ = idB; i ∈ N,

thus ξi ∈ Γ(Ei). On the other hand,

f ji ξj = f ji f j ξ = f i ξ = ξi; j ≥ i,

thus lim←−

ξi exists. Since ξi = f i ξ, Proposition 2.3.5 (applied for B and

Ei) implies that ξ = lim←−

ξi.

Proposition 5.4.2. The C∞(B,R)-modules Γ(E) and lim←−

Γ(Ei) coincidewithin an isomorphism.

Proof. For every j ≥ i, we define the maps

γji : Γ(Ej) −→ Γ(Ei) : ξ 7→ f ji ξ.

Since f ji is continuous linear on the fibres, it follows that γji is a morphismof C∞(B,R)-modules. Moreover, γik γji = γjk (k ≤ i ≤ j). There-fore

Γ(Ej); γji

i,j∈N

is a projective system of C∞(B,R)-modules with limit

lim←−Γ(Ei).On the other hand, the maps

γi : Γ(E) −→ Γ(Ei) : ξ 7→ f i ξ; i ∈ N,

satisfy γji γj = γi (i ≤ j), thus inducing

γ := lim←− γi : Γ(E) −→ lim←−Γ(Ei).

The preceding γ is 1–1, since, for every ξ, η ∈ Γ(E),

γ(ξ) = γ(η) ⇔ γi(ξ) = γi(η); i ∈ N,

⇔ f i ξ = f i η; i ∈ N,

⇔ ξ = lim←− ξi = lim←− η

i = η.[Lemma 5.4.1]

Also, γ is onto, for if(ξi)i∈N

is an arbitrary element of Γ(Ei), then

γji ξj = ξi ⇔ f ji ξj = ξi, j ≥ i;

hence, we obtain the section lim←−

ξi such that

γ(ξ) =(γi(ξ)

)i∈N

=(f i ξ

)i∈N

=(ξi)i∈N

.

Finally, the linearity of every f i on the fibres of E implies that every γi is amorphism of C∞(B,R)-modules, and so is γ.


5.5 The pull-back of plb-vector bundles

The construction of the pull-back of a plb-vector bundle is a useful appli-cation, enlightening the methods expounded so far. Since some of the an-ticipated properties (analogous to the ones described in § 1.4.4) have quitelengthy proofs, we proceed by exhibiting them in separate statements.

Let Ei; f jii,j∈N be a projective system of Banach vector bundles and(E,B, π) the induced plb-vector bundle (see Definitions 5.2.1 and 5.2.4). Ifg : Y → B is a smooth map of Banach manifolds, then, for every i ∈ N, weobtain the pull-back of the bundle (Ei, πi, B). This will be representedby the triplet (g∗(Ei), πi∗, B) (for the sake of conformity, we adorn theprojections—as well as other relevant quantities below—with a star, putas a subscript to avoid double superscripts). Given a trivialization, say,(U, τ i) of Ei, we construct the trivialization (g−1(U), σi) of g∗(Ei), wherethe trivializing map

(5.5.1) σi :(πi∗)−1(

g−1(U))

= g∗(U)×U EjU −→ g∗(U)× Ei

is defined by

(5.5.2) σi(y, u) :=(y, τ ig(y)(u)

), (y, u) ∈ g∗(U)×U E

jU .

Proposition 5.5.1. With the previous notations, for every i, j ∈ N withi ≤ j, there is a morphism of Banach vector bundles f ji∗ : g∗(Ej) → g∗(Ei)such that

g∗(Ei); f ji∗

i,j∈

is a projective system.

Proof. We set

f ji∗ (y, u) :=(y, f ji(u)

), (y, u) ∈ g∗(Ej) = Y ×B E

j .

The map f ji∗ indeed takes values in g∗(Ei) because, for every (y, u) as above,g(y) = πj(u). Since f ji : Ej → Ei is a vb-morphism (over idB), it followsthat g(y) = πi

(f ji(u)

), thus

(y, f ji(u)

)∈ g∗(Ei).

We should first show that(f ji∗ , idY

)is a vb-morphism. To this end, we

check that f ji∗ is smooth: Let any (y0, u0) ∈ g∗(Ej). Since Ei; f jii,j∈Nis a projective system of vector bundles, we can find trivializations (U, τ j)and (U, τ i) of Ej and Ei, respectively, satisfying condition (PVB. 2) ofDefinition 5.2.1, with g(y0) = πj(u0) ∈ U . We consider the correspondingtrivializations (g−1(U), σj) and (g−1(U), σi) of g∗(Ej) and g∗(Ei). As in thebeginning of the proof,

f ji∗

((πj∗)−1(

g−1(U)))⊆(πi∗)−1(

g−1(U)),

5.5. The pull-back of plb-vector bundles 203

from which, by direct application of the definitions and (PVB. 2), we verifythat the next diagram is commutative.

(πj∗)−1(

g−1(U)) f ji∗

-(πi∗)−1(

g−1(U))

g−1(U)× Ej

σj

?

idg−1(U)×ρji

- g−1(U)× Ei

σi

?

Hence, f ji∗ is smooth on the (open) neighborhood(πj∗)−1(

g−1(U))

of (thearbitrary) (y0, u0) ∈ g

∗(Ej).Moreover, for every y ∈ g−1(U), the restriction f ji∗,y of f ji∗ to the fibre(

πj∗)−1

(y) = y ×(πj)−1

(y) = y ×Ejg(y) over y is the map

(y, u) 7−→

(y, f ji∗

∣∣Ej

g(y)

(u)

)

(y fixed each time), thus f ji∗,y is continuous linear. Also, we immediately seethat πi f ji = πi. So we have proved condition (VBM.1) of § 1.4.3.

Next we have to prove condition (VBM. 2). First observe that for atrivialization (g−1(U), σi), as in (5.5.1) and (5.5.2), we define the map σiy :=pr2 σ

i : y×Eig(y) → Ei. Thus σiy(y, u) = τ ig(y)(u), which means that σiy is

continuous linear. The inverse of σiy is given by

(σiy)−1

(e) =(y,(τ ig(y)

)−1(e))

=(y, (τ i)−1(g(y), e)

), e ∈ Ei.

Now, for an arbitrary y0 ∈ g∗(Ej), we consider again the trivializations(g−1(U), σj) and (g−1(U), σi), with y0 ∈ g−1(U). The desired conditionnow translates into showing the smoothness of

(5.5.3) g−1(U) ∋ y 7−→ σiy fji∗,y (σjy)

−1 ∈ L(Ej,Ei).

However, for every e ∈ Ej,

(σiy f

ji∗,y (σjy)

−1)(e) = σiy

(y,(f ji

(τ jg(y)

))−1(e))

= (pr2 σi)(y,(f ji

(τ jg(y)

))−1(e))

=(τ ig(y) f

ji (τ jg(y)

)−1)(e).


Since (PVB. 2), applied to the fibres of the system Ei; f ji, yields

(5.5.4) τ ig(y) fji = ρji τ jg(y),

it follows that(σiy f

ji∗,y (σjy)−1

)(e) = ρji(e). Therefore, (5.5.3) is smooth

since it coincides with the constant map y 7→ ρji.

Having shown that each f ji∗ is a vb-morphism, to complete the proof ofthe statement we should verify the analogs of (PVB. 1) and (PVB. 2) ofDefinition 5.2.1 for the pull-back bundles. The first is satisfied because eachg∗(Ei) has fibre type Ei and Ei; ρji is a projective system. The secondis precisely the commutativity of the previous diagram, obtained from theanalogous trivializations of the pull-back bundles, chosen with respect toevery y ∈ Y .

Before proceeding, we see that

g∗(lim←−E

i)

= Y ×B E = Y ×B lim←−Ei ∼= lim←−(Y ×B E

i) = lim←− g∗(Ei),

a fact implying that the pull-back functor commutes with projective limits.However, the next result gives something more than a simple set-theoreticbijection:

Proposition 5.5.2. Let (g∗(E), π∗, Y ) be the pull-back of the plb-bundle(E = lim←−E

i, B, π = lim←−πi)by g : Y → B. Then g∗(E) and lim←− g

∗(Ei) areisomorphic Frechet vector bundles.

Proof. For every i ∈ N, we define the map

hi : g∗(E) −→ g∗(Ei) : (y, u) 7→ hi(y, u) := (y, f i(u)),

where f i : E = lim←−Ei → Ei, i ∈ N, are the canonical projections. Since

f ji f j = f i (j ≥ i), it follows that f ji∗ hj = hi (see the definition of f ji∗ in

the preceding proof), consequently we obtain the limit map

h := lim←−

hi : g∗(E) −→ lim←−

g∗(Ei).

We intend to show that (h, g) is the desired vb-isomorphism.For this purpose we first prove that h is smooth: Take an arbitrary

(y0, u0) ∈ g∗(E) and choose any trivialization (U, τ) = (U, lim←− τ

i) of E with

g(y0) = π(u0) ∈ U . Then (g−1(U), σ), where

σ : (π∗)−1(U) = g−1(U)×U EU −→ g−1(U)× F :

(y, u) 7→ σ(y, u) :=(y, τg(y)(u)

)


is a trivialization of g∗(E), with y0 ∈ g−1(U). On the other hand, as wehave seen in the proof of Proposition 5.5.1, the trivializations (g−1(U), σi) ofg∗(Ei), for all i ∈ N, satisfy the conditions of Definition 5.2.1, thus we obtainthe plb-trivialization

(g−1(U), lim←−σ

i)

of the bundle(lim←− g

∗(Ei), lim←−πi∗, Y

).

We check that

h(g−1(U)×U EU

)⊆(

lim←−

πi∗)−1(

g−1(U))

= lim←−

((πi∗)−1(

g−1(U))).

This is so, because, for every (y, u) ∈ g−1(U)×U EU ,

(5.5.5)

(lim←−

πi∗)(h(y, u)) =

(lim←−

πi∗)((

hi(y, u))k∈N

)=

=(

lim←−πi∗

) ((y, fk(u)

)k∈N

)=(πi∗(y, f

i(u)))i∈N

= y ∈ g−1(U),

from which we get h(y, u) ∈(

lim←−πi∗

)−1(g−1(U)

). On the other hand, equal-

ity(

lim←−πi∗

)−1(g−1(U)

)= lim←−

((πi∗)

−1(g−1(U)

)), along with (5.5.5), yields

the diagram

(π∗)−1(

g−1(U)) h

-(

lim←−

πi∗)−1(

g−1(U))

g−1(U)× F

lim←−

σi

?

σ-

which is commutative, because

((lim←−

σi) h)(y, u) =

(σi(hi(y, u)

))i∈N

=(σi(y, f i(u)

))i∈N

=(y,(τ ig(y)(f

i(u)))i∈N

)=(y,(

lim←− τig(y)

) ((f i(u))i∈N

))

=(y, τg(y)(u)

)= σ(y, u).

Therefore, h is smooth on the open neighbourhood (π∗)−1(g−1(U)) of thearbitrary (y0, u0) ∈ g ∗ (E).

In the course of the proof of (5.5.5) we found that(

lim←−πi∗

) h = πi∗,

while the previous diagram ensures that the restriction of h to the fibres isa continuous linear map.

Also, for an arbitrary y0 ∈ Y and the trivializations (g−1(U), σ) and(g−1(U), lim

←−σi)

as before, the map

(5.5.6) g−1(U) −→ L(F,F) : y 7→(

lim←−

σi)y hy σ

−1y


is constantly equal to idF, hence (5.5.6) is smooth. Note that we have set

(lim←−σ

i)y

:=(

lim←−σi)∣∣∣(

lim←−

πi∗

)−1

(y), hy := h

∣∣(π∗)−1(y)

, σy := σ|(π∗)−1(y).

The previous arguments altogether show that (h, idY ) is a morphismbetween the Frechet vector bundles g∗(E) and lim←− g

∗(Ei).To complete the proof, it remains to show that h is a bijection, so by

similar arguments and using the same trivializations, we have that (h−1, idY )is also a Frechet vb-morphism. The injectivity of h is almost obvious:

h(y, u) = h(z, w) ⇒ hi(y, u) = hi(z, w), ∀ i ∈ N

⇒ (y, f i(u)) = (z, f i(w)), ∀ i ∈ N

⇒ y = z, u = w.[since u = (ui), w = (wi)]

The surjectivity of h needs a bit of extra work: An arbitrary element oflim←−

g∗(Ei) has then form(yi, ui

)i∈N

, where each (yi, ui) ∈ g∗(Ei) satisfies

f ji∗ (yj , uj) = (yi, ui), for every j ≥ i; equivalently,

(yj , f ji(uj)

)= (yi, ui); i, j ∈ N, j ≥ i.

This implies that y1 = · · · = yi = · · · =: y ∈ Y and u = (ui) ∈ lim←−Ei.

Moreover, π(u) = πi(ui) = g(ui) = g(y). Therefore, (y, u) ∈ g∗(E). Thelatter element is mapped to

(yi, ui

)i∈N

:

h(y, u) =(hi(y, u)

)i∈N

=(y, f i(u)

)i∈N

=(yi, ui

)i∈N

,

which ends the proof.

Proposition 5.5.3. The pair (g∗, g), with g∗ := pr2 |g∗(E), is a vb-morphismof g∗(E) into E.

Proof. Every pair (pri2, g), i ∈ N, with pri2 : g∗(Ei)→ Ei, is a vb-morphismbetween the vector bundles g∗(Ei) and Ei, such that the diagram

g∗(Ei)prj2 - Ej

g∗(Ei)

f ji∗

?

pri2

- Ei

f ji

?


is commutative; hence, lim←−

pri2 is defined and(

lim←−

pri2, g)

is a vb-morphismof g∗(E) and E. But the diagram

g∗(E)pr2 - E

g∗(Ei)

hi

?

pri2

- Ei

f i

?

is also commutative, since

(pri2 h

i)

(y, u) = pri2(y, f i(u)

)= f i(u) =

(f i pr2

)(y, u),

for every (y, u) ∈ g∗(E). Therefore, by Proposition 2.3.5, pr2 = lim←−

pri2, i.e.

(g∗, g) = (pr2, g) =(

lim←−

pri2, g)

is a vb-morphism.

Proposition 5.5.4. The pull-back of a plb-bundle satisfies the followinguniversal property: Let

(E = lim

←−Ei, Y

), π = lim

←−πi be a plb-bundle and

g : E → E a smooth map such that (g, g) is a plb-vector bundle morphism,i.e. g = lim

←−gi, with gi : Ei → Ei (i ∈ N). Then there exists a unique smooth

map π : E → g∗(E) such that (π, idY ) is a vb-morphism of E into g∗(E),and g∗ π = g, where g∗ = pr2|g∗(E).

E

g∗(E)g∗

-

π

-

E

g

-

Y

pr1

?

g-

π

-

B

π

?


Proof. By the universal property of g∗(Ei), for every i ∈ N, there is a uniquevector bundle morphism (πi, idY ) : (Ei, Y , πi)→ (g∗(Ei), Y pri1), where πi =(πi, gi). Since, by assumption, the limits lim

←−πi and lim

←−gi exist, we set

π := lim←−

πi ≡(

lim←−

πi, lim←−

gi). Then

(π, idY ) =(

lim←−

πi, idY)≡ lim←−

(πi, idY

)

is a plb-morphism. Note that the existence of lim←−

πi is ensured by thecommutativity of the diagram:

Ejf ji

- Ei

g∗(Ej)

πj

?

f ji∗

- g∗(Ei)

πi

?

Indeed, for every j ≥ i and u ∈ Ej,

(f ji∗ π

j)(u) = f ji∗

(πj(u), gj(u)

)

=(πj(u), f ji(gj(u))

)

or, by the existence of lim←− πi and lim←− g

i,

=(πi(f ji(u)

), gi(f ji(u)

))

=(πi, gi

)(f ji(a)

)=(πi f(ji)

)(u).

The uniqueness of π (as in the ordinary case) is immediate.

Chapter 6

Examples of projectivesystems of bundles

We elaborate a number of examples of projective systems of vector andprincipal bundles. The most important among them are the infinite bundleof jets of sections of a Banach vector bundle, the generalized bundle offrames of a projective limit bundle, and a generalized bundle associated to anarbitrary Frechet principal bundle (P,G,B, π) by means of an appropriaterepresentation of G in a Frechet space F. The jet bundle is of particularinterest because essentially it motivates the requirements of a limit vectorbundle. On the other hand, the two aforementioned generalized bundles arenon trivial examples of limit bundles with structure group H0(F).

6.1 Trivial examples of plb-vector bundles

i) Every Banach vector bundle is obviously a plb-vector bundle.

ii) Every trivial bundle (B × F, B,pr1), where B is a Banach manifold andF a Frechet space, is a plb-vector bundle. This is the case, because, byTheorem 2.3.8, F = lim

←−Ei, where Ei; ρji is a projective system of Banach

spaces. Considering the trivial Banach vector bundles ℓi = (B×Ei, B,pr1),i ∈ N, and the vb-morphisms (idB ×ρ

ji, idB) : ℓj → ℓi, j ≥ i, we obtain theprojective system of vector bundles

B × Ei; idB ×ρ

ji

[it suffices now touse the trivializations (B, idB × idEi)]. This produces the plb-vector bundle

(B × F, B,pr1) =(B × lim

←−Ei, B,pr1

).

iii) If (E,B, π) is a plb-vector bundle and U0 ⊆ B any open set of thebase, then

(π−1(U0), π|π−1(U0), U0

)is a plb-vector bundle as the limit of the

209

210 Chapter 6. Examples of projective systems of bundles

projective system of vector bundles

((πi)−1

(U0), πi∣∣(πi)−1(U0)

, U0

); i ∈ N,

with connecting morphisms

f ji0∣∣(πj)−1(U0)

:(πj)−1

(U0) −→(πi)−1

(U0), j ≥ i.

This is so because both Ei and(πi)−1

(U0) have the same fibre type F =lim←−Ei, and (PVB. 2) is easily checked using the trivializations

(U, lim←− τ

i)

and(U ∩ U0, lim←− τ

i∣∣(πi)−1(U∩U0)

).

6.2 Plb-vector bundles of maps

Let B be a Banach manifold with model B, and let E be a Banach space.By what have we seen in Examples (c)–(e) of § 1.4.4, we can construct theBanach vector bundle (L(TB,E), B, π), where

L(TB,E) =⋃

x∈B

L(TxB,E),

π(f) := x, if f ∈ L(TxB,E).

The fibre type of this bundle is the Banach space L(B,E).If we replace E by a Frechet space F, the usual construction of the vector

bundle structure on L(TB,E) cannot be applied to the case of L(TB,F).One serious obstacle is the differentiability of the transition functions nowtaking values in L(L(B,F),L(B,F)). The latter is a Hausdorff locally convexspace, not necessarily a Frechet one. However, by exploiting the represen-tation of a Frechet space as a projective limit of Banach spaces, we shallendow L(TB,F) with the structure of a plb-vector bundle, hence with thatof a Frechet bundle.

To this end assume that F = lim←−

Ei, where Ei, ρjii,j∈N is a projectivesystem of Banach spaces. For each i ∈ N, we consider the Banach vectorbundle Li = (L(TB,Ei), B, πi), of fibre type L(B,Ei). For every i, j ∈ N

with j ≥ i, we define the map

lji : L(TB,Ej) −→ L(TB,Ei) : f 7→ ρji f.

Each pair (lji, idB) is a vb-morphism of Lj into Li: First, the smoothness oflji is checked at an arbitrary f0 ∈ L(B,Ei) as follows. If, in particular, f0 ∈

6.2. Plb-vector bundles of maps 211

L(Tx0B,Ei), by appropriate restrictions, if necessary, we may choose a chart

(U, φ) at x0, and trivializations (U, σj), (U, σi) of Lj and Li, respectively,with

σj(g) :=(x, g φ

−1x

); g ∈ L(TxB,E

j),

σi(h) :=(y, h φ

−1y

); h ∈ L(TyB,E

i),

where φz : TzB → B is the isomorphism defined by (1.1.4). Without diffi-culty, we check that lji

((πj)−1(U)

)⊆ (πi)−1(U), and the diagram

(πj)−1(U)lji

- (πi)−1(U)

U × L(B,Ej)

σj

?

idU ×rji- U × L(B,Ei)

σi

?

is commutative, where rji : L(B,Ej)→ L(B,Ei) is given by rji(f) := ρji f .

The above diagram implies the smoothness of lji at f0. Also, lji πi = πj,and the restriction of lji to the fibre (πj)−1(x), for every x ∈ B, is continuouslinear. Therefore, condition (VBM. 1) of a vb-morphism is satisfied.

Moreover, with respect to the previous trivializations, we consider themap

(6.2.1) U ∋ x 7−→ σix ljix

(σjx)−1∈ L

(L(B,Ej),L(B,Ei)

),

where ljix is the restriction of lji to the fibre over x. For every f ∈ L(B,Ej),we have that

(σix l

jix

(σjx)−1)

(f) =(σix l

jix

)(f φx

)=

= σix(ρji f φx

)= ρji f = rji(f);

that is, (6.2.1) is smooth as a constant. This proves (VBM. 2); hence,(lji, idB) is indeed a vb-morphism.

We further show thatLi = L(TB,Ei); lji

i,j∈N

is a projective system

of vector bundles in the sense of Definition 5.2.1: Condition (PVB. 1) istrue, since

L(B,Ei); rji

i,j∈N

is a projective system of Banach spaces, thus

lim←−L(B,Ei) exists. We verify (PVB. 2) by taking, for every x ∈ B, a chart

(U, φ) at x and the trivializations (U, σj), (U, σi) considered earlier, which


yield the above commutative diagram. As a result, we obtain the plb-vectorbundle

L := lim←−

Li =(

lim←−

L(TB,Ei), B, lim←−

πi).

Now, for every i ∈ N, we define the map

li : L(TB,F) −→ L(TB,Ei) : f 7→ ρi f,

where ρi : F = lim←−

Ei → Ei is the canonical projection. Then, for everyg ∈ L(TB,F),

(lji lj

)(g) = lji

(ρj g

)= ρji ρj g = ρi g = li(g);

hence, the following limit of maps exists

l := lim←−

li : L(TB,F) −→ lim←−

L(TB,Ei).

We shall prove that l is a bijection. Before this, let us remark that an ar-bitrary element g ∈ lim

←−L(TB,F) has the form (gi)i∈N, with gi ∈ L(TB,Ei)

such that lji(gj) = gi. Since

gi ∈ L(TB,Ei) =⋃

x∈B

L(TxB,Ei),

there is an x ∈ B such that gi ∈ L(TxB,Ei). Similarly, gj ∈ L(TyB,E

j), forsome y ∈ B. But

x = πi(gi) =(πi lji

)(gj) = πj(g) = y;

that is, the equality lji(gj) = gi, for all i, j with j ≥ i, implies that all gi’shave the same domain, TxB.

We can now proceed to the injectivity of l: For f, f ′ ∈ L(TB,F),

l(f) = l(f ′) ⇒(lim←− l

i)

(f) =(lim←− l

i)

(f ′)

⇒(li(f)

)i∈N

=(li(f ′)

)i∈N

⇒ ρi f = ρi f ′, ∀ i ∈ N,

or, since f and f ′ have the same domain, TxB,

⇒ ρi(f(u) = ρi(f ′(u)), ∀ i ∈ N, u ∈ TxB

⇒ f(u) = f ′(u), ∀ u ∈ TxB

⇒ f = f ′.

6.2. Plb-vector bundles of maps 213

On the other hand, let any a ∈ lim←−

L(TB,Ei). As before, a = (gi)i∈N.

Since, by the above remark, all gi’s have the same domain, we obtain thecontinuous linear map g := lim

←−gi ∈ lim

←−L(TxB,F) ⊂ L(TB,F). Then

l(g) =(li(g)

)i∈N

=((ρi g)

)i∈N

=(gi)i∈N

= a;

that is, l is also surjective.

By means of the bijection l we transfer the plb-vector bundle structureof lim←−L(B,Ei) to L(TB,F) so that (l, idB) is an isomorphism of Frechetvector bundles. We note that the diagram

L(TB,F)l

- lim←−L(TB,Ei)

B

lim←−

πiπ-

is commutative because, for every f ∈ L(TxB,F),

(lim←−π

i l)

(f) =(

lim←−πi)((

lj(f))j∈N

)

=(lim←−π

i) ((

ρj f)j∈N

)

=(πi(ρj f)

)j∈N

= x = π(f)

Without particular difficulty we extend the previous constructions to thecase of k-linear map bundle

Lk(TB,F) =⋃

x∈B

Lk(TxB,F),

and to the k-alternating linear map bundle

Ak(B,F) =⋃

x∈B

Ak(TxB,F).

All of them become plb-bundles (in the sense of Definition 5.2.1); therefore,they are Frechet vector bundles.


6.3 The infinite jet bundle

In § 1.4.4(f) we dealt with the structure of the bundle Jk(ℓ) := (JkE,B, πk)of k-jets of the local sections of a Banach vector bundle ℓ = (E,B.π). Herewe want to show that

J∞(E) := lim←− JkE

is a Frechet vector bundle, as a plb-bundle.For our purpose, we consider the maps

J lk : J l(E) −→ Jk(E) : jlxξ 7→ jkxξ, l ≥ k.

and obtain the projective systemJk(E);J lk

l,k∈N

, whose limit is J∞(E).We shall prove that this is in fact a projective system of Banach vectorbundles.

Following the main lines of earlier proofs of the same nature, we firstshow that every J lk is smooth. Indeed, if jlx0 is an arbitrary jet of J l(E),we choose a vb-chart (U, φ,Φ) of E, with x0 ∈ U , as well as the trivial-izations (U, σl) and (U, σk) of J l(E) and Jk(E), respectively. We checkthat J lk

((πl)−1(U)

)⊆ (πk)−1(U) (l ≥ k), thus we obtain the commutative

diagram

(πl)−1(U)J lk

- (πk)−1(U)

U × P l(B,E)

σl

?

idU ×Plk- U × P k(B,E)

σk

?

where P lk : P l(B,E) → P k(B,E) is the continuous linear map given byP lk(f1, . . . , fk, . . . , fl) := (f1, . . . , fk). The commutativity of the diagramis checked as follows: For every jlxξ ∈ (πl)−1(U),(σk J lk

)(jlxξ)

= σk(jlkx ξ

)=(x; ξφ(φ(x)),Dξφ(φ(x)), . . . ,Dkξφ(φ(x))

)

=(

idU ×Plk)(x; ξφ(φ(x)),Dξφ(φ(x)), . . . ,Dlξφ(φ(x))

)

=((

idU ×Plk) σl

)(jlxξ).

Consequently, J lk is continuous on the neighborhood (πl)−1(U) of jlx0 .Obviously, πk J lk = πl, while the restriction of the previous diagram

to each fibre (πl)−1(x) yields

J lkx := J lk∣∣(πl)−1(x)

=(σkx)−1 P lk σlx.

6.4. The tangent bundle of a plb-bundle 215

Therefore, for any x ∈ B, we choose the previous trivializations (U, σl) and(U, σk) and define the map

J : U −→ L(P l(B,E), P k(B,E)

): y 7→ σky J

lky

(σly)−1

.

We immediately see that J(y) = P lk (constantly), for every y ∈ U , thus J issmooth, thus every (J lk, idB) is a vb-morphism, and

Jk(E);J lk

l,k∈N

is acountable family of vector bundles connected by vb-morphisms. It remainsto show that it is a projective system of vector bundles in the sense ofDefinition 5.2.1.

Indeed, the maps P lk : P l(B,E)→ P k(B,E) satisfy the equalities

P km P lk = P lm, ∀ k, l,m ∈ N : l ≥ k ≥ m.

ThusP k(B,E);P lk

is a projective system of Banach spaces, whose limit

lim←−Pk(B,E) is isomorphic to the Frechet space P∞(B,E) = E × L(B,E) ×

L2s(B,E) × · · · [see Example 2.3.3(2)]. This implies condition (PVB. 1) of

Definition 5.2.1. Moreover, for an arbitrary x ∈ B, condition (PVB. 2) isa consequence of the commutativity of the previous diagram with the sametrivializations (U, σl), (U, σk). Therefore,

Jk(E);J lk

l,k∈N

is a projectivesystem of Banach vector bundles with corresponding limit the plb-bundle(J∞(E), B, lim

←−πk). In particular, this is a Frechet vector bundle.

Remark. It is worth noting that the smooth structure on J∞(E) =lim←−

Jk(E), derived from the previous approach (see also Proposition 5.2.2) iswider than the one defined by F. Takens in [Tak79]. The latter is obtained bydeclaring that a map g : J∞(E)→ R is smooth if, locally, there exist k ∈ N,Uk ⊆ Jk(E) open and gk : Uk → R smooth, so that g

∣∣J−1k

(Uk)= gk Jk,

where Jk : J∞(E) → Jk(E) is the natural projection. This condition issatisfied if and only if g is the projective limit of the smooth maps gk J lk

(l ≥ k); therefore, Taken’s R-valued smooth maps on J∞(E) are necessarilyprojective limits of smooth maps. However, in our framework, smoothnessis not restricted only to pls-maps (compare with Remark 3.1.9(1); see alsoDefinition 3.1.7).

6.4 The tangent bundle of a plb-bundle

Let Ei; f jii,j∈N be a projective system of vector bundles ℓi = (Ei, B, π),with connecting morphisms the vb-morphisms (f ji, idB) : ℓj → ℓi (j ≥ i),with limit the plb -bundle ℓ = lim←− ℓ

i =(E := lim←−E

i, B, π := lim←−πi). Apply-

ing the tangent operator, for every i ∈ N, we obtain the vector bundle Tℓi :=


(TEi, TB, Tπi). Accordingly, we have the smooth maps Tf ji : TEj → TEi,for all j ≥ i.

In view of later applications, we want to show that TEi, T f jii,j∈N is aplb-bundle. It is clear that, by Tf ji in the system, we mean the vb-morphism(Tf ji, idTB) of Tℓj → Tℓi.

From πi f ji = πj, it follows that Tπi Tf ji = Tπj. On the other hand,for an arbitrary v ∈ TxB, where TB is the base of all the bundles TEi,we choose a trivialization

(U, τ = lim←− τ

i)

of E, with b ∈ U , and consider

the trivializations(τ−1B (U), σi

)of TEi, respectively (for all i ∈ N), where

τB : TB → B is the projection of the tangent bundle TB (see § 1.1.5), and

σi :(Tπi

)−1 (τ−1B

(U))−→ τ−1

B(U)× Ei × Ei :

[(α, u)] 7−→ σi([(α, u)]) :=(Tπi([(α, u)]), τ iπi(u)(u), (pr2 τ

i α)′(0)).

Recall that [(α, u)] is the equivalence class of a smooth curve α : (−ǫ, ǫ) →TEi with α(0) = u. Also, (pr2 τ

i α)′(0) = D(pr2 τi α)(0).(1).

Let (τ−1B (U), σj), (τ−1B (U), σi) (j ≥ i) be two trivializations, as above,and σjv, σiv their restrictions to the fibres (Tπj)−1(v) and (Tπj)−1(v), respec-tively. Note that σjv : (Tπj)−1(v)→ Ej ×Ej is given by σjv = (pr2,pr3) σ

j .We show that the diagram

(Tπj)−1(v)Tf ji

- (Tπi)−1(v)

Ej × Ej

σjv

?

ρji × ρji- Ei × Ei

σiv

?

is commutative (of course, Tf ji is also restricted to the fibres). Indeed, forevery [α, u] ∈ (Tπj)−1(v),

(σiv Tf

ji)([(α, u)]) = σiv

([(f ji α, f ji(u)

)])

=(τ iπi(fji(u))

(f ji(u)

),(pr2 τ

i f ji α)′

(0))

or, by condition (PVB. 2) and the equality πi(u) = πj(u),

=(ρji(τ jπj(u)

(u)), (ρji pr2 τ

j α)′(0)),

6.4. The tangent bundle of a plb-bundle 217

and, by the linearity of ρji,

=(ρji(τ jπj(u)

(u)), ρji (pr2 τ

j α)′(0))

=(ρji × ρji

) (τ jπj(u)

(u), (pr2 τj α)′(0)

)

=((ρji × ρji

) σjv

) ([(α, u)]

).

Therefore, the restriction of Tf ji to the fibres of TEj is a continuous linearmap. With the same trivializations, we see that the map

G : τ−1B (U) −→ L(Ej × Ej,Ei × Ei) :

w 7−→ σiw Tfji∣∣(Tπj)−1(w)

(σjw)−1

yields G(w) = ρji × ρji, for all w, i.e. G is constant, thus smooth. Hence,we conclude that (Tf ji, idTB) is a vb-morphism of TEj into TEi (j ≥ i).

Each TEi is of fibre type Ei×Ei and, obviously, the plb-space lim←−(Ei×Ei)is defined, thus (PVB. 1) of Definition 5.2.1 is satisfied. Finally, we verifycondition (PVB. 2) by using the above trivializations (τ−1B (U), σi)i∈N andfollowing the procedure applied to the proof of the commutativity of theprevious diagram. As a result, we obtain the plb-vector bundle

lim←−

Tℓi =(lim←−

TEi, TB, lim←−

Tπi)

Wishing to show that the latter bundle is isomorphic to (TE, TB, Tπ),we induce the maps Tf i : TE → TEi, where f i : E = lim

←−Ei → Ei are the

canonical projections, for all i ∈ N. We observe that f ji f j = f i impliesTf ji Tf j = Tf i, thus we obtain the limit map h := lim

←−Tf i : TE →

lim←−

TEi. We check that h is the diffeomorphism R of Theorem 3.2.8 (holding

for arbitrary plb-manifolds). Indeed, for every x = (xi) ∈ E,

h∣∣TxE

=(lim←−Tf

i) ∣∣TxE

= lim←−

(Tf i

) ∣∣T ixE

i = lim←−TixE

i = Rx,

since the morphism Rx is given by Corollary 3.2.6 with µi = f i, i ∈ I.

Next we immediately see that(

lim←−Tπi) h = Tπ. On the other hand,

if v ∈ TB is an arbitrary element of the base space, we consider the trivial-ization (U, τ = lim←− τ

i) of E, with τB(v) ∈ U , the corresponding trivialization(τ−1B (U), lim←−σ

i)

of lim←−TEi, and the trivialization

(τ−1B (U), σ

)of TE derived


from (U, τ). We check that the diagram

(Tπ)−1(U)h- lim←−

((Tπi)−1(U)

)

F× F

lim←−

σiv

?

σv-

is commutative, because

(lim←−

σiv h) (

[(α, u)])

=(σiv(Tf i

([(α, u)]

)))i∈N

=(σiv([(

f i α, f i(u))]))

i∈N

=(τ iπi(f i(u))(f

i(u)), (pr2 τi f i α)′(0)

)i∈N

=((τ iπi(f i(u)) f

i)(u),

(pr2 (idU ×ρ

i) τ α)′

(0))i∈N

=(ρi(τπ(u)(u)

),(ρi pr2 τ α

)′(0))i∈N

=(

(ρi × ρi)(τπ(u)(u), (pr2 τ α)′(0)

))i∈N

= σv([(α, u)]

).

This means that the restriction of h to the fibres of TE is a continuouslinear map. Using once more the trivializations

(τ−1B (U), σ

)of TE and(

τ−1B (U), lim←− σi)

of lim←−TEi, we see that the map

F : τ−1B (U) −→ L(F× F,F× F) : v 7→ lim←−

σiv h∣∣(Tπ)−1(v)

σ−1v

is constantly idF×F, thus F is smooth. As a result, (h, idTB) is a morphismbetween the Frechet vector bundles TE and lim←−TE

i. With the same trivial-

izations, we prove, in a reverse way, that (h−1, idTB) is also a vb-morphism;hence, TE ≡ lim←−TE

i by means of the vb-isomorphism (h, idTB).

6.5 The generalized frame bundle

Let Ei, f jii,j∈N be a projective system of Banach vector bundles with limitthe (Frechet) plb-vector bundle (E,B, π). We want to define the frame bun-dle of (E,B, π). The pathology of the GL(F) compels us to a radical revision

6.5. The generalized frame bundle 219

of the frame bundle by considering H0(F) as the appropriate structure groupwithin our framework.

Before proceeding, we introduce the following notation, combining (2.3.5)and (5.1.2): For two Frechet spaces F1 and F2, we set

(6.5.1) Hi0(F1,F2) := Hi(F1,F2)⋂ i∏

j=1

Lis(Ej1,Ej2)

Accordingly, we define the space

P (Ei) :=⋃

x∈B

Hi0(F, Ex),

which is meaningful because Ex = lim←−Eix. Thus an element of P (Ei) has

the form(q1, . . . , qi

), where the isomorphisms qj : Ej → Ejx (j = 1, . . . , i)

satisfy the equalities f jk qj = qk ρjk, for every 1 ≤ j, k ≤ i with k ≤ j.Recall that f jk : Ej → Ek and ρjk : Ej → Ek.

The bold typeface is used to distinguish P (Ei) from the ordinary bun-dle of frames mentioned in § 1.6.5, which in the present context would beP (Ei) :=

⋃x∈B Lis(F, Ex).

Proposition 6.5.1. Each P (Ei) (i ∈ N) is a Banach principal bundle overB, with structure group Hi0(F), and projection pi : P (Ei)→ B given by

pi(q1, . . . , qi

):= x, if

(q1, . . . , qi

)∈ Hi0(F, Ex).

Proof. The smooth structure of P (Ei) is defined as follows: For an arbitrary(g1, . . . , gi

)∈ P (Ei) with pi

(g1, . . . , gi

)= x, we choose a local trivialization(

U, lim←− τi)

of E, x ∈ U , and define the bijection

Φi : (pi)−1(U) −→ U ×Hi0(F) :(q1, . . . , qi

)7−→

(x; τ1x q

1, . . . τ ix qi),

where τ jx : Ejx → Ej (j = 1, . . . , i) is the isomorphism induced by the corre-sponding trivializations.

For another trivialization(V, lim←−

σi), with x ∈ U ∩ V , and the corre-

sponding bijection Ψi : (pi)−1(V ) → V × Hi0(F), we see that, on the over-lapping,

(6.5.2)

(Ψi (Φi)−1

) (x;h1, . . . , hi

)=

=(x;σ1x (τ1x)−1 h1, . . . , σix (τ ix)−1 hi

),


which is a diffeomorphism. Then, in virtue of the gluing process (see, e.g.,[Bou67, no 5.2.4]) P (Ei) is indeed a Banach manifold turning the quadruple(P (Ei),Hi0(F), B,pi) into a Banach principal bundle, where Hi0(F) acts on(the right of) P (Ei) in the obvious way, i.e.

(q1, . . . , qi

)·(g1, . . . , gi

)=(q1 g1, . . . , qi gi

),

for every(q1, . . . , qi

)∈ P (Ei) and

(g1, . . . , gi

)∈ Hi0(F).

For later use we prove:

Corollary 6.5.2. The transition functionsgiαβ : Uαβ → H

i0(F)

α,β∈I

of

P (Ei), over a trivializing cover Uαα∈I of B, are given by

(6.5.3) giαβ(x) =(g1αβ(x), . . . , giαβ(x)

); x ∈ Uαβ = Uα ∩ Uβ,

wheregjαβ : Uαβ → GL(Ej)

α,β∈I

(j = 1, . . . , i) are the transition functions

of both Ej and the ordinary frame bundle P (Ej).

Proof. This is a direct consequence of (6.5.2) and the fact thatgjαβαβ∈I

are the transition functions of both Ej and P (Ej) [see (1.6.15)].

For every j ≥ i, we define the following connecting morphisms:

rji : P (Ej) −→ P (Ei) :(q1, . . . , qj

)7→(q1, . . . , qi

),

hji0 : Hj0(F) −→ Hi0(F) :(g1, . . . , gj

)7→(g1, . . . , gi

),

(see also (2.3.6) and the notations of Proposition 5.1.1).

Lemma 6.5.3. For every j ≥ i, the triplet(rji, hji0 , idB

)is a principal

bundle morphism of (P (Ej),Hj0(F), B,pj) into (P (Ei),Hi0(F), B,pi)

Proof. Immediate consequence of the preceding definitions.

Proposition 6.5.4. The following assertions hold true:i) The collection

(P (Ei),Hi0(F), B,pi

);(rji, hji0 , idB

)i,j∈N

is a projective system of Banach principal bundles.ii) The set P (E) := lim←−P (Ei) is the total space of a locally trivial prin-

cipal bundle over B, with structure group H0(F), called the generalizedframe bundle of E.

6.6. Generalized associated bundles 221

Proof. In virtue of Definition 4.1.1 and the preceding lemma, using the triv-ializations U,Φii∈N defined in the proof of Proposition 6.5.1, we obtainthe first conclusion, thus P (E) exists.

For ii) take any x ∈ B and consider the trivializations U,Φii∈N, x ∈ U ,as before. It is easily checked that the diagram

(pj)−1(U)Φj

- U ×Hj0(F)

(pi)−1(U)

rji

?

Φi- U ×Hi0(F)

idU ×hji0

?

is commutative. As a result, the morphism

(6.5.4) Φ := lim←−Φi : lim←−

((pi)−1(U)

)−→ U ×H0(F)

exists and determines a topological trivialization of P (E) over U . Theprojection of P (E) is p = lim←−pi, while the action of H0(F) on (the right) ofP (E) is the projective limit of the actions on the factors.

Remarks 6.5.5. 1) The elements of P (E) are of the form (gi)i∈N, withgi ∈ P (Ei), since lim←− g

i exists. In this respect see also the identifications(2.3.9), (2.3.9′).

2) The homomorphism Φ defined by (6.5.4) is not smooth in the ordinarysense, since H0(F) is not a Frechet-Lie group. However, if Φ is consideredas a (U ×H(F))-valued map (H(F) is a Frechet space), then it is smooth inthe sense of the differentiability defined in §2.2. Therefore, Φ is generalizedsmooth in the sense of Remark 5.2.6(3). By the same token, the action ofH0(F) on P (E) can be thought of as smooth.

3) In view of the preceding remark, P (E) is a smooth Frechet principalbundle, justifying the term generalized frame bundle of E.

4) In the next section we shall show that the original vector bundle E isassociated with the generalized bundle of frames P (E) (see Corollary 6.6.5).

6.6 Generalized associated bundles

Motivated by the construction of § 1.6.6, we want to answer the followingquestion: Given a Frechet principal bundle (P,G,B, π) and a Frechet spaceF, is it possible to construct an associated vector bundle (of fibre type F),


from an arbitrary representation of G into F? The answer is negative, if wetry to imitate the classical pattern, since such a representation amounts toa homomorphism of the form ϕ : G → GL(F), while, as we have explainedon many occasions, GL(F) is too problematic. In fact, although the associ-ated bundle P ×ϕ F exists set-theoretically, it has in general no differentialstructure. As in the previous section, the replacement of GL(F) by H0(F)is the key to a (partial) affirmative answer.

So, starting with a (not necessarily a projective limit) Frechet principalbundle (P,G,B, π) over a Banach base, we consider a representation of Ginto F; that is, a topological group homomorphism

(6.6.1) : G −→ H0(F),

which is also smooth if considered as taking values in the Frechet spaceH(F) ⊃ H0(F) [see (2.3.12) and (2.3.3)].

As usual, F = lim←−Ei, where Ei, ρji is a projective system of Banachspaces. Then, by the definition of H0(F) [see (5.1.3)], we obtain the ordinaryrepresentations

(6.6.2) i : G −→ GL(Ei) : g 7→ pri((g)), i ∈ N.

Here pri : H0(F) → Lis(Ei) denotes the projection to the i-th factor andGL(Ei) is identified with Lis(Ei). Therefore, in the notations of § 1.6.6, weinduce the Banach vector bundles (Ei, B, πi), where

Ei := P ×i Ei =

[(p, ui)]i | (p, u

i) ∈ P × Ei

; i ∈ N,

and πi([(p, ui)] = π(p). Obviously, [(p, ui)]i is the orbit of (p, ui) with respectto the action (p, ui) · g = (p · g, i(g−1)(ui)). Notice the use of a matchingindex in the corresponding equivalence class.

The Banach vector bundle structure of each Ei will be apparent in theproof of the next result.

Proposition 6.6.1. The limit E := lim←−Ei exists and admits the structure

of a Frechet vector bundle over B.

Proof. For every i, j ∈ N, with j ≥ i, we can define the map

f ji : Ej −→ Ei : [(p, uj)]j 7→ [(p, ρji(uj))]i,

since lim←−(i(g))) exists for every g ∈ G. Also, f ik f ji = f jk, for every

j ≥ i ≥ k, as a result of the analogous equalities for ρji. Therefore,


Ei; f jii,j∈N is a projective system inducing E := lim←−

Ei, though not yetsatisfying the conditions of Definition 5.2.1.

The local structure of E is determined as follows: Assume that C is theopen cover of B over the sets of which P is trivial. Let x0 ∈ B be anarbitrary point, and let (U,Φ) be a local trivialization of P , with x0 ∈ U .Then, as in § 1.6.6, for each Ei we have the corresponding local trivialization

Ψi : U × Ei −→ (πi)−1(U) : (x, ui) 7→ [(s(x), ui)]i,

where s is the natural section of P over U (with respect to Φ; see thebeginning of § 1.6.3). Immediate computations imply that Ψi is a bijection,and f ji

((πj)−1(U)

)⊆ (πi)−1(U); thus the diagram

U × EjΨj

- (πj)−1(U)

U × Ei

idU ×ρji

?

Ψi- (πi)−1(U)

f ji

?

is commutative. Indeed,

(f ji Ψj)(x, uj) =[(s(x), ρji(uj)

)]i

=(Ψi (idU ×ρ

ji))(x, uj); j ≥ i,

for every (x, uj) ∈ U×Ej. Hence, taking the inverse maps

Φi = (Ψi)−1i∈N

,

we see that Eii∈N is a projective system of Banach vector bundles (in thesense of Definition 5.2.1), thus Theorem 5.2.5 concludes the proof.

To proceed further, we define the homomorphism

(6.6.3) ϕ := ε ,

where now

ε : H0(F) −→ GL(F) : (f i)i∈N 7→ lim←− fi,

[compare with the general case of (2.3.4)], and the action of G on (the right)of P × F, determined by (p, u) · g :=

(p · g, ϕ(g−1)(u)

).

Proposition 6.6.2. The quotient (with respect to ϕ) F := P ×ϕ F, being inbijective correspondence with E = lim←−E

i, inherits the structure of a Frechetvector bundle over B.


Proof. We define the mapping

f : P × F −→ E : (p, u) 7→([(p, ρi(u))]i

)i∈N

,

where ρi : F→ Ei (i ∈ N) are the canonical projections of F. Since, for every(p, u) ∈ P × F and g ∈ G,

f((p, u) · g) = f(p · g, ϕ(g−1)(u)

)=([(

p · g, ρi(ϕ(g−1)(u)))]i

)i∈N

=([(

p · g, i(g−1) (ρi(u)

) ]i

)i∈N

=([(p, ρi(u))]i

)i∈N

= f(p, u),

there is a well-defined mapping f induced on the quotient F , i.e. f([(p, u)]) =([(p, ρi(u))]i

)i∈N

, for every [(p, u)] ∈ E. We check that:

i) f is 1–1: If we assume that f([(p, u)]) = f([(q, v)]), then [(p, ρi(u))]i =[(q, ρi(v))]i, for every i ∈ N. Since π(p) = π(q), there exists a unique g ∈ Gsuch that q = p · g and ρi(v) = i(g−1)

(ρi(u)

), for every i ∈ N. Therefore,

v =(ρi(u)

)i∈N

=(i(g

−1)(ρi(u)))i∈N

=(lim←−

i(g−1)) (

(ρi(u)))i∈N)

= ε((g−1)

)(u) = φ(g−1)(u),

implying that [(p, u)] = [(q, v)].

ii) f is onto: Let an arbitrary a ∈ E = lim←−Ei. Then a =

([(pi, ui)]i

)i∈N

such that, by the property of the elements of the limit with respect to theconnecting morphisms,

f ji([(pj , uj)]j

)= [(pi, ui)]i; j ≥ i,

while, by the definition of f ji,

f ji([(pj , uj)]j

)= [(pj, ρji(uj))]i, j ≥ i.

Thus, [(pi, ui)]i = [(pj , ρji(uj))]i implies the existence of a gji ∈ G such that

(6.6.4) pj = pi · gji, and ρji(uj) = i(g−1ji)(ui).

We set p := p1 and gk := g−1k1 (k ∈ N). Then, the first of (6.6.4) yields (forj = k, i = 1)

(6.6.5) p = pk · g−1k1 = pk · gk, ∀ k ∈ N.


Applying (6.6.5) again to the first of (6.6.4), we see that

pj = pi · gji ⇒ p · g−1j = p · g−1i · gji,

and, because G acts freely on P ,

(6.6.6) gj = g−1ji · gi, j ≥ i.

Furthermore, by the compatibility of the actions with the connecting mor-phisms of Eii∈N, i.e. pji j = i (since ii∈N is a projective system),we have that

ρji(j(g−1j)(uj)

)= i

(g−1j) (ρji(uj)

)

or, by the second equality of (6.6.4), and equality (6.6.6),

ρji(j(g−1j)(uj)

)=(i(g−1i) i(g−1ji))

(ui) = i(g−1i)(ui).

Hence, the element v :=(i(g−1i)(ui)

)i∈N

belongs to F, and

f([(p, v)]) =([(

pi · gi, i(g−1i)(ui)

)]i

)i∈N

=([(pi, ui)]i

)i∈N

= a.

Consequently, f is the desired bijection which proves the statement.

Remark 6.6.3. The preceding proposition implies that

P ×ϕ(lim←−

Ei)∼= lim←−

(P ×ρi E

i)

as vector bundles. This formula generalizes the set-theoretical commutativ-ity between inverse limits and cartesian products.

Propositions 6.6.1 and 6.6.2, combined together, are summarized in thefollowing main result.

Theorem 6.6.4. Let (P,G,B, π) be a Frechet principal bundle over a Ba-nach base, F ∼= lim←−Ei a Frechet space and ϕ : G→ GL(F) a representationof G in F. If ϕ can be factored as in (6.6.3), then F := P ×ϕ F admitsthe structure of a Frechet vector bundle associated with P . In particular, Fis identified with the projective limit of a system of Banach vector bundles(Ei, B, πi)i∈N of fibre type Ei, respectively.

We conclude with the following result mentioned in Remark 6.5.5 (4).

Corollary 6.6.5. Applying the technique of this section to the case of thegeneralized principal bundle of frames P (E) of a plb-vector bundle E, dis-cussed in § 6.5, we readily verify that the associated vector bundle P (E)×ϕF,where ϕ = ε idH0(F), coincides with E.

Chapter 7

Connections on plb-vectorbundles

The objective of this chapter is to study projective systems of (linear) con-nections on plb-vector bundles. It will be shown that the derived limits areconnections in the classical sense, characterized, however, by a generalizedtype of Christoffel symbols. The present category of connections entailsimportant relevant geometric notions, like the parallel displacement alongcurves in the base space. The former cannot be approached, in general,because of the inherent difficulties in the study of differential equations inFrechet spaces. The corresponding holonomy groups are also studied. Thesegroups seem to live in the borders of the categories of plb-manifolds and al-gebraic groups as we explain at the end of § 7.2.

7.1 Projective limits of linear connections

For the convenience of the reader, we recall from § 1.5.1 that a (not necessar-ily linear) connection on a Banach vector bundle ℓ = (E,B, π), of fibre typeE and base space model B, is a bundle morphism K : TE −→ E. Fixing avb-chart (Uα, φα,Φα) of E and the induced vb-chart of the tangent bundleTE, the local representation of K,

(7.1.1) Kα : φα(Uα)× E× B× E −→ φα(Uα)× E,

is given by

(7.1.2) Kα(x, λ, y, µ) = (x, µ + κα(x, λ).y),

227

228 Chapter 7. Connections on plb-vector bundles

where κα : φα(Uα)× E→ L(B,E) is the (smooth) local component of K. IfK is linear, then κα is linear with respect to the second variable and inducesthe Christoffel symbols

Γα : φα(Uα)→ L(E,L(B,E))

α∈I

by setting

(7.1.3) Γα(x).λ = κα(x, λ), (x, λ) ∈ φα(Uα)× E.

Focusing now on the category of projective limits of vector bundles,we consider a plb-vector bundle ℓ ≡ lim←− ℓ

i = lim←−(Ei, B, πi) with connecting

morphisms f ji : Ej → Ei (j ≥ i) and fibre type the Frechet space F = lim←−Ei.As in the case of plb-principal bundles, we assume that B is a Hausdorffspace admitting smooth partitions of unity.

A projective system of connections is a sequence of connections Ki

on ℓi (i ∈ N) commuting with the connecting morphisms of the plb-bundlesE = lim←−E

i and TE = lim←−TEi, i.e.

(7.1.4) f ji Kj = Ki Tf ji, j ≥ i.

In virtue of (1.5.19), the preceding equality means that Kj and Ki are(f ji, idB)-related connections.

To show that such projective systems of connections lead to connectionson the limit bundle, we need the equivalent of (7.1.4) in terms of the localcomponents κiα : φα(Uα)×Ei → L(B,Ei), i ∈ N, of the connections Kii∈N,respectively, in analogy to the general formula (1.5.27). To this end, we firstsee that the connecting morphisms

(7.1.5) ρji : Ej −→ Ei

of Eii∈N induce the connecting morphisms

(7.1.6) rji : L(B,Ej) −→ L(B,Ei) : f 7→ ρji f, j ≥ i

of the system L(B,Ei)i∈N yielding L(B,F) ≡ lim←−L(B,Ei).

Furthermore, from the definition of a projective system of vector bundles(see, in particular, condition (PVB. 2) of Definition 5.2.1), and the proof ofProposition 5.2.2, we obtain the trivializations (Uα, τ

iα) of Ei, i ∈ I, and

their corresponding vb-charts(Uα, φα,Φ

iα := (ϕα × idEi) τ iα

), over the

local charts (Uα, φα) of B. Regarding these vb-charts and the commutativediagram in the proof of Proposition 5.2.2, we verify that the local principalpart of each connecting morphism f ji,

(f jiα)#

: φα(Uα) −→ L(Ej,Ei),

7.1. Projective limits of linear connections 229

[see (1.5.22) and (1.5.23) together with (1.5.31)] is a constant map; namely,

(f jiα)#

(x) = ρji, x ∈ φα(Uα).

Therefore, (1.5.27) now becomes

(7.1.7) rji κjα = κiα (

idφα(Uα)×ρji), j ≥ i.

In the case of a linear connection, taking the Christoffel symbols of the formΓkα : φα(Uα)→ L(Ek,L(B,Ek))

α∈I

(k = j, i), (7.1.7) is equivalent to

(7.1.8) rji Γjα(x) = Γiα(x) ρji; x ∈ φα(Uα), j ≥ i.

In summary,(7.1.4) ⇔ (7.1.7) ⇔ (7.1.8).

Proposition 7.1.1. If Kii∈N is a projective system of connections, thenthe limit K := lim←−K

i is a connection on the plb-vector bundle ℓ ≡ lim←− ℓi. K

is called a plb-connection.

Proof. As we have proved in Proposition 5.2.2, the vb-charts (Uα, φα,Φiα)

converge projectively to the plb-vector chart (Uα, φα,Φα ≡ lim←−

Φiα) of ℓ. On

the other hand, the induced chart of the limit tangent bundle TE is(τ−1E

(π−1(U)

)= lim←− τ

−1Ei

((πi)−1(U)

),Φ, Φ = lim←− Φi

),

where each Φi is defined as in (1.5.5).An immediate consequence of (7.1.7) is that the local components

κiα : φα(Uα)× Ei → L(B,Ei)

i∈N

,

of Ki form (for each α ∈ I) a projective system of smooth maps; hence,

(7.1.9) κα := lim←−κiα : φα(Uα)× F −→ L(B,F)

is a well-defined pls-map (and therefore smooth), such that

κα(x, λ) =(κiα(x, ρi(λ)

))i∈N

⇔ ρi κα(x, λ) = κiα(x, ρi(λ)

); i ∈ N,

where

(7.1.10) ρi : F = lim←−Ei −→ Ei; i ∈ N,


are the canonical projections of the fibre type. Based on the latter equiv-alence, and using the limit charts of E mentioned in the beginning of theproof, we check that

Kα(x, λ, y, µ) =(Φ K|τ−1

E(π−1(U)) Φ−1

)(x, λ, y, µ)

=((

Φi Ki|(τ iEi

)−1((πi)−1(U)) (Φi)−1)(x, λi, y, µi)

)i∈N

=(Kiα(x, λi, y, µi)

)i∈N

=(x, µi + κiα(x, λi).y

)i∈N

=(x,(µi)i∈N

+(ρi(κα(x, λi.y

)i∈N

)

=(x, µ+ κα(x, λ).y

),

for every x ∈ φα(Uα), y ∈ B, λ = (λi)i∈N, and µ = (µi)i∈N ∈ F. As a result,the local characterization of a connection (via local components) impliesthat K is indeed a connection on the Frechet vector bundle E = lim←−E

i.

In particular, we have:

Corollary 7.1.2. Let K = lim←−

Ki be a projective limit of linear connections

on a plb-vector bundle ℓ = lim←−

ℓi. Then K is also a linear connection.

Proof. The linearity of the factor connections Kii∈N is equivalent to thefact that the local components

κiα : φα(Uα)× Ei −→ L(B,Ei); i ∈ N,

are linear with respect to the second variable, for every α ∈ I. Since thelocal components of K are projective limits, i.e. κα = lim←−κiα, the latterbecome also linear with respect to their second variable, thus K turns to bea linear connection.

Next we look at the Christoffel symbols of plb-connections. It will beshown that these connections are characterized by Christoffel symbols whosevalues are restricted to continuous linear maps represented by projectivelimits. More precisely, in accordance with the previous formalism [see also§ 1.5.3 and (7.1.3)], the Christoffel symbols of a plb-connection K = lim

←−Ki,

over the vb-charts(Uα, φα,Φα ≡ lim

←−Φiα

)of ℓ, are the maps

Γα : φα(Uα) −→ L(F,L(B,F)),


determined, as usual, by

(7.1.11) Γα(x).λ = κα(x, λ); (x, λ) ∈ φα(Uα)× F.

Then we obtain the following preliminary result:

Proposition 7.1.3. Γα(x) = lim←−Γiα(x), for every x ∈ φα(Uα) and α ∈ I.

Proof. Because Γiα(x)i∈N is a projective system, as an immediate conse-quence of (7.1.8), we need only to show that Γiα(x)i∈N converges to Γα(x).Indeed, if we denote by

(7.1.12) ri : L(B,F) −→ L(B,Ei) : f 7→ ρi f

the canonical projections of L(B,F) ≡ lim←−L(B,Ei), then, by (7.1.9),

(ri Γα(x)

)(λ) = ri

(κα(x, λ)

)= κiα

(x, ρi(λ)

)=(Γiα(x) ρi)

)(λ),

for every λ ∈ F and every i ∈ N. Therefore, Proposition 2.3.5 implies theassertion.

It is worth noticing here that, despite the previous “point-wise” conver-gence, the Christoffel symbols of a linear plb-connection themselves are notnecessarily projective limits, i.e. equalities Γα = lim

←−Γiα are not in general

true. However, each Γα is associated to a limit of Christoffel-like maps inthe following way: For every limit vb-chart (Uα, φα,Φα) ≡ lim

←−

(Uα, φα,Φ

iα

)

of ℓ ≡ lim←− ℓi as before, we define the maps

(7.1.13)Γ∗iα : φα(Uα) −→ Hi(Ei,L(B,Ei)) :

x 7→(Γ1α(x),Γ2

α(x), . . . ,Γiα(x)),

for all i ∈ N. Referring to Theorem 2.3.10, in particular to equalities (2.3.3)and (2.3.4), we obtain the limit space

H(F,L(B,F)) = lim←−Hi(F,L(B,Ei))

and the continuous linear embedding

(7.1.14) ε : H(F,L(B,F)) → L(F,L(B,F)) : (gi)i∈N 7→ lim←− gi.

Note that the previous considerations are meaningful because L(B,F) ≡lim←−L(B,Ei) while in (7.1.13) we have also applied (2.3.9), (2.3.9′).

A direct consequence of Theorem 2.3.10 is now the next result.


Proposition 7.1.4. With the previous notations we have:i) The pls-map

Γ∗α := lim←−Γ∗iα : φα(Uα) −→ H(F,L(B,F))

can be defined.ii) The ordinary Christoffel symbols Γα : φα(Uα) → L(F,L(B,F))α∈I

of the plb-connection K = lim←−Ki factorize in the form

Γα = ε Γ∗α.

The maps Γ∗αwill be called generalized or plb-Christoffel symbols.They are important because they characterize plb-connections, as shown inthe next result.

Proposition 7.1.5. Let K be an arbitrary linear connection on a plb-vectorbundle ℓ ≡ lim

←−ℓi. If the ordinary Christoffel symbols of K,

Γα : φα(Uα) −→ L(F,L(B,F))α∈I ,

factor into Γα = ε Γ∗α, where

Γ∗α := lim←−Γ∗iα : φα(Uα) −→ H(F,L(B,F))

are pls-maps, for all vector bundle charts (Uα, φα,Φα) ≡ lim←−

(Uα, φα,Φ

iα

),

then K coincides with a linear plb-connection, i.e. K = lim←−Ki.

Proof. The assumption that Γ∗α = lim←−

Γ∗iα is a pls-map means that

Γ∗iα : φα(Uα) −→ Hi(Ei,L(B,Ei));

thus, for every x ∈ φα(Uα),

Γ∗iα (x) =(q1(x), . . . , qi(x)

)∈

i∏

k=1

L(Ek,L(B,Ek)

),

so that rjk qj(x) = qk(x) ρjk holds true for every j, k = 1, . . . , i, withj ≥ k. If

Prk :i∏

j=1

L(Ej,L(B,Ej)

)−→ L

(Ek,L(B,Ek)

)

is the k-th projection, we define the maps

Γiα := Pri Γ∗iα ; i ∈ N,


which, by their construction, satisfy

(7.1.15) rji Γjα(x) = Γiα(x) ρji; x ∈ φα(Uα), j ≥ i.

Since Γαα∈I are the Christoffel symbols of K, thus they satisfy the ana-log of the compatibility condition (1.5.12), it is readily verified that a sim-ilar condition (within the Banach framework) holds for the maps Γiαα∈I .Therefore, the factor Banach vector bundles ℓi admit respective linear con-nections Ki (i ∈ N), with Christoffel symbols the given families of maps.Observing that (7.1.15) is precisely (7.1.8), we conclude that Ki is a pro-jective system with limit the plb-connection lim

←−Ki.

We shall show that lim←−Ki coincides with the initial connection K. By

Proposition 2.3.5 and the local characterization of connections via the localcomponents, along with (7.1.10) and (7.1.12), it suffices to verify the equality

ri κα = κiα (

idφα(Uα)×ρi).

Equivalently, it suffices to show that, for every(x, λ = (λi)

)∈ φα(Uα)× F,

ri(κα(x, λ)) = κiα(x, λi)

⇔ ri(Γα(x).λ) = Γiα(x).λi

⇔ ri Γα(x) = Γiα(x) ρi.

The last equality is a consequence of the factorization assumption of Γα. Asa matter of fact,

Γα(x) = ε(Γ∗α(x)

)

⇔ Γα(x) = lim←−Γiα(x)

⇔ ri Γα(x) = lim←−

Γiα(x) ρi.

Thinking of H(F,L(B,F)) as a subspace of L(F,L(B,F)) by the embed-ding (7.1.14), we summarize the preceding two propositions in the followingmain result.

Theorem 7.1.6. A linear connection K on a plb-vector bundle ℓ ≡ lim←−

ℓi

is a linear plb-connection if and only if its Christoffel symbols take values inthe subspace H(F,L(B,F)) of L(F,L(B,F)).


7.2 Parallel displacement and holonomy groups

As we mentioned in the introduction to the present chapter, the paralleldisplacement along curves of the base space cannot be ensured. Conse-quently, holonomy groups in the classical sense cannot even be defined. Theprojective limit approach gives a way out in this case too. If we are re-stricted to plb-vector bundles and connections, the above important groupscan be recovered and yield results very close to those obtained in the Banachcase, bypassing thus the problems concerning differential equations in the(Frechet) models.

Keeping up the formalism of plb-vector bundles and connections appliedin the previous section, we prove the following first result on parallel sectionsof a limit vector bundle. In this respect we also refer to § 1.5.5 for the usualdefinitions.

Lemma 7.2.1. Let β : [0, 1] → B be a smooth curve in the base of a plb-vector bundle ℓ = (E,B, π) ≡ lim←−(Ei, B, πi) = lim←− ℓ

i, and let K = lim←−Ki be

a linear plb-connection on ℓ. Theni) Every section ξ : [0, 1] → E along β (: ξ ∈ Γβ(E)) is realized as a

projective limit of corresponding sections on the factor bundles ℓi, i.e.

ξ = lim←− ξi, ξ ∈ Γβ(Ei).

ii) A section ξ ∈ Γβ(E) is parallel (with respect to K), if and only ifξi ∈ Γβ(Ei) is parallel (with respect to Ki), for every i ∈ N.

Proof. i) The factor sections are obtained by projecting ξ to the factor bun-dles; that is,

ξi := f i ξ : [0, 1]→ Ei; i ∈ N,

where f i : E = lim←−

Ei → Ei are the canonical projections. They are smoothas composites of smooth maps and are projected to β since

πi ξi = πi f i ξ = π ξ = β.

Besides, their relation with the connecting morphisms f ji : Ej → Ei, namely

f ji ξj = f ji f j ξ = f i ξ = ξi; j ≥ i,

ensures that lim←− ξi is defined and coincides with ξ by the very definition of

(ξi)i∈N.

ii) Assume first that each ξi ∈ Γβ(Ei) is parallel, thus by (1.5.16),

Ki Tξi ∂ = 0.

7.2. Parallel displacement and holonomy groups 235

Hence, in virtue of (3.2.8),

K Tξ ∂ = lim←−Ki lim←−Tξ

i ∂ = lim←−(Ki Tξi ∂) = 0,

which means that K is parallel.Conversely, if ξ ∈ Γβ(E) is parallel, then

K(Tξ(∂t)) = 0 ∈ Eπ(ξ(t)) = Eβ(t) = π−1(β(t)); t ∈ [0, 1],

and, by Theorem 3.2.8,

0 = f i(K(Tξ(∂t))

)= Ki

(Tf i(Tξ(∂t))

)= Ki

(Tξi(∂t)

)∈ Eiβ(t),

for all t ∈ [0, 1]. This shows that every ξi is parallel.

Using the preceding lemma we obtain now the following main result ofthis section.

Theorem 7.2.2. Let ℓ = lim←−

ℓi be a plb-vector bundle endowed with a linear

plb-connection K = lim←−

Ki. If β : [0, 1]→ B is a smooth curve and u ∈ Eβ(0)an arbitrarily chosen point, then there exists a unique parallel section of ℓalong β, satisfying the initial condition (0, u).

Proof. By the first part of the proof of Theorem 5.2.5 (referring the fibresof a plb-bundle), Eβ(0) = lim←−E

iβ(0). Therefore, any u ∈ Eβ(0) takes the form

u = (ui)i∈N, where ui ∈ Eiβ(0) and f ji(uj) = ui (j ≥ i). Since each Ei

is a Banach vector bundle, there is a unique parallel ξi ∈ ΓβEi such that

ξi(0) = ui. Then, for every j ≥ i, the map f ji ξj : [0, 1] → Ei is a parallelsection along β, since

πi (f ji ξj) = πj ξj = β,

Ki T (f ji ξj) ∂ = Ki Tf ji Tξj ∂ =

= f ji Kj Tξj ∂ = f ji 0 = 0,

the last equality being a consequence of the fibre-wise linearity of the con-necting morphisms. Moreover,

(f ji ξj)(0) = f ji(uj) = ui = ξi(0).

Hence, by the uniqueness of the parallel section with initial condition (0, ui),

f ji ξj = ξi; j ≥ i,


from which we deduce that the smooth map ξ := lim←−

ξi : [0, 1]→ E is defined.The latter is a section of E along β, because

(π ξ)(t) = π((ξi(t))i∈N

)=(πi(ξi(t))

)i∈N≡ β(t).

It is also parallel according to Lemma 7.2.1 and satisfies the desired initialcondition

ξ(0) =((ξi)(0)

)i∈N

= (ui)i∈N = u.

Finally, assume that there is another parallel section η of E along β suchthat η(0) = u. Then η determines a family (ηi)i∈N of analogous parallelsections on the factor bundles Ei with ηi(0) = ui (i ∈ N). Once again, theuniqueness of parallel sections in Banach vector bundles, yields ηi = ξi, forall i ∈ N. Thus ξ = lim

←−ξi = lim

←−ηi = η, which completes the proof.

Definition 7.2.3. Let β : [0, 1] → B be a smooth curve in the base of aplb-vector bundle E = lim

←−Ei endowed with a linear connection K (thus

K ≡ lim←−

Ki). Then, analogously to (1.5.18), the parallel displacement ortranslation along β is the map

τβ : Eβ(0) −→ Eβ(1) : u 7→ ξu(1),

where ξu is the unique parallel section of E along B, with ξu(0) = u.

The parallel displacement remains also within the category of projectivelimits because of the following result.

Proposition 7.2.4. Let E = lim←−

Ei be plb-vector bundle endowed with a

linear connection K = lim←−

Ki. For every smooth curve β : [0, 1] → B inthe base of E, the parallel displacement along β in E coincides with theprojective limit of the corresponding parallel displacements in the Banachfactor bundles, i.e. τβ = lim←− τ

iβ

Proof. In conjunction with Lemma 7.2.1 and Theorem 7.2.2, it suffices tocheck the compatibility of

(τ iβ)

with the connecting morphisms f ji : Ej → Ei

and the canonical projections f i : E = lim←−Ei → Ei, restricted to the fibres

Eβ(0) = lim←−Eiβ(0) and Eβ(1) = lim←−E

iβ(1); in other words, we should verify

the commutativity of the diagrams


Ejβ(0)

τjβ

- Ejβ(1) Eβ(0)τβ

- Eβ(1)

Eiβ(0)

f ji

?

τ iβ

- Eiβ(1) Eiβ(0)

f ji

?

f i

?

τ iβ

- Eiβ(1)

f i

?

Let any v ∈ Ejβ(0). If ξjv : [0, 1] → Ej is the (unique) parallel section

of the bundle Ej along β with ξjv(0) = v, then, as we pointed out in theproof of Theorem 7.2.2, the corresponding section of Ei along β with initialcondition (0, f ji(v)) is precisely f ji ξjv. Therefore,

(7.2.1)

(f ji τ jβ

)(v) = f ji

(ξjv(1)

)= ξifji(v)(1)

= τ iβ(f ji(v)

)=(τ iβ f

ji)(v),

thus proving the commutativity of the first diagram and the existence of thelimit lim←− τ

iβ.

Similarly, for any u = (ui) ∈ E, ξu = lim←− ξiui . In particular, for every

u = (ui) ∈ Eβ(0),

(7.2.2)(f i τβ

)(u) = f i(ξu(1)) = ξiui(1) = τ iβ(ui) = τ iβ

(f i(u)

),

implying the commutativity of the second diagram and the equality of thestatement.

We define the holonomy group Φb of a plb-connection K = lim←−

Ki,with reference point b ∈ B, by setting

Φb :=τβ : Eb

≃−−→ Eb (toplinear isomorphism)

for all smooth curves β : [0, 1] → B with β(0) = β(1) = b. Since we aredealing with a fixed linear connection K, and there is no danger of confusionwith connections on principal bundles, here we simply write Φb instead ofKΦb, the latter originally being defined in § 1.5.5.

The restricted holonomy group Φ0b is defined analogously, by con-

sidering curves closed at b, homotopic to zero.


To obtain substantial properties of the holonomy groups we need toassume the following strong condition, which is not in general true:

(7.2.3)

The connecting morphisms f ji : Ej → Ei and the canonical pro-

jections f i : E = lim←−

Ei → Ei of the projective system of vector

bundles Eii∈N are surjective maps.

Then we are in a position to prove:

Proposition 7.2.5. If Φibi∈N is the family of corresponding holonomy

groups of the factor connections, the following assertions are true:i) The projective limit group lim←−Φi

b exists.

ii) Φb is a subgroup of lim←−Φib by means of an isomorphism.

Proof. For the first assertion we define the maps

σji : Φjb −→ Φi

b : τ jβ 7→ τ iβ, j ≥ i.

They are well-defined, for if

(7.2.4) τjβ = τ jγ ,

we have to show that

τ iβ = σji(τ jβ) = σji(τ jγ ) = τ iγ ,

for any smooth curves β and γ closed at b. Indeed, because of the existenceof the limits τβ = lim←− τ iβ and τγ = lim←− τ iγ , equality (7.2.4) yields

f ji τ jβ = f ji τ jγ ,

which, in virtue of (7.2.1), turns into

τ iβ fji = τ iγ f

ji.

This proves the desired equality τ iβ = τ iγ as a result of the surjectivity of theconnecting morphisms [recall condition (7.2.3)].

The maps σji are also group morphisms, since the properties of theordinary (in Banach bundles) parallel displacement (1.5.18) imply that

σji(τjβ τ

jγ

)= σji

(τjβ∗γ

)= τ iβ∗γ = τ iβ τ

iγ = σji

(τjβ

) σji

(τ jγ), j ≥ i.

On the other hand, for every triplet of indices j ≥ i ≥ k,

(σik σji

)(τjβ

)= σik

(σji(τjβ

))= σik

(τ iβ)

= τ kβ = σjk(τjβ

).


The previous arguments prove that the projective limit of groups lim←−

Φib can

be defined.

For the second assertion we project the holonomy group Φb onto thecorresponding factor groups by means of the maps

hi : Φb −→ Φib : τβ 7→ τ iβ , i ∈ N.

Working as in the proof of the first assertion, using (7.2.2) and the surjectiv-ity of the canonical projections restricted to the fibres, we prove that (hi)i∈Nare well-defined group morphisms yielding the projective limit

(7.2.5) h := lim←−

hi : Φb −→ lim←−

Φib.

Concerning the kernel of h we observe that

τβ ∈ Kerh ⇔ h(τβ) =(

idEib

)i∈N⇔ hi(τβ) = idEi

b⇔ τ iβ = idEi

b,

for all i ∈ N. However, according to Proposition 7.2.4, τβ = lim←−

τ iβ; there-fore, τβ = idEb

, i.e. the kernel of h is the trivial group and h an injectivehomomorphism by which Φb can be identified with a subgroup of lim←−Φi

b.

Remarks 7.2.6. 1) From the proof of assertion ii) it is clear that Φb co-incides, up to a group isomorphism, with a subgroup of a projective limitof Banach-Lie groups. However, Φb fails to be a Frechet-Lie group itselfbecause the existence of appropriate limit charts cannot be assured. Thesame remark applies to the holonomy groups of connections on plb-principalbundles whose structure has been described in Theorem 4.3.5.

Both cases clarify now the claim that the holonomy groups of projectivelimit connections live between the categories of plb-manifolds and topologi-cal groups, as commented in the introduction to the present chapter.

2) Under the identification induced by h [see (7.2.5)], Φb becomes atopological group, in contrast to GL(F) (in which Φb embeds) that does notadmit any reasonable topological group structure.

3) The assumption (7.2.3) is always fulfilled in the case of topologicalspaces via the construction of an appropriate system of topological spaceswhose limit is homeomorphic to the initial limit space (for details we referto [Dug75, Appendix Two, § 2.8]. This is not necessarily true for morecomplicated structures such as (Banach) vector bundles.


7.3 Connections on plb-vector and frame bundles

Linear connections on finite-dimensional or Banach vector bundles can beapproached by means of the general theory of (infinitesimal) connections onprincipal bundles by associating vector bundles with their bundles of frames.The same is true in our setting if we consider the generalized frame bundlesdefined in § 6.5, where we have also explained the reasons necessitating theintroduction of these bundles.

More precisely, we establish here a correspondence between linear con-nections on a limit vector bundle E and connections (in the sense of § 4.2)on the generalized bundle of frames P (E), defined in Proposition 6.5.4 [seealso Remark 6.5.5(3)]. To this end we fix a plb-vector bundle

ℓ = (E,B, π) ≡ lim←−

ℓi = lim←−

(Ei, B, πi)

of fibre type the Frechet space F = lim←−

Ei, with connecting morphisms

f ji : Ej → Ei, j ≥ i, and canonical projections f i : E → Ei, i ∈ N. Aswe have seen in Proposition 6.5.4,


); (rji, hji0 , idB)

i,j∈N

is a projective system of Banach principal bundles inducing the plb-principalbundle (

P (E) = lim←−P (Ei),H0(F), B,p = lim←−pi).

Referring also to equalities (2.3.5), (2.3.6) and (5.1.2), we recall that

P (Ei) =⋃

x∈B

Hi0(F, Ex),

whereas

(rji, hji0 , idB) :(P (Ej),Hj0(F), B,pj

)−→


)

is a morphism of principal bundles with

rji(q1, . . . , qj

)=(q1, . . . , qi

)and hji0

(g1, . . . , gj

)=(g1, . . . , gi

).

With the notion of related connections in mind (see § 1.7.5), we firstexamine the effect that the presence of a linear plb-connection on E has onthe projective system P (Ei)i∈N:

Lemma 7.3.1. A linear plb-connection K = lim←−Ki on E determines a

family of(rji, hji0 , idB

)-related connections (θi)i∈N on the Banach principal

bundles P (Ei), respectively.

7.3. Connections on plb-vector and frame bundles 241

Proof. As described in §1.7.4, each linear connection Ki on Ei induces aconnection form

ωi ∈ Λ1(P (Ei),GL(Ei) ≡ L(Ei)

)

on the ordinary frame bundle P (Ei) of Ei. In fact, ωi is completely de-termined by the corresponding local connection forms ωiα ∈ Λ1(Uα,L(Ei)),α ∈ I, defined in turn by the Christoffel symbols Γiαα∈I of Ki by theanalog of equality (1.7.16), namely

ωiα,x(v) = Γiα(φα(x))(φα(v)

), x ∈ Uα, v ∈ TxB.

Here (Uα, φα) are the charts inducing local trivializations of the involvedbundles. As a result, for each α ∈ I, we may define the differential formsθiα ∈ Λ1(Uα,H

i(F)) by setting

(7.3.1) θiα =(ω1α, . . . , ω

iα

);

more explicitly,

(θiα)x(v) :=((ω1α)x(v), . . . , (ωiα)x(v)

); x ∈ Uα, v ∈ TxB,

for every i ∈ N. We shall show that(θiα)i∈N

determine a connection on the

principal bundle P (Ei).Towards this end, first we readily check that the adjoint representation

of Hi0(F),Adi : Hi0(F) −→ Aut(Hi(F)),

is given by

Adi(g1, . . . , gi

)= Ad1

(g1)× · · · ×Adi

(gi);

(g1, . . . , gi

)∈ Hi0(F),

where the operator Adi on the right-hand side denotes the usual adjointrepresentation of GL(Ei), for any Banach space Ei; hence,

[Adi

(g1, . . . , gi

)](q1, . . . , qi

)=(g1 q1

(g1)−1

, . . . , gi qi (gi)−1)

,

for every(q1, . . . , qi

)∈ Hi(F).

Similarly, the left Maurer-Cartan differential of Hi0(F)-valued maps onB (see § 1.2.6) is given by

(H1, . . . ,H i

)−1d(H1, . . . ,H i

)=((H1)−1

dH1, . . . ,(H i)−1

dH i),

for every(H1, . . . ,H i

)∈ C∞

(B,Hi0(F)

), with

(Hk)−1

dHk denoting the leftMaurer-Cartan differential of Hk ∈ C∞

(B,GL(Ek)

), k = 1, . . . , i.


With the previous notations, the compatibility condition (1.7.10), adapt-ed to the case of P (Ei), along with (6.5.3) concerning the transition func-tions of the latter bundle, leads directly to

(7.3.2) θiβ = Adi(g−1αβ

).θiα + g−1αβ · dgαβ

on Uαβ . Consequently, for each index i ∈ N, the forms θαα∈I indeeddetermine a connection form θi on P (Ei), with θi ∈ Λ1

(P (Ei

),Hi(F)) and

local connection forms θαα∈I .

Finally, to prove that θj and θi are(rji, hji0 , idB

)-related (j ≥ i), it is

sufficient (by Proposition 1.7.1) to verify the analog of (1.7.20), i.e. equality

(7.3.3) hji0 .θjα = Adi

((hiα)−1

)θjα + (hiα)−1dhiα,

over Uα. Before proving this claim, recall that hji0 : Hj(F) → Hi(F) is the

Lie algebra morphism induced by the Lie group homomorphism hji0 , andhiα : Uα → H

i0(F) is defined by equality

(7.3.4) rji(σjα(x)) = σiα(x) · hiα(x),

where σkα (k = j, i ; j ≥ i) are the natural local sections of P (Ek). Now,because the local structure of P (Ek) (see Proposition 6.5.1) implies that

σkα(x) =((τ1x)−1

, . . . ,(τkx)−1)

,

it follows from (7.3.4) that

hiα(x) =(

idE1 , . . . , idEi

),

for all x ∈ Uα; in other words, σiα is constant, thus (7.3.3) reduces to

(7.3.5) hji0 .θjα = θiα.

Taking into account that hji0 coincides with

hji : Hj(F) −→ Hi(F) :(g1, . . . , gj

)7→(g1, . . . , gi

); j ≥ i,

(after the identification of the Lie algebra of Hi0(F) with Hi(F)), it followsthat (7.3.5) is an obvious consequence of (7.3.1).


Based on the preceding result, we define the H(F)-valued 1-form θ onP (E) by setting

(7.3.6) θ((gi)i∈N

):= lim←−

(θi(g1, g2, ..., gi

)).

Note that, after the identification

(gi)i∈N≡(g1, (g1, g2), . . . (g1, . . . , gi), . . .

)= lim←−

((g1, . . . , gi)

)

[see (2.3.9), (2.3.9′)], (g1, . . . , gi) can be thought of as the i-th projection of(gi)i∈N. Also, according to the comments following Definition 4.2.1, equality(7.3.6) takes the symbolic expression θ = lim←− θ

i.Besides, using the generalized smooth structure of P (E) discussed in

Remarks 6.5.5(2) and 6.5.5(3), θ is a generalized smooth form on P (E), sowe may write θ ∈ Λ1(P (E),H(F))); therefore, in virtue of Definition 4.2.1and Theorem 4.2.5, θ may be considered as a connection form on P (E).

The previous arguments actually prove:

Proposition 7.3.2. A linear plb-connection K = lim←−

Ki on the plb-vector

bundle E = lim←−

Ei determines a (generalized) connection form θ on P (E).

To find the local connection forms of θ, we easily see that the naturalsections of σαα∈I of P (E), with respect to an open cover Uαα∈I ofB overwhich all the bundles involved are locally trivial, are given by σα = lim

←−σiα,

where σiαα∈I are the corresponding natural sections of P (Ei), for everyi ∈ N. In particular, for every x ∈ Uα,

σα(x) =((τ ix)−1)

i∈N

≡((τ1x)−1

,((τ1x)−1

,(τ2x)−1)

, . . . ,((τ1x)−1

, . . . ,(τ ix)−1)

, . . .)

≡(σ1α(x), σ2α(x), . . . , σiα(x), . . .

)=(σiα(x)

)i∈N

.

Accordingly, the local connection forms θα ∈ Λ1(Uα,H(F)) (α ∈ I) satisfy,as expected, equality

(7.3.7) θα = lim←−

θiα.

Indeed, for every x ∈ Uα, we have:

θα(x) = θα,x =(σ∗αθ

)x

= θσα(x) Txσα

= θ((σiα(x)

))i∈N Txσα


or, by (7.3.6) and Proposition 3.2.5,

= lim←− θi((τ1x)−1

, . . . ,(τ ix)−1)

lim←−Txσi

= lim←−

(θiσiα(x) Txσ

i)

= lim←−

(((σiα)∗θi

)x

)

= lim←−

(θiα, x

)= lim←−

(θiα(x)

)=(

lim←−

θiα)(x),

which yields (7.3.7).

Completing Proposition 7.3.2, we prove the following theorem, gener-alizing the usual association of linear connections on vector bundles withconnections on the bundle of frames discussed in § 1.7.4.

Theorem 7.3.3. There is a bijective correspondence between linear plb-connections on a plb-vector bundle E = lim←−E

i and connections on the gen-eralized bundle of frames P (E).

Proof. We have already seen that a connection K = lim←−

Ki on E induces aconnection form θ on P (E).

Conversely, a connection form θ ∈ Λ1(P (E),H(F)) determines a plb-connection on E: Clearly, θ = lim

←−θi, where θi ∈ Λ1(P (E),Hi(F))). By

the general theory of connections on plb-principal bundles, we may writeθα = lim←− θ

iα, where θα = σ∗αθ and θiα = (σiα)∗θi, for every α ∈ I and i ∈ N.

Recall from previous proofs that σα and σiα are natural sections of P (E)and P (Ei), respectively.

As a first step to our goal, we check that, for a fixed i ∈ N,(θiα)α∈I

are

the local connection forms of a connection on P (Ei), inducing in their turn aconnection on the ordinary bundle of frames. Indeed, since θ is a connectionform on P (E), its local connection forms satisfy the compatibility condition

(7.3.8) θβ = Ad(g−1αβ

).θα + g−1αβdgαβ .

Therefore, taking into account the equalities θα = lim←−

θiα (α ∈ I) and gαβ =

lim←−

giαβ (see Proposition 4.1.8), the i-th projection of (7.3.8) leads to

(7.3.9) θiβ = Adi((giαβ)−1

).θiα +

(giαβ)−1

dgiαβ,

which proves that(θiα)α∈I

determine a connection on P (Ei). On the other

hand, for every x ∈ Uα and every v ∈ TxB, we have that θiα,x(v) ∈ Hi(F);hence, we may write

θiα,x(v) =(ω1α,x(v), . . . , ωiα,x(v)

),


which implies that ωkα ∈ Λ1(Uα,L(Ek)) (k = 1, . . . , i) and

(7.3.10) ρjk ωjα,x(v) = ωkα,x(v) ρjk, j, k = 1, . . . , i; j ≥ k,

as a consequence of the structure of Hi(F) [see (5.1.1) in conjunction with(2.3.5)]. Since

ωiα,x(v) = Pri(θiα,x(v)

); Pri : L(E1)× · · · × L(Ei) −→ L(Ei),

equality (7.3.9) leads to

ωiβ = Ad(giαβ).ωiα +

(giαβ)−1

dgiαβ ;

more explicitly,

ωiβ,x(v) =(giαβ(x)

)−1 ωiα g

iαβ(x) +

(giαβ)−1 Txg

iαβ(v),

for every x ∈ Uα and v ∈ TxB. This means that ωiαα∈I define a connectionon the ordinary bundle of frames P (Ei) of Ei.

Now, following the procedure of § 1.7.4 and setting

(7.3.11)(Γiα(y).λ

)(h) :=

(ψ∗αω

iα

)y(h).λ; y ∈ φα(Uα), λ ∈ Ei, h ∈ B,

where ψα = φ−1α , we obtain the family

Γiα : φα(Uα)→ L(Ei,L(B,Ei))α∈I

.

Then (7.3.9) implies the compatibility condition of

Γiαα∈I

(for each i ∈ N),

thus the latter are the Christoffel symbols of a linear connection Ki on Ei.Furthermore, translating (7.3.10) in terms of Christoffel symbols, we seethat

ρji((Γjα(y).λ)(h)

)=(Γiα(y).ρji(λ)

)(h); y ∈ φα(Uα), λ ∈ Ei, h ∈ B,

or, equivalently [in virtue of (1.5.8′) and (7.1.3)],

ρji Γjα(y)(λ) = Γiα(y)(ρji(λ)

); y ∈ φα(Uα), λ ∈ Ei,

⇔ rji(κjα(y, λ)

)= κiα

(y, ρji(λ)

); y ∈ φα(Uα), λ ∈ Ei,

⇔ rji κjα = κiα (

idφα(Uα)×ρji);

that is, we obtain (7.1.6) which guarantees that the linear connections Kj ≡(Γjα)α∈I

and Ki ≡(Γiα)α∈I

are (f ji, idB)-related. Hence, (7.1.3) is fulfilled

and K := lim←−

Ki is a plb-linear connection on E.The desired bijectivity is a direct consequence of the association of linear

connections with connections on principal bundles of frames, and vice-versa,by relating Christoffel symbols with local connection forms as in (7.3.11) andits inverse in the proof of Lemma 7.3.1.


Corollary 7.3.4. Let K ≡

Γαα∈I

be an arbitrary linear connection on

E = lim←−Ei. If the Christoffel symbols of K are related with the forms

ωiαα∈I

[derived from θ ≡ (θα)α∈I ] by (7.3.11), then K is necessarily aplb-linear connection.

Proof. The connection form θ ≡ (θα)α∈I determines a plb-linear connectionK = lim

←−Ki, with Ki determined by (θiα)α∈I . Since the Christoffel sym-

bols of K satisfy also (7.3.11), we conclude that K = K. This proves theassertion.

Chapter 8

Geometry of second ordertangent bundles

The second order tangent bundle T 2B of a smooth manifold B consists ofthe equivalence classes of curves in B that agree up to their acceleration,and arises in a natural way in several problems of theoretical physics anddifferential geometry (cf., for instance, [DG05], [DR82]). However, the vectorbundle structure on T 2B is not as straightforward as that of the ordinary(viz. first order) tangent bundle TB of B; in fact, it relies on the choice ofa linear connection on B.

Aiming at the reader’s convenience, in §§ 8.1.1–8.1.3 we specialize to theordinary tangent bundle a few facts from the theory of linear connections onvector bundles, exhibited in Chapter 1. In § 8.2 we proceed to the details ofthe structure of T 2B, for a Banach manifold B. Our next target is to find outthe extent of the dependence of the vector bundle structure of T 2B on thechoice of the linear connection on B. This naturally leads us to the notionof second order differentials (§ 8.3). With their help, we prove (in § 8.4) thatrelated (or conjugate) connections induce—up to isomorphism—the samevector bundle structure on T 2B.

The last two sections are devoted to the projective limits of second or-der tangent bundles (§ 8.5), and the generalized second order frame bundle(§ 8.6). Note that, whereas the tangent vectors of curves and velocities ofparticles naturally form vector bundles, their derivatives, which yield cur-vatures and accelerations, do not. In order to cover this gap, second or-der vector bundle structures are constructed for projective limits of Banachmodelled manifolds.

247

248 Chapter 8. Geometry of second order tangent bundles

8.1 The (first order) tangent bundle in brief

We transcribe a few features of the general theory of linear connections tothe particular case of the tangent bundle of a Banach manifold. This willpave the way to the main topics of the present chapter.

8.1.1 Linear connections on manifolds

Let B be a Banach manifold with atlas (Uα, φα)α∈I . We have seen in§ 1.1.4, in conjunction with § 1.4.1, that the structure of the tangent bundle(TB,B, τB) of B is determined by the vb-charts (Uα, φα,Φα)α∈I , wherethe (trivializing) diffeomorphism Φα : TB|Uα ≡ τ−1B (Uα) → φα(Uα) × B isdetermined by

(8.1.1) Φα(v) =(φα(x), φα(v)

)= (φα(x),Φα,x(v)) =

(φα(x), (φα γ)′(0)

),

for every x ∈ B and every v = [(γ, x)] ∈ TxB, where γ is a smooth curve inB passing through x.

Analogously, the double tangent bundle (T (TB), TB, τTB) has a localstructure induced by the vb-charts

(τ−1TB (TB|Uα),Φα, Φα

), the diffeomor-

phismΦα : τ−1TB (TB|Uα) −→ φα(Uα)× B× B× B

being given by the analog of (1.5.5), namely

(8.1.2)Φα(X) =

(Φα(τTB(X)),Φα(X)

)=(Φα(v),Φα(X)

)=

=(φα(x), φα(v),Φα(X)

)=(φα(x), (φα γ)′(0), (Φα c)

′(0)),

for every X = [(c, v)] ∈ Tv(TB), v = [(γ, x)] ∈ TxB, where c is a smoothcurve in TB through v.

For the sake of brevity we shall write

(8.1.3) Φα ≡ (Uα, φα,Φα), Φα ≡(τ−1TB(TB|Uα),Φα, Φα

).

In the same vein, specializing the material of §§ 1.5.1–1.5.3 to the case ofthe tangent bundle, we see that a linear connection on TB is a vb-morphismK : T (TB)→ TB whose local representation (with respect to Φα and Φα),

Kα = Φα K Φ−1α : φα(Uα)× B× B× B −→ φα(Uα)× B,

has the form Kα(x, h, y, k) = (x, k+κα(x, h).y), where the local component

κα : φα(Uα)× B→ L(B)

8.1. The (first order) tangent bundle in brief 249

is linear with respect to the second variable. As is the custom, K is brieflycalled a linear connection on B.

The Christoffel symbols Γαα∈I of K are given by the general formulasof § 1.5.3, according to the form of their range. The compatibility conditionof the symbols Γα : φα(Uα) → L2(B,B;B)α∈I reduces to the followingvariant of (1.5.12):

(8.1.4)Dφαβ(x) Γβ(x) = D2φαβ(x) +

+ Γα(φαβ(x)) (Dφαβ(x)×Dφαβ(x)

),

for every x ∈ φβ(Uαβ); α, β ∈ I, with φαβ = φα φ−1β . This is the case,

because now

Gαβ(x) = φα,b φ−1β,b = Dφαβ(x); b = φ−1β (x) ∈ Uαβ.

Analogous conditions hold for the other expressions of the Christoffel sym-bols.

8.1.2 First order differentials

We fix two Banach manifolds B and B′, with respective atlases

(Uα, φα) ≡ (Uα, φα,B)α∈I , and (Vβ , φβ) ≡ (Vβ, φβ ,B′)β∈J .

Let f : B → B′ be a smooth map. As in (1.5.21), fβα = φβ f φ−1α denotes

the local representation of f with respect to the charts (Uα, φα) and (Vβ, φβ)such that f(Uα) ⊆ Vβ. Then the local representation of the ordinary (firstorder) differential (or tangent map) Tf : TB → TB′,

(Ψβ f Φ−1α

): φα(Uα)× B −→ φβ(Vβ)× B′,

relative to the charts Φα and Ψβ [see convention (8.1.3)], is given by theanalog of (1.5.20), which now is

(8.1.5)(Ψβ f Φ−1α

)(x, h) =

(fβα(x),Dfβα(x).h

),

for every (x, h) ∈ φα(Uα)× B.For a smooth map f as before, the pair of differentials (T (Tf), T f), with

T (Tf) : T (TB) → T (TB′), is a vb-morphism between the double tangentbundles (T (TB), TB, τTB) and (T (TB′), TB′, τTB′). The local representa-tion of T (Tf) [in terms of the vb-charts Φα and Ψα as in (8.1.2)] is themap

Ψβ T (Tf) Φ−1α : φα(Uα)× B× B× B −→ φβ(Vβ)× B′ × B′ × B′,


given by [see also (1.5.25)]

(8.1.6)

(Ψβ Tf Φ−1α

)(x, h, y, k) =

=(fβα(x),Dfβα(x).h,Dfβα(x).y,Dfβα(x).k +D2fβα(x)(y, h)

),

for every (x, h, y, k) ∈ φα(Uα)× B× B× B.

8.1.3 Related linear connections on manifolds

Let B and B′ be Banach manifolds and f : B → B′ a smooth map. Assumethat B, B′ are equipped with the connection K and K ′, respectively. Wesay that K and K ′ are f-related if they are (Tf, f)-related in the sense of§ 1.5.6.

The general formula (1.5.27), expressing related connections in terms oflocal components, now becomes

(8.1.7)κ′β(fβα(x),Dfβα(x).h

)(Dfβα(x).y) =

= Dfβα(κα(x, h).y

)−D2fβα(x)(y, h),

for every (x, h, y, k) ∈ φα(Uα)×B×B×B. Therefore, if K and K ′ are linearconnections, then, in virtue of (1.5.30), they are f -related if and only if

(8.1.8) Γ′β(fβα(x)) (Dfβα(x)×Dfβα(x)

)= Dfβα Γα(x)−D2fβα(x),

for every charts (Uα, φα) and (Vβ , φβ) with f(Uα) ⊆ Vβ, and every x ∈φα(Uα). In the preceding equality we have considered Christoffel symbolsof the form Γα : φα(Uα)→ L2(B,B;B) and Γ′β : φβ(Vβ)→ L2(B

′,B′;B′).

8.2 Second order tangent bundles

We fix throughout this section a smooth manifold B, modelled on a Banachspace B, with atlas (Uα, φα)α∈I . For every x ∈ B,

Cx = γ : (−ε, ε)→ B | γ smooth with γ(0) = x; ε > 0,

obviously denotes the set of smooth curves through x. We define the follow-ing equivalence relation in Cx:

(8.2.1) γ1 ≈x γ2 ⇔.γ1(0) =

.γ2(0) and

..γ1(0) =

..γ2(0);

that is, the curves are tangent of second order or equivalent up toacceleration

8.2. Second order tangent bundles 251

Here, for an arbitrary γ ∈ Cx, the curves.γ and

..γ are, respectively, the

first and second derivatives (or velocity and acceleration), defined by

.γ : (−ε, ε) −→ TB : t 7−→ Ttγ(∂t),..γ : (−ε, ε) −→ T (TB) : t 7−→ Tt

.γ(∂t),

where ∂ = d/dt is the basic vector field of R, thus ∂t coincides with 1 ∈ R

under the natural identification TtR ≡ R.

In accordance with the first order equivalence of curves given in § 1.1.4,we also have that

γ1 ≈x γ2 ⇔ there is a chart (U, φ) of B such that:

(φ γ1)′(0) = (φ γ2)

′(0) and (φ γ1)′′(0) = (φ γ2)

′′(0), or

D(φ γ1)(0) = D(φ γ2)(0) and D2(φ γ1)(0) = D2(φ γ2)(0).

It is a matter of routine checking to see that the latter conditions are inde-pendent of the choice of charts at x.

In analogy to the ordinary (first order) tangent space, the second ordertangent space or tangent space of order two at x ∈ B is defined by

T 2xB := Cx/ ≈x ,

while the second order tangent bundle or tangent bundle of ordertwo of B is

T 2B :=⋃

x∈B

T 2xB.

It is worth noting here that T 2xB can be always thought of as a topological

vector space isomorphic to B× B via the bijection

φ2α : T 2

xB≃−−→ B× B : [(γ, x)]2 7−→

((φα γ)′(0), (φα γ)′(0)

),

where [(γ, x)]2 stands for the equivalence class of γ with respect to ≈x.However, this structure depends on the choice of the chart (Uα, φα); hence,we cannot define a vector bundle structure on T 2B based on the aforemen-tioned bijections, for all x ∈ B. A convenient way to overcome this obstacleis to assume that B is endowed with the additional structure of a linearconnection (see § 8.1.1).

Theorem 8.2.1. If B admits a linear connection K, then T 2B becomes aBanach vector bundle with structure group GL(B× B).


Proof. Let us denote by π2 : T 2B → B the natural projection given byπ2([(γ, x)]2) = x. If Γα : φα(Uα) → L2(B,B;B)a∈I are the Christoffelsymbols of K with respect to the atlas (Ua, φa)a∈I of B, then, for eachα ∈ I, we define the map τ2α : π−12 (Uα)→ Uα × B× B with

(8.2.2)τ2α([(γ, x)]2) :=

(x, (φα γ)′(0), (φα γ)′′(0)+

+ Γα(φα(x))((φα γ)′(0), (φα γ)′(0))

),

for every x ∈ π−12 (Uα). The maps τ2αα∈I are obviously well-defined andinjective. They are also surjective, since any element (x, h, k) ∈ Uα × B× B

can be obtained, via τ2α , as the image of the equivalence class [(γ, x)]2 of thesmooth curve γ = φ−1α σ : (−ε, ε)→ B, where

(8.2.3) σ(t) := φα(x) + ht +1

2

(k − Γα(φα(x))(h, h)

)t2,

with ε small enough so that σ((−ε, ε)

)⊂ φα(Uα).

On the other hand, since the diagram

π−12 (Uα)τ2α - Uα × B× B

Uα

pr1π2

-

is commutative, it is clear that the pairs (Uα, τ2α), for all α ∈ I, determine on

T 2B the structure of a locally trivial fibre bundle, such that the restrictionsof τ2α to the fibres, τ2α,x : TxB → B × B, are linear isomorphisms, for everyx ∈ B.

The question now is whether the previous structure is that of a vectorbundle. The answer will follow from the behaviour of τ2α on overlappings. Inthis respect, let us consider two trivializations (Uα, τ

2α) and (Uβ , τ

2β) of T 2B

with Uαβ = Uα ∩ Uβ 6= ∅. For every x ∈ Uαβ we check that

(τ2α,x (τ2β,x)

−1)(h, k) = τα,x([(γ, x)]2),

where, by the analog of (8.2.3), γ : (−ε, ε)→ Uαβ is the smooth curve troughx with

γ(t) = φ−1β

(φβ(x) + ht+

1

2

(k − Γβ(φβ(x))(h, h)

)t2).

8.2. Second order tangent bundles 253

As a result, setting φαβ = φα φ−1β , (8.2.2) implies that

(τ2α,x (τ2β,x)−1

)(h, k) = τα,x([(γ, x)]2)

=(

(φα γ)′(0), (φα γ)′′(0) + Γα(φα(x))((φα γ)′(0), (φα γ)′(0))

)

=(Dφαβ(φβ(x)).h,Dφαβ(φβ(x)).k −Dφαβ(φβ(x)) Γβ(φβ(x)).(h, h) +

+D2φαβ(φβ)(x)(h, h)),

or, taking into account the compatibility condition of the Christoffel symbols(8.1.4) (and noting also the difference between the present x and that in theaforementioned equality),

(8.2.4)(τ2α,x (τ2β,x)

−1)(h, k) =

(Dφαβ(φβ(x)).h,Dφαβ(φβ(x)).k

).

Therefore, τ2α,x (τ2β,x)−1 ∈ Lis(B× B), and the transition maps

T 2αβ : Uαβ −→ Lis(B× B) : x 7−→ τ2α,x (τ2β,x)−1

are smooth, because

(8.2.5) T 2αβ =

(Dφαβ φβ

)×(Dφαβ φβ

)= Tαβ × Tαβ,

where Tαβ : Uαβ → Lis(B)α,β∈I are the transition functrions of TB. Con-sequently, (T 2B,B, π2) is indeed a vector bundle over B, of fibre type B×B,with structure group GL(B× B).

The vb-charts of T 2B are the triplets (Uα, φα,Φ2α), where the diffeomor-

phisms Φ2α : π−12 (Uα)→ φα(Uα)× B× B are given by

(8.2.6) Φ2α := (φα × idB× idB) τ2α.

A byproduct of the preceding proof is the following:

Corollary 8.2.2. The second order tangent bundle T 2B is vb-isomorphicto TB ×B TB.

Proof. In virtue of (8.2.4), the cocycles of both bundles coincide.

From the preceding constructions it is also clear that the vector spacestructure of T 2

xB is given by

[(γ1, x)]2 + λ [(γ2, x)]2 =(τ2α,x

)−1(τ2α,x([(γ1, x)]2) + λ τ2α,x[(γ2, x)]2

),

which is independent of the choice of charts in virtue of (8.2.4)


We conclude this section by proving the converse of Theorem 8.2.1 if theisomorphism of Corollary 8.2.2 has an explicit expression. More precisely,we consider the trivializations (π−12 (Uα), τ2α)α∈I of T 2B (see the proof ofTheorem 8.2.1)), and assume that their restrictions to each fibre over x ∈ B,π−12 (x) ∼= τ−1B (x)× τ−1B (x), is written in the form

(8.2.7) τ2α,x ≡ τα,x × τ′α,x,

such that

(8.2.8) τ2α,x([(γ, x)]2

)=(τα,x([(γ, x)]), τ ′α,x([(γ, x)])

),

for every smooth curve γ through x. We recall that τB : TB → B is the pro-jection of the tangent bundle, τα, τ

′α : τ−1B

≃−−→ Uα×B are the trivializations

of TB over Uα and τα,x, τ′α,x : TxB

≃−−→ B the induced isomorphisms on the

fibres.

Theorem 8.2.3. Let B be a smooth manifold modelled on the Banach spaceB, and assume that the second order tangent bundle T 2B of B has a vectorbundle structure of fibre type B×B, isomorphic to TB×B TB. Then, underthe conditions (8.2.7) and (8.2.7), B admits a linear connection.

Proof. Let (π−12 (Uα), τ2α)α∈I be the trivializations of T 2B satisfying (8.2.7)and (8.2.8). Then, we may construct a chart (U, φα) of B such that

Txφα(γ(0)) =(

pr1 τ2α

)([γ, x]2

)= τα([(γ, x)]).

Indeed, if (U, φ) is an arbitrarily chosen chart of B with U ⊆ Uα, we maytake φα := τ1α,x (Txφ)−1 φ. Using charts of the previous form, we definethe Christoffel symbols Γα : φα(Uα)→ L2(B,B;B) of the desired connectionby setting

Γα(y)(u, u) := τ2α,x([(γ, x)]2)− (φα γ)′′(0); y ∈ φα(Uα),

where γ is a curve of B representing the tangent vector u, with respect tothe chart (Uα, φα). The remaining values of Γα(y) on elements of the form(u, v), with u 6= v, are defined by demanding Γα(y) to be symmetric bilinearmaps. Γαα∈I satisfy the necessary compatibility condition (8.1.4) sincethe trivializations (π−12 (Uα), τ2α)αǫI agree, via the transition functions ofT 2B, on all common areas of their domains. Hence, Γαα∈I give rise to alinear connection on B.

8.3. Second order differentials 255

8.3 Second order differentials

To examine the dependence of the vector bundle structure of the secondorder tangent bundle on the choice of linear connections on the base space,we need the notion of the differential of second order. To define it, weconsider two smooth manifolds B and B′ modelled on the Banach spacesB and B′, respectively, with corresponding atlases A = (Uα, φα)α∈I andB = (Vβ , ψβ)β∈J . We fix two linear connections K and K ′ on B andB′, respectively, with Christoffel symbols Γα : φα(Uα) → L2(B,B;B)α∈Iand Γβ : φβ(Uβ) → L2(B

′,B′;B′)β∈J . As proved in the preceding section,the pairs (B,K) and (B′,K ′) induce the second order tangent bundles T 2Band T 2B′ with vector bundle atlases A2 = (π−1B,2(Uα), φα,Φα)α∈I and

B2 = π−1B′,2(Vβ), ψβ ,Ψβ)β∈J .

Definition 8.3.1. If f : B → B′ is a smooth map, the second orderdifferential of f is the map T 2f : T 2B → T 2B′, given by T 2f([(γ, x)]2) :=[(f γ, f(x))]2.

T 2f is well-defined. Indeed, if γ1, γ2 ∈ Cx such that γ1 ≈x γ2, then takingany charts (Uα, φα) ∈ A2 and (Vβ, φβ) ∈ B2 with x ∈ Uα and f(Uα) ⊆ Vβ(provided by the smoothness of f at x), we check that

(f γ1)(0) = (f γ2)(0) = f(x),

(f γ1).(0) = T0(f γ1)(∂0) = Txf(

.γ1(0))

= Txf(.γ2(0)) = (f γ2).(0),

(f γ1)..(0) = T.γ1(0)

(Tγ1(0)f)(.γ1(0))

= T.γ2(0)(Tγ2(0)f)(

..γ2(0))

= (f γ2)..(0).

The same is true if we use the definition of equivalence classes by means oflocal charts.

Lemma 8.3.2. The pair (T 2f, f) determines a fibre bundle morphism be-tween (T 2B,B, π2) and (T 2B′, B′, π′2).

Proof. Equality π′2 T2f = f π2 is immediately verified. It remains to

ascertain the differentiability of T 2f . For an arbitrary [(γ0, x0)]2 ∈ T 2B,the smoothness of f implies the existence of two charts (Uα, φα) ∈ A and(Vβ, ψβ) ∈ B with x0 ∈ Uα and f(Uα) ⊆ Vβ, inducing the smooth localrepresentation fβα := ψβ f φ

−1α of f . Considering now the corresponding


charts π−12 (Uα) ≡ T 2B|Uα and (π′2)−1(Vβ) ≡ T 2B′|Vβ of T 2B and T 2B′, wecheck that the following diagram is commutative:

T 2B|Uα

T 2f- T 2B′|Vβ

φα(Uα)× B× B

Φ2α

?

Ψ2β T

2f (Φ2α)−1

- ψβ(Vβ)× B′ × B′

Ψ2β

?

Therefore, for every (y, h, k) ∈ φα(Uα)× B× B,

(Ψ2β T

2f (Φ2α)−1

)(y, h, k) =

(Ψβ T

2f)

([(γ, x)]2),

with [(γ, x)]2 determined by the smooth curve γ(t) = φα(σ(t)), where

x = φ−1α (y); σ(t) = y + ht+1

2(k − Γα(k, k)) t2, t ∈ (ǫ, ǫ).

This means that

(φα γ)(0) = y, (φα γ)′(0) = h, (φα γ)′′(0) = k − Γα(k, k).

As a result, the local representation of T 2f turns into

(Ψ2β T

2f (Φ2α)−1

)(y, h, k) = Ψβ ([(f γ, f(x))])

=(ψβ(f(x)), (ψβ f γ)′(0)), (ψβ f γ)′′(0)) +

+ Γ′β(ψβ(f(x)))((ψβ f γ)′(0), (ψβ f γ)′(0)

))

=(

(fβα (φα γ))(x),Dfβα(φα(x)).(φα γ)′(0) +

+Dfβα(φα(x)).(φα γ)′′(0) +

+D2fβα(φα(x))((φα γ)′(0), (φα γ)′(0)

))

=(fβα(y),Dfβα(y).h,Dfβα(y).k −Dfβα(y).Γα(y)(h, h) +

+D2fβα(y).(h, h) + Γ′β(fβα(y))(Dfβα(y).h,Dfβα(y).h

)),

which proves the smoothness of T 2f at an arbitrary element of T 2B andconcludes the proof.

8.3. Second order differentials 257

From the previous computations and equality

(Ψ2β T

2f (Φ2α)−1

)(y, h, k) =

=(fβα(y),

(Ψ2β,f(x) T

2xf (Φ2

α,x)−1)

(h, k)),

with x = φ−1α (y), it follows that

(8.3.1)

(Ψ2β,f(x) T

2xf (Φ2

α,x)−1)

(h, k) =

=(Dfβα(y).h,Dfβα(y).k −Dfβα(y).Γα(y)(h, h) +

+D2fβα(y).(h, h) + Γ′β(fβα(y))(Dfβα(y).h,Dfβα(y).h

))

Hence, despite the fact that T 2f is a fibre bundle morphism, the presence ofthe Christoffel symbols and the derivatives of second order prevents it frombeing necessarily linear on the fibres, in contrast to the case of ordinary(first order) differentials. Here, related (or conjugate) connections providean efficient way to surmount this obstacle.

Indeed, if K and K ′ are f -related, then (8.1.8) transforms (8.3.1) into

(8.3.2)

(Ψ2β,f(x) T

2xf (Φ2

α,x)−1)

(h, k) =

=(Dfβα(φα(x).h,Dfβα(φα(x)).k),

for every (x, h, k) ∈ Uα × B× B. Therefore, one infers the following:

Corollary 8.3.3. If the connections K and K ′ on B and B′, respectively,are f -related, then the second order differential T 2f : T 2B → T 2B′ is linearon the fibres

Remark 8.3.4. The assumption that K and K ′ are f -related, employedin Corollary 8.3.3, is a sufficient, but not necessary, condition ensuring thelinearity of T 2f on the fibres. The optimal (necessary and sufficient) choicewould be to assume that the “problematic” part appearing in (8.3.1), namely

B ∋ u 7−→ −Dfβα(φα(x)).Γα(φα(x))(u, u) +D2fβα(φα(x))(u, u) +

+ Γ′β(fβα(φα(x)))(Dfβα(φα(x))(u),Dfβα(φα(x))(u))

)∈ B

is a linear, not necessarily zero, map. The possible geometric consequencesof such an assumption remain an open problem.


Example 8.3.5. We give here some particular examples of related connec-tions in order to clarify a bit the preceding remark.

1) In the case of a constant map f , equality (8.1.8) collapses to a trivialidentification of zero quantities, since fβα is constant. As a result, all linearconnections are related through constant maps.

2) If we consider the map f = idB , then necessarily K = K ′. This agreeswith the fact that, in this case, equality (8.1.8) yields

Dφαβ(φα(x)).Γα(φα(x))(h, k)) =

D2φβα(φα(x))(h, h) + Γ′β(φβ(x))(Dφβα(φα(x))(h),Dφβα(φα(x))(h)),

which is precisely the compatibility condition (1.8.4) of the Christoffel sym-bols of a connection on B.

Completing Lemma 8.3.2 and Corollary 8.3.3, we obtain the first mainresult of this section:

Theorem 8.3.6. Let T 2B, T 2B′ be the second order tangent bundles deter-mined by the pairs (B,K), (B′,K ′), and let f : B → B′ be a smooth map.If the connections K and K ′ are f -related, then (T 2f, f) is a vector bundlemorphism.

Proof. The only thing we need to check is (see, e.g., [Lan99]) the smoothnessof the map

Uα −→ L(B× B,B′ × B′) : x 7→ Ψ2β,f(x) T

2xf (Φ2

α,x)−1,

which is an obvious consequence of (8.3.2)

8.4 Connection dependence of second order tan-

gent bundles

As we have seen in § 8.2, the vector bundle structure of T 2B depends heavilyon the choice of a linear connection K on the base manifold B. The resultsof the preceding section allow us to estimate the extent of this dependence.In fact, we obtain:

Theorem 8.4.1. Let K, K ′ be two linear connections on a Banach manifoldB. If f is a diffeomorphism of B such that K and K ′ are f -related, then thevector bundle structures on T 2B, determined by K and K ′, are isomorphic.

Proof. In virtue of Theorem 8.3.6, (T 2f, f) is a vb-isomorphism.

8.5. Second order Frechet tangent bundles 259

We introduce the following terminology: Two pairs (B,K) and (B′,K ′),where the linear connections K, K ′ are f -related with respect to a diffeo-morphism f of B, as in Theorem 8.4.1, are called equivalent. The cor-responding equivalence class of (B,K) is denoted by [(B,K)]f . Then wehave:

Corollary 8.4.2. The elements of [(B,K)]f determine, up to isomorphism,the same vector bundle structure on T 2B. Consequently, the latter structuredepends not only on a pair (B,K) but also on the entire class [(B,K)]f .

From the preceding discussion, it is now clear that the vb-identificationof two bundles T 2B, T 2B′ is not ensured by the existence of a mere diffeo-morphism f : B

≃−−→ B. One has to take into account the geometry of B

and B′, as it is expressed by the linear connections on them.

An interesting question is whether it is possible to characterize the iso-morphism classes of second order tangents using systems of connections. Abrief description of these systems and relevant comments are given in theAppendix (item 6, p. 275).

8.5 Second order Frechet tangent bundles

As in § 3.1, we consider a plb-manifold M , realized as the projective systemof Banach manifolds M i;µjii,j∈N and modelled on the Frechet space F,where F ≡ lim

←−Ei; ρjii,j∈N, with Ei being the Banach space models of M i,

respectively, for all i ∈ N. We shall prove that the corresponding secondorder tangent bundle of M belongs to the category of plb-vector bundles.The first result towards this direction is:

Proposition 8.5.1. The second order tangent bundles T 2M ii∈N form aprojective system with limit set-theoretically isomorphic to T 2M .

Proof. For any pair of indices (i, j) with j ≥ i, we define the map

µji2 : T 2M j −→ T 2M i : [(γ, x)]j2 7→[(µji γ, µji(x)

)]i2,

where the brackets [·, ·]j2, [·, ·]i2 denote the second order equivalence classesof curves in M j and M i, respectively, defined in § 8.2. These maps are well-defined, since two equivalent curves γ1, γ2 on M j [in the sense of (8.2.1)]yield

T (n)µji2(γ(n)1 (0)

)=(µji2 γ1

)(n)(0) =

(µji2 γ2

)(n)(0) = T (n)µji2

(γ(n)2 (0)

),


for every n = 0, 1, 2, where

T (1)µji2 : TM j −→ TM i and T (2)µji2 : T (TM j) −→ T (TM i)

are the first (ordinary) and second order differential of µji2 , respectively, and

γ(n)(0) =

γ(0), n = 0.γ(0), n = 1..γ(0), n = 2

The family T 2M i;µji2 i,j∈N determines a projective system, since the nec-essary conditions

µik2 µji2 = µjk2 , j ≥ i ≥ k

are immediately verified from the analogous conditions of the initial con-necting morphisms µjii,j∈N.

On the other hand, the second order differentials of the canonical pro-jections µi : M →M i of M , namely

T 2µi : T 2M −→ T 2M i : [(γ, x)]2 7→[(µi γ, µi(x)

)]i2

; i ∈ N,

can be taken as the canonical projections µi2 of the limit space lim←−T2M i;

that is, µi2 := T 2µi, since the equality µji2 µj2 = µi2 holds for any j ≥ i. As

a result, we obtain the map

F := lim←−

T 2µi : T 2M −→ lim←−

(T 2M i) : [(γ, x)]2 7→([(

µi γ, µi(x))]i

2

)i∈N

.

This is an injection because F ([(γ1, x)]) = F ([(γ2, x)]) implies

T (n)µi(γ(n)1 (0)

)=(µi γ1

)(n)(0) =

(µi γ2

)(n)(0) = T (n)µi

(γ(n)2 (0)

),

for every n = 0, 1, 2; therefore, γ(n)1 (0) = γ

(n)2 (0) (n = 0, 1, 2), since

TM ≡ lim←−

TM i and T (TM) ≡ lim←−

T (TM i),

in virtue of Theorem 3.2.8.

The surjectivity of F is a bit more complicated and goes as follows:Given any element

a =([(

γi, xi)]i

2

)i∈N∈ lim←−(T 2M i),


the definition of µji2 implies that

(8.5.1)[(µji γj , µji(xj)

)]i2

=[(γi, xi

)]i2

; j ≥ i,

thus x = (xi) ∈ M = lim←−

M i. Moreover, if(U = lim

←−U i, φ = lim

←−φi)

is aprojective limit chart of M at x, and

(π−1M (U) = lim←−π

−1M i(U

i),Φ = Tφ = lim←−Tφi),

(π−1TM

(π−1M (U) = lim

←−π−1TM i(π

−1M i(U

i)), Φ = T (Tφ) = lim

←−T (Tφi)

)

are the corresponding charts of TM and T (TM), respectively, it turns outthat the equality

((φi µji γj)(0), Tφi

((µji γj).(0)

))=((φi γi)(0), Tφi

((γi).(0)

))

implies that

(ρji((φj γj)(0)

), Tφi

(Tµji((γj).(0))

))=((φi γi)(0), Tφi((γi).(0))

).

Therefore, the vectors u =((φi γi)(0)

)i∈N

and v =((φi γi).(0)

)i∈N

be-

long to F ∼= lim←−

Ei.

Similarly, relations (8.5.1) ensure that (µji γj)..(0) = (γi)..(0) which,by means of the charts of T (TM) defined above, yields

T (Tφi)((µji γj)..(0)

)= T (Tφi)

((γi)..(0)

)

or, equivalently,

ρji((φj γj)..(0)

)= (φi γi)..(0),

for every j ≥ i. Hence,

w =((φi γi)..(0)

)i∈N∈ F ∼= lim←−Ei.

Considering now the curve h in F with

h(t) = u+ tv +t2

2w; t ∈ R,

and the curve γ := φ−1 h of M (with an appropriate restriction of the


domain of h, if necessary), we easily check that

(µi γ)(0) = µi(x) = xi = γi(0),

(µi γ).(0) =((φi)−1 ρi h

).(0) = T ((φi)−1)

((ρi h).(0)

)

= T ((φi)−1)(ρi(v)) = T ((φi)−1)((φi γi).(0)

)

= (γi).(0),

(µi γ)..(0) =((φi)−1 ρi h

)..(0) = T (T ((φi)−1))

((ρi h)..(0)

)

= T (T ((φi)−1))(ρi(w)) = T (T ((φi)−1))((φi γi)..(0)

)

= (γi)..(0),

for all indices i, j with j ≥ i. Consequently, the curves µi γ and γi are

(second order) equivalent, thus F ([(γ, x)]2) =([(

γi, xi)]i

2

)i∈N

= a. This

completes the surjectivity of F and establishes the desired set-theoreticalisomorphism between T 2M and lim←−(T 2M i).

Based on the preceding identification of T 2M and lim←−(T 2M i), we may

define a Frechet vector bundle structure on T 2M by means of an appropriatelinear connection on M . The problems concerning the structure group of thisbundle are surmounted by replacing, once again, the pathological GL(F×F)by the topological group (see also § 5.2)

H0(F× F) :=

(li)i∈N ∈

∞∏

i=1

GL(Ei × Ei) : lim←−

li exists

.

More precisely, we prove the following main result.

Theorem 8.5.2. If a Frechet manifoldM = lim←−

M i is endowed with a linear

plb-connection K = lim←−

Ki, then T 2M is a Frechet vector bundle over Mwith structural group H0(F× F).

Proof. Let

(Uα = lim←−

U iα, φα = lim←−

φiα)α∈I

be a plb-atlas of M and letΓiα : φiα(U iα)→ L2(Ei,Ei;Ei)

α∈I

be the Christoffel symbols of each fac-

tor linear connection Ki on M i (i ∈ N). Then, as proved in Theorem 8.2.1,each T 2M i is a Banach vector bundle over M i of fibre type Ei. The corre-sponding local trivializations

τ iα :(πi2)−1

(U iα) −→ U iα × Ei × Ei; α ∈ I,


are given by

τ iα([γ, x]i2

)=(x, (φiα γ)′(0), (φiα γ)′′(0)

+ Γiα(φiα(x))((φiα γ)′(0)), (φiα γ)′(0)

)).

Taking into account that the families µji2 i,j∈N, µjii,j∈N and ρjii,j∈Nare connecting morphisms of the projective systems T 2M = lim

←−(T 2M i),

M = lim←−

M i and F = lim←−

Ei, respectively, we check that the projectionsπi2 : T 2M i →M i

i∈N

satisfy the equality

µji πj2 = πi2 µji2 ; j ≥ i,

while the trivializations τ iαi∈N satisfy

(µji × ρji × ρji) τ jα = τ iα µji2 , j ≥ i.

As a consequence, conditions (PVB. 1) and (PVB. 2) of Definition 5.2.1 arefulfilled and the family

T 2M i;µji2

i,j∈N

is a projective system of Banach

vector bundles, thus the derived limit T 2M = lim←−

(T 2M i) is a (Frechet)plb-vector bundle. In particular, its projection to the base is

π2 = lim←−πi2 : T 2M −→M,

and the local trivializations have the form

(8.5.2) τα = lim←− τiα : π−12 (Uα) −→ Uα × F× F,

for every α ∈ I. The corresponding transition functions

(8.5.3) Tαβ = τα,x τ−1β,x; α, β ∈ I,

can be considered as taking values in the generalized Lie group H0(F × F),since Tαβ = ε T ∗αβ, where T ∗αβα,β∈I are the smooth maps

T ∗αβ : Uα ∩ Uβ −→ H0(F× F) : x 7→(τ iα,x

(τ iβ,x

)−1)i∈N

and ε is the natural inclusion

ε : H0(F× F) −→ L(F× F) : (li)i∈N 7→ lim←− li.

In this way, T 2M is endowed with a vector bundle structure over M , withfibres of type F×F and structure groupH0(F×F). This bundle is isomorphicto TM ×M TM since both bundles have identical transition functions:

Tαβ(x) = τα,x τ−1β,x =

(D(ψa ψ

−1β

) ψβ

)(x)×

(D(ψa ψ

−1β

) ψβ

)(x).


In analogy to Theorem 8.2.3), we obtain the following converse of The-orem 8.5.2.

Theorem 8.5.3. Let T 2M be a Frechet plb-vector bundle over M , withstructure group H0(F × F), as in Theorem 8.5.2. If T 2M is isomorphic toTM×MTM under the analogs of (8.2.7) and (8.2.8), thenM admits a linearconnection which can be realized as a projective limit of linear connections.

Proof. We have seen that the vector bundle structure on T 2M is definedby a family of trivializations τα : π−12 (Uα) → Uα × F × Fα∈I , realized asprojective limits of the trivializations τ iα : (πi2)−1(U iα)→ U iα×E

i×Ei of T 2M i

(i ∈ N). Consequently, every factor T 2M i is a vector bundle isomorphic toTM i ×M i TM i, and, following the proof of Theorem 8.2.3, M i admits alinear connection Ki, with Christoffel symbols determined by

Γiα(y)(ui, ui) = τα,iα,x)([γi, x]i2

)− (φiα γ

i)′′(0),

where γi is the curve of M i representing the vector ui, with respect to thechart (U iα, φ

iα). It is now readily checked that lim

←−

(Γiα(yi)(ui, ui)

)exists, for

every y = (yi) ∈ φ(U) = lim←−

φi(Ui) and (ui) ∈ F = lim←−

Ei. This ensures that

the connections Kii∈N form a projective system with projective limit thedesired linear connection K = lim

←−Ki on M .

Regarding the structure group of T 2M and TM ×M TM , we note that,in virtue of Remarks 5.2.6 and Definition 5.2.7 (for E = TM), we have theidentification

H0(F)×H0(F) ≡ H0(F× F).

8.6 Second order frame bundles

As already discussed in §§ 1.6 and 6.5, several geometric properties of vectorbundles can be studied by using the corresponding bundle of linear frames.The purpose of the present section is to exhibit the structure of the secondorder frame bundles associated with Banach and plb-manifolds.

We first consider a Banach manifold B with model B. We further assumethat B is endowed with a linear connection K, thus T 2B admits a corre-sponding vector bundle structure (see Theorem 8.2.1). Then, the secondorder frame bundle of B is defined by

(8.6.1) P 2(B) :=⋃

x∈B

Lis(B× B, T 2

xB).

(Compare with the ordinary frame bundle defined in § 1.6.5.)

8.6. Second order frame bundles 265

Proposition 8.6.1. P 2(B) is principal bundle over B, with structure groupGL(B× B).

Proof. Let (Uα, φα)α∈I be a smooth atlas of B and the correspondingtrivializations τ2α : π−12 (Uα)→ Uα×B×B)α∈I of T 2B (see Theorem 8.2.1).We denote by τ2α,x ∈ Lis(T

2xB,B× B) the restriction of τ2α to the fibre over

x ∈ B, and by p : P 2(B) → B the projection given by p(h) = x, for everyh ∈ Lis(B× B, T 2

xB).The group GL(B × B) acts on (the right of) P 2(B) in a natural way;

namely, h · g := h g, for every (h, g) ∈ P 2(B)×GL(B×B). Then the localstructure of P 2(B) is obtained by the obvious bijections

Fα : p−1(Uα) −→ Uα ×GL(B× B) : h 7→(p(h), τ2α,p(h) h

), α ∈ I.

Indeed, each Xα := p−1(Uα), a ∈ I, can be endowed with a smooth manifoldstructure modelled on the Banach space B×GL(B×B). Since Fα(Xa∩Xβ) =Uαβ×GL(B×B) is an open subset of Fα(Xα), it follows that Xα∩Xβ is openinXα. Moreover, the differential structure of Xα∩Xβ as a submanifold of Xα

coincides with that obtained via Xβ because Fβ F−1α is the diffeomorphism

(Fβ F−1α )(x, g) =

(x,(comp (T 2

βα × idL(B×B)))(x, g)

),

for every (x, g) ∈ Uαβ × GL(B × B). Here, comp: L(B) × L(B) → L(B)denotes the composition map, and T 2

αβα,β∈I are the transition functions

of T 2B [see (8.2.5)]. Therefore, by the gluing Lemma ([Bou67, No 5.2.4]),P 2(B) turns to be a principal Banach bundle, with transition functions

G2αβ : Uαβ −→ GL(B× B) : x 7→ Fα,x F

−1β,x.

They are related with the transition functions of T 2B by

G2αβ(x)(g) =

(Fα,x F

−1β,x

)(g) = T 2

αβ(x) g.

The bundle T 2B is associated with P 2(B) by means of the action ofGL(B× B) on the right of P 2(B)× B× B,

(h, (u, v)

)· g =

((h g), g−1(u, v)

).

More precisely, we prove the following:

Theorem 8.6.2. The quotient E =(P 2(B)× B× B

)/GL(B×B) coincides,

up to isomorphism, with T 2B.


Proof. With the notations of the proof of Proposition 8.6.1, we define theprojection

π : E −→ B : [h, (u, v)] 7→ p(h),

and the local trivializations

Φα : π−1(Uα) −→ Uα × B× B :[(h, (u, v)

)]7→(p(h),

(pr2 Fα

)(h).(u, v)

)=(p(h), τ2α,p(h) h

),

for all α ∈ I, where pr2 denotes the projection of Uα × GL(B,B) to thesecond factor.

Each Φα is injective, because

Φα

([(h, (u, v)

)])= Φα

([(h1, (u1, v1

)])

⇒ p(h) = p(h1) := x ∈ B,(τ2α,x h

)(u, v) =

(τ2α,x h1

)(u1, v1).

Therefore, h(u, v) = h1(u1, v1) and the classes [h, (u, v)], [h1, (u1, v1)] coin-cide via the isomorphism g := h−11 h.

Also, Φα is surjective: If (x, (u, v)) is an arbitrary element of Uα×B×B,then

Φα

([(h, (u, v)

)])=(x,(τ2α,x h

)(u, v)

)=(x, (u, v)

),

where h :=(τ2α,x

)−1∈ Lis(B× B, T 2

xB).

The restrictions of the previous trivializations to the fibres, Φα,x :=

pr2 Φα|π−1(x), x ∈ B, imply that Φα,x Φ−1β,x = τ2α,x (τ2β,x

)−1. Hence, it is a

matter of routine checking to verify thatE admits a vector bundle structurewith corresponding transition functions

Tαβ(x) = Φα,x Φ−1β,x = τ2α,x (τ2β,x

)−1= T 2

αβ(x), x ∈ Uαβ.

The preceding identification of the transition functions of E and T 2B estab-lishes a vector bundle isomorphism.

Remark 8.6.3. For the sake of completeness, we describe a concrete iso-morphism G : E → T 2B, by setting G

([(h, (u, v)

)]):= h(u, v).

G is well-defined: If[(h, (u, v)

)]=[(h1, (u1, v1)

)], then there exists a

g ∈ GL(B × B) such that h g = h1 and g(u1, v1) = (u, v), thus h(u, v) =h1(u1, v1).

G is injective: Equality G([(

h, (u, v))])

= G([(

h1, (u1, v1))])

impliesthat h(u, v) = h1(u1, v1); therefore, for g := h−11 h,

(h1, (u1, v1)

)· g =

(h1 g, g

−1(u1, v1))

=(h, (u, v)

).


G is surjective: Indeed, for an arbitrary w ∈ T 2xB, if (Uα, φα) is a chart

of B at x, and τ2α is the corresponding trivialization of T 2B, then, for

h :=(τ2α,x

)−1∈ Lis(B× B, T 2

xB) = P 2(B)x, (u, v) := τ2α,x ∈ B× B,

it follows that G([(

h, (u, v))])

= w.

Finally, we verify that

τ2α G Φ−1α = id∣∣Uα×B×B

; α ∈ I,

while, for each α ∈ I,

τ2α,x Gx Φ−1α,x = idB×B , x ∈ Uα.

The preceding equalities imply immediately conditions (VBM. 1) and (VBM.2) of § 1.4.3 and ensure that G is a vb-isomorphism.

We want to study now the frame bundle of a plb-manifold

M = lim←−Mi;µjii,j∈N,

modelled on the Frechet space F ∼= lim←−Ei; ρjii,j∈N, where Ei are the Banach

space models of M i (i ∈ N). To this end, we further assume that

(8.6.2) the canonical projections µi : M →M i(i ∈ N) are surjective.

(Recall that ρi : F → Ei are surjective maps, see Remarks 2.3.9.) Then weset

P 2(M i) :=⋃

xi∈M i

(hk)1≤k≤i

,

where the maps hk ∈ Lis(Ek × Ek, T 2µik(xi)

Mk) satisfy the equalities

µmk2 hm = hk (ρmk × ρmk), i ≥ m ≥ k.

(Compare with the bundles P (Ei) of Proposition 6.5.1.)

Proposition 8.6.4. Each P 2(M i) (i ∈ N) is a principal fibre bundle overM i, with structure group the Banach-Lie group Hi0(F× F).

Proof. By (8.6.2), for every xi ∈M i there is an x ∈M such that µi(x) = xi.Let

(Uα = lim←−U


iα

), a ∈ I, be a family of plb-charts of M

that cover all the possible selections of elements xi and x with x ∈ Uα,


and let(π−12 (Uα) = lim

←−(πi2)

−1(U iα), τα = lim←−

τ iα)

be the corresponding triv-

ialization of T 2M as defined by (8.5.2). Recall that the diffeomorphismsτα : π−12 (Uα) −→ Uα × F × F induce fibre-wise the linear isomorphismsτα,x : T 2

xM → F× F, with

τα,x = pr2 τα|π−12 (x) = lim

←−τ iα,x = lim

←−

(pr2 τ

iα

∣∣(πi

2)−1(xi)

),

where now pr2 : Uα × F2 → F2 denotes the projection to the second factor.Next, we define the projections

pi : P 2(M i) −→M i :(h1, h2, . . . , hi

)7→ xi,

for all (h1, h2, . . . , hi) with hi ∈ Lis(Ei × Ei, T 2

xiMi), as well as the action

of Hi0(F× F) on the right of P 2(M i):(h1, h2, . . . , hi

)·(g1, g2, . . . , gi

):=(h1 g1, h2 g2, . . . , hi gi

).

Following the general pattern of the proof of Proposition 8.6.1 (with theappropriate modifications), we define the bijections

(8.6.3)Φiα :(pi)−1

(U iα) −→ U iα ×Hi0(F× F) :

(h1, . . . , hi

)7−→

(pi(h1, . . . , hi

), τ1α,xi h

1, . . . , τ iα,xi hi),

if pi(h1, . . . , hi

)= xi.

The injectivity of each Φiα (a ∈ I) is obvious, while any (xi, g1, . . . , gi) ∈

U iα ×Hi0(F× F) can be written as

Φiα

((τ1α,xi

)−1 g1, . . . ,

(τ iα,xi

)−1 gi),

thus showing that Φiα is a surjection. As a result, each Xα := (pi)−1(U iα)

can be endowed with the structure of a Banach manifold modelled on Ei ×Hi0(F× F). Moreover, for every (α, β),

Φiα(Xα ∩Xβ) = (U iα ∩ U

ib)×H

i0(F× F)

is open in Φiα(Xα), thus Xα ∩Xβ is open in Xα.

The differential structure of Xa ∩Xb, as an (open) submanifold of Xα,coincides with the one induced by Xβ , since Φi

β (Φiα)−1 is a diffeomorphism

of (U iα ∩ Uiβ)×Hi0(F× F). Indeed,

(Φiβ (Φiα

)−1) (xi, g1, . . . , gi

)=

=(xi,((

comp (T kβ × prk))(xi, g1, ..., gi)

)k=1,2,...,i

),


wherecomp: L(Ek)× L(Ek) −→ L(Ek) : (f, g) 7→ f g,

(T kαβ

)a,b∈I

are the transition functions of T 2Mk, and

prk :i∏

k=1

L(Ek) −→ L(Ek)

is the projection to the k-th factor. Hence, by the gluing Lemma,

P 2(M i) =⋃

a∈I

Xa,

is a Banach principal bundle with local trivializations given by (8.6.3).

The principal bundlesP 2(M i)

i∈N

form a projective system. More

precisely, for every i, j ∈ N2 with j ≥ i, the connecting morphisms are givenby

rji : P 2(M j) −→ P 2(M i) :(h1, h2, . . . , hj

)7→(h1, h2, . . . , hi

),

obviously satisfying the necessary relations rik rji = rjk (j ≥ i ≥ k).Similarly, the connecting morphisms

hji0 : Hj0(F× F) −→ Hi0(F× F) :(g1, g2, . . . , gj

)7→(g1, g2, . . . , gi

),

satisfy hik0 hji0 = hjk0 (j ≥ i ≥ k). Consequently, lim←−P 2(M i) exists and can

be endowed with a principal bundle structure, as asserted by the followingresult.

Theorem 8.6.5. P 2(M) := lim←−P 2(M i) is a (Frechet) plb-principal bundleover M , with structure group H0(F× F).

Proof. The trivializations (8.6.3) of P 2(M i), i ∈ N, form a projective sys-tem, because (

µji × hji)Φj

α = Φiα r

ji, j ≥ i.

Taking into account that

lim←−

U iα = Uα and lim←−

(Hi0(F× F)

)= H0(F× F),

we see that the isomorphisms

(8.6.4) Φα := lim←−Φiα : p−1(Uα)→ Uα ×H0(F× F); a ∈ I,


are well-defined, for p = lim←−

pi. These isomorphisms provide local topolog-

ical trivializations on P 2(M), which can be also thought of as differentialones under the conventions of § 5.1, regarding the generalized differentialstructure of H0(F × F). Moreover, each Φα is equivariant with respect tothe action of H0(F × F) on the right of P 2(M) = lim←−P 2(M i), induced by

their counterpart actions on the factors. Therefore, P 2(M) becomes a plb-principal bundle over M , with structure group H0(F × F) and transitionfunctions given by

g2αβ = Φα Φ−1β = lim

←−

(Φiα (Φiβ

)−1); α, β ∈ I.

The principal bundle P 2(M) := lim←−

P 2(M i) is called the generalized

second order frame bundle of the Frechet manifold M = lim←−

M i.

The preceding definition is a natural generalization of the standard sec-ond order frame bundle within the framework of Frechet manifolds. As amatter of fact, if M is a Banach manifold, modelled on E, then the projectivesystems M i;µji and P 2(M i); rji reduce to the trivial ones M ; idMand P 2(M); idP 2(M), respectively, where P 2(M) is the second order framebundle of M defined by (8.6.1). Analogously, H0(F × F) coincides withGL(E × E). Therefore, in this case, P 2(M) reduces precisely to the secondorder frame bundle of M , i.e. P 2(M) = lim

←−P 2(M i) = P 2(M).

It is worth noticing that the generalized second order frame bundleP 2(M) of M = lim

←−M i, apart from being a projective limit of Banach prin-

cipal bundles, it can be expressed also in a form analogous to that of itsfactors; namely,

P 2M ≡⋃

x∈M

(hi)i∈N

∣∣hi ∈ Lis(Ei × Ei, T 2µi(x)M

i) : lim←−

hi exists,

in view of the identification((h1, h2, ..., hi)

)i∈N≡(hi)i∈N

[see also (2.3.9)and (2.3.9′)].

Analogously to the case of ordinary frame bundles, the second ordertangent bundle T 2M of a plb-manifold M = lim←−M

i is associated with the

generalized bundle P 2M . More precisely, we prove the following:

Theorem 8.6.6. The quotient space E :=(P 2M × (F× F)

)/H0(F × F),

derived from the action of H0(F× F) on the right of P 2M × (F× F),

((hi), (ui, vi)

)i∈N·(gi)i∈N

:=((hi gi

),(gi)−1(

ui, vi))

i∈N,

is isomorphic to the second order tangent bundle T 2M .


Proof. Let π be the natural projection of E to the base manifold M , i.e.

π : E −→M :[(

(hi), (ui, vi))]7→ p

((hi)

):= lim←−

(pi(hi)

),

if p and pi are the projections of the bundles P 2(M) and P 2(M i), re-spectively (see the relative notations in the proofs of Proposition 8.6.4 andTheorem 8.6.5). Working with an open plb-covering

(Uα = lim←−U


iα

)α∈I

of M and the corresponding trivializations [see also (8.6.2)]

Φα = lim

←−Φiα : p−1(Uα)→ Uα ×H0(F× F)

α∈I

of P 2(M), we define the maps

Φα : π−1(Uα) −→ Uα × F× F :[(

(hi), (ui, vi))]7−→

(p((hi)

),Φα,2

((hi)

)((ui, vi)

)),

for all α ∈ I, where Φα,2 denotes the projection of Φα to H0(F× F).

Each Φα is injective: First observe that

Φα

([((hi), (ui, vi)

)])= Φα

([((hi), (ui, vi)

)])

⇒ p((hi)

)= p((hi)

)=(xi)∈M = lim

←−M i.

Since(hi),(hi)∈ P 2M , the limits lim←−h

i and lim←−hi can be defined. More-

over, in virtue of (8.6.3),

Φα

([((hi), (ui, vi)

)])= Φα

([((hi), (ui, vi)

)])

⇒ Φα,2

([((hi), (ui, vi)

)])= Φα,2

([((hi), (ui, vi)

)])

⇒((τ iα,xi h

i)(ui, vi)

)i∈N

=((τ iα,xi h

i)(ui, vi)

)i∈N

.

As a result, (hi(ui, vi

))i∈N

=(hi(ui, vi

))i∈N

.

Considering now the isomorphisms gi := (hi)−1hi ∈ GL(Ei×Ei), we obtaintheir limit lim←− g

i. Thus, with respect to the action of g := (gi) ∈ H0(F×F),

we conclude that[(

(hi), (ui, vi))]

=[(

(hi), (ui, vi))]

.


Also, Φα is surjective. Indeed, for any((xi), (ui, vi)

)i∈N∈ Uα × F× F,

each linear isomorphism hi :=(τ iα,xi

)−1belongs to Lis(Ei × Ei, T 2

xiM i). It

is now readily checked that(hi)i∈N∈ P 2(M). Therefore,

Φα

([((hi), (ui, vi)

)])=((xi),

((τ iα h

i)(ui, vi)))

=((xi), (ui, vi)

),

thus proving the desired surjectivity.

Since, for every α, β ∈ I, Φα,x Φ−1β,x = τα,x τ−1β,x ∈ Lis(F × F), where

τα,x = lim←−

τ iα,xi

, with x = (xi), we easily verify that E is indeed a vector

bundle with local trivializations (Uα, Φα)α∈I .On the other hand, if we denote by

Tαβ

αβ∈I

the transition functions

of E, equality (8.5.3) implies that

Tαβ(x) = Φα,x Φ−1β,x = τα,x τ−1β,x = T 2

αβ(x); x ∈ Uαβ

where nowT 2αβ

α,β∈I

are the transition functions of T 2M . Therefore,, E

is isomorphic to T 2M .

As in the case of Theorem 8.6.2 (see also Remark 8.6.3), we can describea concrete vb-isomorphism between E and T 2M , namely

G : E −→ T 2M :[(

(hi), (ui, vi))]7→(hi(ui, vi)

).

The range of G is indeed T 2M because, for every[(

(hi), (ui, vi))]∈ E, the

familyhi : Ei×Ei → T 2

xiMii∈N

is a projective system, and (ui), (vi) belong

to F = lim←−

Ei; therefore,(hi(ui, vi)

)i∈N∈ T 2M = lim

←−T 2M i. Moreover, G is

well-defined, since[(

(hi), (ui, vi))]

=[(

(hi), (ui, vi))]

implies the existenceof (gi) ∈ H0(F× F), such that

hi gi = hi, gi(ui, vi) = (ui, vi); i ∈ N,(hi(ui, vi)

)i∈N

=(hi(gi(ui, vi))

)i∈N

=(hi(ui, vi)

)i∈N

.

The next step is to show that G is a bijection. First we see thatG([(

(hi), (ui, vi))])

= G([(

(hi), (ui, vi))])

yields hi(ui, vi) = hi(ui, vi), for

all i ∈ N. Also, the isomorphisms gi :=(hi)−1 hi (i ∈ N) define a projec-

tive limit, as a consequence of the existence of the projective limits of bothfamilies

(hi)

and(hi). Because

((hi), (ui, vi)

)·(gi)

=((hi gi), (gi)−1(ui, vi)

)=((hi), (ui, vi)

),


we conclude that[(

(hi), (ui, vi)]

=[(

(hi), (ui, vi))]

, which proves that G isinjective.

The surjectivity of G goes as follows: If (wi) ∈ T 2M = lim←−

T 2M i, where

wi ∈ T 2xiM i and x = (xi) ∈ M = lim

←−M i, then, taking a plb-chart

(Uα =

lim←−Uiα, φα = lim←−φ

iα

)of M and the corresponding chart

(Uα = lim

←−U iα,Φα = lim

←−Φiα

)

ofT 2M , the linear isomorphism [see also (8.5.2)]

τα,x := pr2 Φα

∣∣π−12 (x)

= lim←− τiα,xi : T 2

xM≃−−→ F× F

is defined. Hence,

((τ iα,xi

)−1)i∈N∈ P 2M,

(τ iα,xi(w

i))i∈N

=(ui, vi

)i∈N∈ F× F,

and

G([((

τ iα,xi)−1)

i∈N,(ui, vi

)i∈N

])=

((τ iα,xi

)−1(ui, vi)

)

i∈N

=(wi)i∈N

.

To prove that G is a vector bundle isomorphism it suffices to ensure thatit preserves the trivializations of the two bundles involved. This is verifiedby the following equalities, for every α ∈ I:

(Φα G)([(

(hi)i∈N, (ui, vi)i∈N

)])= Φα

((hi(ui, vi)

)i∈N

)

=(

Φiα

(hi(ui, vi)

))i∈N

=(p((hi)i∈N

),((τ iα h

i)(ui, vi))i∈N

)

= Φα

([((hi)i∈N, (u

i, vi)i∈N)])

.

Appendix: Further study

In the following list we select a few problems presenting a research interest,naturally complementing the main ideas and methods expounded in thecourse of this work.

1. State and prove a Chern-Weil theorem in the framework of projectivelimit principal and vector bundles. The later case may be of particularinterest because the ordinary general linear group GL(F) of the Frechetfibre type of the bundle should be replaced by the generalized Lie groupH0(F) with Lie algebra H(F).

2. Investigate the possibility to prove the analog of the holonomy theorem(concerning the algebra of the holonomy group). The Banach case ofthe classical result of W. Ambrose and I.M. Singer is studied in [Mag04]and [Vas78(b)].

3. Many aspects of the ordinary geometry of projective limits of Liegroupoids and Lie algebroids can be extended to the Frechet frame-work by using the methods of this book. A first attempt towards thisdirection appears in [Cab12].

4. Investigate and develop an approach to infinite-dimensional symplecticgeometry within the projective limit framework.

5. The development of a general theory of G-structures, where G is a pro-jective limit Frechet-Lie group, also may be interesting. Applicationsof this approach would provide a Frechet bundle with the analogs ofmany classical structures.

6. Another point of view of the totality of linear connections on a smoothfinite dimensional manifold M that is worth noting here, is that of sys-tem of connections devised by Mangiarotti and Modugno ([MM83],[Mod87]). Namely, whereas the function space of all linear connections

275

276 Appendix: Further study

is infinite dimensional, even in the case of finite dimensional M , it ispossible to obtain a finite dimensional bundle-representation of all lin-ear connections on M in terms of such a connection system. Indeed,there exists a unique universal connection of which every connectionin the system of connections is a pullback. A similar relation holdsbetween the corresponding universal curvature and the curvatures ofthe connections of the system (in this respect see Cordero, Dodson anddeLeon [CDL89]). This is a different representation of an object similarto that introduced by Narasimhan and Ramanan [NR61, NR63] for G-bundles, also allowing a proof of Chern-Weil’s theorem (cf. [CDL89],[Gar72], [KN69]).

The system of all linear connections on a finite dimensional manifoldM has a representation on the tangent bundle via the system space

CT = α⊗jγ ∈ T ∗M⊗MJTM | jγ : TM → TTM projects onto idTM.

Here we view idTM as a section of T ∗M ⊗ TM , which is a subbundleof T ∗M ⊗ TTM , with local expression dxλ ⊗ ∂λ.

The fibred morphism for the system CT is

ξT : CT ×M TM −→ JTM ⊂ T ∗M ⊗TM TTM,

(α⊗ jγ, ν) 7−→ α(ν)jγ.

In coordinates (xλ) on M and (yλ) on TM ,

ξT = dxλ ⊗ (∂λ − γiλ ∂i) = dxλ ⊗ (∂λ − y

j Γijλ ∂i).

Each section of CT → M , such as Γ : M → CT : (xλ) → (xλ, γµϑ),determines the unique linear connection Γ = ξT (Γ πT , idTM ) withChristoffel symbols Γλµϑ.

On the fibred manifold π1 : CT×MTM → CT , the universal connectionis given by:

ΛT : CT ×M TM −→ J(CT ×M TM) ⊂ T ∗CT ⊗ T (CT ×M TM)

(xλ, vλµν , yλ) 7−→ [(Xλ, V λ

µν)→ (Xλ, V λµν , Y

µV λµνX

ν)].

In coordinates,

ΛT = dxλ ⊗ ∂λ + dva ⊗ ∂a + yµviµν dxν ⊗ ∂i.


Explicitly, each Γ ∈ Sec(CT /M) gives an injection (Γ πT , idTM), ofTM into CT × TM , which is a section of π1, Γ coincides with therestriction of ΛT to this section:

ΛT |(ΓπT ,ITM )TM = Γ,

and the universal curvature of the connection Λ is given by:

ΩT = dΛTΛT : CT ×M TM → ∧2(T ∗CT )⊗TM V (TM).

So, here the universal curvature ΩT has the coordinate expression:

ΩT =1

2

(ykvjkλ ∂jy

mvimµ dxλ ∧ dxµ + 2 ∂ay

mvimµ dxa ∧ dxµ

)⊗ ∂i.

For more details of the corresponding universal calculus see Dod-son and Modugno [DM86]. In the case of Riemannian and pseudo-Riemannian manifolds, Canarutto and Dodson [CD85] used systemsof principal connections to establish certain incompleteness stabilityproperties; Del Riego and Dodson [DD88] established certain topo-logical and universal properties of sprays and Lie algebras, obtainingassociated completeness criteria.

The system of linear connections provides a bundle framework in whichchoices of linear connection may be made, and hence vector bundlestructures on T 2M are determined. It would be interesting to extendto infinite dimensional Banach and even Frechet manifolds the systemsof connections approach of [MM83] and its associated universal con-nections [CDL89]. That might make it possible to characterize furtherthe isomorphism classes of second order tangent bundles. Specifically,since all connections are pullbacks of the universal connection, in whatway are these pullbacks characterized through the conjugacy classes?

7. A natural question is the study of tangent bundles of higher order,extending the properties and results of Chapter 8.

8. The possibility to obtain metrics on the projective limits of Hilbertor Finsler bundles could give many important results related with themetric and leading to a wealth of applications.

9. It would be interesting to extend the finite-dimensional results of Dod-son and Vazquez-Abal [DV90, DV92] for bundle projection and liftingof harmonicity. This could apply to the infinite dimensional case of a


projective limit Hilbert manifold E = lim∞←s Es, of a projective sys-

tem of smooth Hilbert manifolds Es, consisting of sections of a tensorbundle over a smooth compact finite dimensional Riemannian man-ifold (M,g). Such spaces arise in geometry and physical field theoryand they have many desirable properties but it is necessary to establishexistence of the projective limits for various geometric objects. Smo-lentsev [Smo07] gives a detailed account of the underlying theory weneed—that paper is particularly concerned with the manifold of sec-tions of the bundle of smooth symmetric 2-forms on M and its criticalpoints for important geometric functionals. We may mention the workof Bellomonte and Trapani [BT11]who investigated directed systemsof Hilbert spaces whose extreme spaces are the projective and the in-ductive limit of a directed contractive family of Hilbert spaces. Via thevolume form on (n-dimensional compact) (M,g) a weak induced met-ric on the space of tensor fields is

∫M g(X,Y ) but there is a stronger

family [Smo07] of inner products on Es, the completion Hilbert spaceof sections. For sections X,Y of the given tensor bundle over M weput

(X,Y )g,s =

s∑

i=0

∫

Mg(∇(i)X,∇(i)Y ) s ≥ 0.

Then the limit E = lim∞←s Es with limiting inner product gE is a

Frechet space with topology independent of the choice of metric gon M. In particular it is known, for example see Omori [Omo70,Omo97] and Smolentsev [Smo07], that the smooth diffeomorphismsf : (M,g) → (M,g) form a strong projective limit Lie group Diff(M)modelled on the projective limit manifold

Γ(TM) = lim∞←s

Γs(TM)

of smooth sections of the tangent bundle. Moreover, the curvature andRicci tensors are equivariant under the action of Diff(M) which yieldsthe Bianchi identities as consequences.

10. A large body of work has concerned the properties of operators, par-ticularly linear ones, on infinite dimensional spaces, because of theirimportance in representing ordinary differential equations on functionspaces. The fibred equivalent of such problems yields partial differ-ential equations on manifolds. A common problem in applications oflinear models is the characterization and solution of continuous linearoperator equations on Hilbert, Banach and Frechet spaces. However,


there are many open problems. For example, it is known that for a con-tinuous linear operator T on a separable Banach space E there may beno non-trivial closed subspace nor non-trivial closed subset such thatA ⊂ E with TA ⊂ A ([Rea88, AAB94, Enf87]). Atzmon [Atz83] pro-vided what turned out to be an example of an operator on a somewhatartificial space but Golinski [Gol12, Gol13] gave an operator withoutinvariant subset and showed that operators without nontrivial invari-ant subspaces exist on C∞(K) for an arbitrary smooth compact man-ifold K.

There has been substantial interest from differential geometry in hy-percyclic operators, whose iterations generate dense subsets, a reviewis given in [Dod12]. A continuous linear operator T on a topologicalvector space E is cyclic if for some f ∈ E the span of T nf, n ≥ 0is dense in E. On finite-dimensional spaces there are many cyclicoperators but no hypercyclic operators. The operator T is calledchaotic [GEM11] if it is hypercyclic and its set of periodic points isdense in E. Each operator on the Frechet space of analytic functionson CN , which commutes with all translations and is not a scalar multi-ple of the identity, is chaotic. On the Frechet space H(C) of functionsanalytic on C, the translation by a fixed nonzero α ∈ C is hypercyclicand so is the differentiation operator f 7→ f ′. All infinite-dimensionalseparable Banach spaces admit hypercyclic operators but finite dimen-sional spaces do not. In particular a Frechet space admits a hyper-cyclic operator if and only if it is separable and infinite-dimensional,Ansari [Ans97], and the spectrum of a weakly hypercyclic operatormust meet the unit circle, eg. Dilworth and Troitsky [Dil03]. Suchcontexts indicate a number of areas of potential application of theprojective limit approach to study the corresponding Frechet differen-tial geometry and its operators.

11. The quantum completion A of the space of connections in a manifoldcan be viewed as the set of morphisms from the groupoid of the edgesto the compact gauge group G and Velhinho [Vel02] used this to gen-eralize the description of the gauge-invariant quantum configurationspace A/G. The definition of functional calculus on A/G relies on therepresentation of A and A/G as projective limits of families of finite-dimensional compact manifolds, offering means to construct measuresand vector fields. This groupoid approach is applied in [Vel02] toshow that the quotient of A by the gauge group is homeomorphic toA/G, clarifying the relation between the two spaces. See also Thie-


mann [Thi07] for a detailed discussion of A/G, the development ofmeasures and functional calculus on projective limits and their role inloop quantum gravity.

12. In his thesis of 1967, Ebin [Ebin67] gave a detailed study of the spaceof Riemannian metrics and in particular those on a compact smoothmanifold M ; he gave a summary of those results in [Ebin68]. TheRiemannian metrics M j C∞(S2T ∗), being sections of the bundle ofsmooth symmetric covariant 2-tensors on M induce positive definitebilinear forms on the tangent spaces of M. The group D of diffeomor-phisms of M with C∞ topology acts on the right of C∞(S2T ∗) bypull-back

A : C∞(S2T ∗)×D → C∞(S2T ∗) : (γ, η) 7→ η∗(γ)

and M is invariant under this action. The restriction of this action isa right action on M

A :M×D →M : (γ, η) 7→ η∗(γ).

because (ξη)∗γ = η∗ξ∗(γ). For λ ∈ M the isotropy group of λ is

Iλ = η ∈ D|η∗(λ) = λ.

Theorem ([[Ebin67, Ebin68]]). A induces a homeomorphism of D/Iγonto the orbit Oγ of D through γ by ηIγ 7→ η∗(γ). Then there is asubspace S ⊆M containing γ with the following properties:(1) A(Iγ , S) = S,(2) If η ∈ D with η∗(S)

⋂S 6= ∅, then η ∈ Iγ ,

(3) There exists a neighbourhood U of the identity coset in D/Iγ anda local section χ : U → D such that

F : U × S →M : (u, s) 7→ χ(u)∗(s)

is a homeomorphism onto a neighbourhood of γ.

Then for all λ ∈ M sufficiently near γ there exists η ∈ D such thatIλ ⊆ ηIγη

−1. Now, M is locally like a Frechet space and does notdirectly admit a manifold structure so in order to prove the theoremEbin ([Ebin67]) enlarged it to belong to the Sobolev space Hs(S2T ∗)used by Palais [Pa65], denoted the enlarged space by Ms. Next heenlarged D to Ds+1, the Hs+1 maps M → M with Hs+1 inverses, as


in Palais [Pa68]. A Ds+1-invariant Riemannian metric was constructedonMs. Then using the normal bundle to Osγ and the exponential mapon Ms the properties (3) of S in the above theorem were established.

We know that the space M can be represented as a projective limitof Banach manifolds from the earlier papers [Gal96, Gal98], as de-scribed in Chapter 4. Ghahremani-Gol and Razavi [GGR13] used thisprojective limit of Banach manifolds to represent the infinite dimen-sional space of Riemannian metrics on a compact manifold. Using thework of Galanis and coworkers on existence and uniqueness of integralcurves of a projective system of vector fields, described here in Chap-ter 2, they applied it to the parabolic partial differential equations forthe Ricci flow and its integral curves. They found short-time solutionsthat are locally unique and, in particular, showed that the Ricci flowcurve starting from an Einstein metric is not a geodesic. This workhas a number of potential lines of further development because of theimportance of spaces of metrics in many physical applications.

13. Information geometry and in particular quantum information the-ory increasingly make use of infinite-dimensional spaces of probabil-ity density functions and geometrical constructions thereon. Cur-rent research in this context may be found in [NBh12, NBa13, Nil14]and there are many applications in the sciences. Banach manifoldscan be used to represent an infinite dimensional family of probabil-ity density functions of exponential type, however, unfortunately theall-important likelihood function is not continuous on this manifoldand Fukumizu [Fuk05] turned to the weaker topology arising from re-producing kernel Hilbert space structure, cf. also [SM94]. It seemslikely that these developments may benefit from the projective limitapproach to geometric structures on infinite-dimensional manifolds.

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List of Notations

The list contains most of the notations used throughout this work, togetherwith a brief description and the page of their first appearance.

Chapter 1

L(E,F) space of continuous linear maps from Eto F, 1

L(E) abbreviation of L(E,E), 1

Lis(E,F) space of invertible elements in L(E,F), 1

Lis(E) abbreviation of Lis(E,E), 1

GL(E) general linear group of E, 1

Df(x) ∈ L(E,F) (Frechet) derivative of f : U ⊆ E→ Fat x ∈ U , 2

Df : U → L(E,F) total (Frechet) derivative of f , 2

Dkf kth derivative of f , 2

C∞ symbol of smoothness, 2

(U, φ) chart of a manifold, 2

(U, φ,B) chart with specific model B, 2

A (maximal) atlas of a manifold, 2

fV U , fψφ, fβα expressions of local representations off : M → N , 2

[(α, x)] or [α, x] equivalent class of tangent curves, 3

TxM tangent space of M at x, 3

φ : TxM → B isomorphism induced by a chart (U, φ,B),3

τM : TM →M projection of the tangent bundle, 4

(π−1(U),Φ), chart of TM induced by (U, φ), 4

Txf differential or tangent map of f : M → N

297

298 List of Notations

at x, 4

α(t) velocity vector of a smooth curveα at t, 5

Tf (total) differential or tangent map off : M → N , 6

X (M) space of smooth vector fields on M , 6

X(f) map defined by X(f)(x) := Txf(Xx), 6

Φ X φ−1 local representation of X ∈ X (M), 6

Xφ principal part of Φ X φ−1, 7

Xα abbreviation of Xφα, 7

γ : G×G→ G multiplication of a group G, 8

α : G→ G inversion of a group G, 8

λg : G→ G left translation by g ∈ G, 8

ρg : G→ G right translation by g ∈ G, 8

L(G) Lie algebra of a Lie group G, 9

h : L(G)→ TeG identification of L(G) with TeG, 9

g TeG considered as a Lie algebra via h, 9

exp ≡ expG exponential map of G, 9

Ad: G→ Aut(g) adjoint representation of G, 10

Ak(TxB, g) space of k-alternating maps TxB → g, 10

Ak(TB, g) (total space of the) bundle of k-alternatingmaps, 10, 22

Λk(B, g) space of g-valued k-forms on B, 10

Dlf ≡ f−1df left Maurer-Cartan or logarithmic differen-tial, 11

Drf ≡ df.f−1 right Maurer-Cartan or logarithmic differen-tial, 11

Fθ : M → G fundamental solution of Drz = π∗θ, 12

θ# : π1(M)→ G monodromy homomorphism of Drx = θ, 12

δ : M ×G −→M action of G on (the right of) M , 13

X∗ fundamental or Killing vector field, 13

Rg = δg : M →M right translation of M by g ∈ G, 13

Ex = π−1(x) fibre, over x, of a vector bundle E, 14

Uαβ abbreviation of Uα ∩ Uβ, 15

τα,x : π−1(x)→ E isomorphism induced by (Uα, τα), 14

(Uα, τα) trivialization of a vector bundle, 15

ℓ = (E,B, π) or E vector bundle, 15


(Uα, φα,Φα) vector bundle chart, 15

Φα,x : Ex → E isomorphism induced by (Uα, φα,Φα), 16

Tαβ transition map (function) of a vector bundleover Uαβ, 16

fx restriction of f on the fibre over x, 17

(f, h) : ℓ1 → ℓ2 morphism of vector bundles, 17

f : E1 → E2 another notation for morphisms of vectorbundles over the same base, 18

VBB category of vector bundles over B, 18

VBB(E) category of vector bundles over B and of fibretype E, 18

GL(E) sheaf of germs of smooth GL(E)-valued maps,19

H1(B,GL(E)) 1st cohomology group of B with coefficients inGL(E), 19

(E1 ×B E2, B, π) fibre product of vector bundles over B, 19

(E1 ⊕ E2, B, π) direct/Whitney sum of vector bundles over B,20

f∗(ℓ) = (f∗(E), Y, f∗(π)) pull-back of ℓ = (E,Bπ) by f : Y → B, 20

(L(E,E′), B, L) linear map bundle, 21(Lk(E1 × · · · × Ek, E′), B, Lk

)k-linear bundle map, 22

Ak(E,E′) k-alternating map bundle, 22

Ak(TB, g) k-alternating map bundle with fibresAk(TxB, g), 23

Γ(E) ≡ Γ(B,E) C∞(B,R)-module of global smooth sections of(E,B, π), 23

Γ(U,E) module of smooth sections of E over U ⊆ B,23

Φ ξ φ−1 local representation of ξ ∈ Γ(E) relative toa vector bundle chart (U, φ,Φ), 23

ξφ : φ(U)→ E local principal part of Φ ξ φ−1, 23

ξα : φα(Uα)→ E abbreviation of ξφα, 23

P k(B,E) space of E-valued k-polynomials on B, 23

pkf(a) polynomial (f(a), Df(a), . . . , Dkf(a)), 23

jkxξ k-jet of ξ at x, 24

Jk(ℓ) = (JkE,B, πk) k-jet bundle of sections of ℓ = (E,B, π), 24

K := r V connection map, 27

Kα ≡ KUαK relative to (Uα, φα,Φα), 28


κα local component of a connection K, 29

Γα Christoffel symbol (map) over (Uα, φα), 30

∇ covariant derivation, 32

Γγ(E) set of sections of E along γ, 33

∂ ≡d

dtbasic vector field of R, 33

τγ : Eγ(0) −→ Eγ(1) parallel displacement or translation along γ,33, 70

KΦb, holonomy group of (a linear connection) Kwith reference point b, 33

KΦ0b , restricted holonomy group of K with reference

point b, 33

f#βα local principal part of a vector bundle mor-

phism (f, h), 35

f# = f#αα local principal part of a vector bundle mor-

phism (f, idB), 37

(U,Φ) or (U,Ψ) local trivialization of a principal bundle(cf. vector bundle charts in p. 15), 38

Φx : π−1(x)→ G isomorphism induced by a trivialization (U,Φ)(cf. the vector bundle analog in p. 16), 39

ℓ = (P,G,B, π) or P principal bundle, 37

Γ(U, P ) set of smooth sections of P over U ⊆ B, 40

sα ∈ Γ(Uα, P ) (natural) section of P over Uα, 40

k : P ×B P → G map defined by q = p · k(p, q), , 40

gαβ : Uαβ → G transition map or function of a principalbundle P over Uαβ, 41

h∗(ℓ) = (h∗(P ), G,B′, π∗) pull-back of ℓ = (P,G,B, π) by h : B′ → B,43

ℓ(E) ≡ P (E) frame bundle of a vector bundle (E,B, πE),44

P ×G H quotient of P ×H induced by a Lie groupmorphism φ : G→ H , 46

ϕ(ℓ) principal bundle (P ×G H,H,B, πH) associ--ated to (P,G.B, π) by ϕ : G→ H , 47

ν : P × g→ TP vector bundle morphism with ν(p,Xe) = X∗

p ,52

V P vertical subbundle of TP , 52

νp : g→ VpP linear isomorphism identifying the Lie algebrag with VpP , 52


HP horizontal subbundle of TP , 55

uh horizontal component of u ∈ TpP , 55

Λ1(P, g) space of g-valued 1-forms on P , 55

ω ∈ Λ1(P, g) connection form on P , 55

uv vertical component of u ∈ TpP , 56

ωα ∈ Λ1(Uα, g) local connection form over Uα, 56

gα : π−1(Uα)→ G the map with gα(p) = (pr2 Φα)(p), 57

Ω ∈ Λ2(P, g) curvature form of ω, 64

Ωα ∈ Λ2(Uα, g), local curvature form over Uα 65

ωo canonical flat connection on B ×G, 67

αp horizontal lift of α with αp(0) = p, 68

Φx holonomy group of ω with reference pointx ∈ B, 70

Φ0x restricted holonomy group of ω with reference

point x ∈ B, 70

Cx loop group at x ∈ B, 70

C0x group of 0-homotopic loops at x ∈ B, 70

kp the map given by τα(p) = p · kp(τα), 70

Φp holonomy group of ω with reference pointp ∈ P , 70

Φ0p restricted holonomy group of ω with reference

point p ∈ P , 70

P [p] holonomy bundle of P at p, 71

H(B,G) set of classes of equivalent flat bundles withbase B and group G, 71

hω : π1(B)→ G holonomy homomorphism of a flat bundle(P, ω), 71

S(B,G) set of classes of similar homomorphismsh : π1(B)→ G, 72

Chapter 2

p : F→ R seminorm, 76

Γ = pαα∈I family of seminorms, 76

TΓ topology induced by Γ above, 76

BΓ neighborhood basis of TΓ, 76

F often a Frechet space (mainly from Chapter 2onwards), 77

Df : U × F1 → F2 total derivative of f (a la Leslie, in Frechet


spaces), 82

Ei, ρjii,j∈I projective system, 84

ρji : Ej → Ei connecting morphisms of the preceding, 84

lim←−

Ei projective limit of Ei, ρjii,j∈I , 84

ρi : lim←−Ei → Ei i-th canonical projection of the preceding, 84

(xi) = (xi)i∈I equivalent expression of elements in projectivelimits, 84

lim←−

f i projective limit of maps, 86

H(F1,F2) Frechet space of particular continuous linearmaps from F1 to F2, 90

Hi(F1,F2) Banach space of particular continuous linearmaps from F1 to F2, , 90

ε : H(F1,F2)→ L(F1,F2) the map (f i) 7→ lim←−

f i, 90

LI(F) a particular subspace of L(F) = L(F,F), 97

H(F) abbreviation of H(F,F), 97

comp the composition map LI(F)× L(F)→ L(F) :(f, g) 7→ f g, 98

ev the evaluation map L(E,F)× E→ F :(f, a) 7→ f(a), 101

Chapter 3

M i;µjii,j∈N projective system of manifolds, 108

µi : M = lim←−

M i →M i the i-th canonical projection of M , 110(

lim←−

U i, lim←−

ϕi)

typical chart of a projective limit manifold,

109

Gi; gjii,j∈N projective system of Lie groups, 124

gi : G = lim←−

Gi → Gi i-th canonical projection of G, 123

Chapter 4ℓi;F jii,j∈N projective system of principal bundles, with

ℓi = (P i, Gi, B, πi), F ji =(pji, gji, idB

), 142

pji : P j → P i connecting morphisms of P i, pjii,j∈N, 142

pi : P = lim←−

P i → P i i-th canonical projection of P , 143

(Uα,Φα)i∈Nabbreviation of a trivializing cover(

lim←−i∈N

U iα, lim←−i∈N

Φiα

)

i∈N

of lim←−

ℓi, 142


ki : (πi)−1 → Gi the map determined by u = si(πi(u)) · ki(u),147

k : π−1(U)→ G the projective limit of the preceding, 148

ωi(ui) same as ωiui , 154

ωi(ui)(w) the same as ωiui(w), 154

ω = lim←−

ωi projective limit of connection forms, 154

ωiαα∈I (for fixed i ∈ N) the local connection forms ofωi, 158

Chapter 5

Hi(F) abbreviation of Hi(F,F), 185

Hi0(F) the group Hi(F)⋂∏i

j=1 Lis(Ej), 186

H0(F ) the group H(F)⋂∏

∞

j=1 Lis(Ej), 186

(Ei, B, πi); f jii,j∈N projective system of vector bundles, 187

f ji : Ej → Ei connecting morphisms of the preceding, 187

EiU variant of (πi)−1(U), 187

f i : E = lim←−

Ei → Ei i-th canonical projection of E, 196

(T ∗

αβ) H0(F)-valued cocycle, 192

Chapter 6

J∞(E) infinite jet bundle lim←−

JkE, 214

Hi0(F1,F2) the space Hi(F1,F2)⋂∏i

j=1 Lis(Ej1,E

j2),

219

(P (Ei),Hi0(F), B,pi) Banach principal bundle with total spaceP (Ei) :=

⋃x∈B Lis(F, Ex), 219

giαβαβ∈I transition functions of P (Ei), 220

P (E) by definition the limit lim←−P (Ei), 220

Chapter 7

Γ∗

α generalized Christoffel symbol over (Uα, φα),232

Chapter 8

γ1 ≈x γ2 equivalence of curves up to acceleration, 250..γ acceleration of curve γ, 251

T 2xB second order tangent space of B at x, 251


T 2M second order tangent bundle of B, 251

π2 : T 2B → B natural projection of T 2B, 252

(Uα, τ2α) local trivialization of T 2B, 252

T 2f : T 2B → T 2B′ second order differential of f : B → B′, 255

P 2(B) second order frame bundle of a Banachmanifold B, 264

P 2(M) second order frame bundle of a Frechetmanifold M = lim

←−M i, 270

Subject index

acceleration of a curve, 251action

– effective, 13– free, 13– freely transitive, 13– right translation of, 13– smooth, 13– transitive, 13

adjoint representation, 10ambient space, 2associated bundle

– principal, 47– vector, 48

atlas (maximal)– of a manifold, 2– of a vector bundle, 15

B-morphism, 40Banach

– Lie group, 8– exponential of, 9

– manifold, 2– B-modelled, 2

– vector bundle, 15base (space)

– of a principal bundle, 38– of a vector bundle, 15

basic vector field, 33Bianchi identity, 65bundle

– flat, 71– of (linear) frames, 44– tangent, 4

C1-differentiability in l.c.s, 82Cn-differentiability in l.c.s, 82canonical

– flat connection, 67– form

– left on G, 10– right on G, 10

– map of V E, 27– projections of projective limit, 84

Cauchy sequence, 77

chaotic operator, 279Christoffel

– symbols– generalized, 232– of a covariant derivative, 32– of a linear connection, 30

cocycle– condition, 16– of a principal bundle, 41– of a vector bundle, 16–H0(F)-valued, 192

cocycles– cohomologous

– of plb-bundles, 198– of principal bundles, 42– of vector bundles, 18

connecting morphisms, 84connection

– flat, 67– canonical, 67

– form, 55– linear, 29– local forms of, 56– map, 27– on a principal bundle, 54– on a vector bundle, 27– plb on a vector bundle, 229

connections– (f, ϕ, h)-related, 59– (f, h)-related, 34– f -related, 250– projective system of, 154, 228– system of, 275

covariant derivation, 32– Leibniz condition of, 32

curvature form, 64– of a plb-connection, 167

curve– acceleration of, 251– horizontal lifting of, 68– tangent vector of, 5– velocity of, 251– velocity vector of, 5

curves

305

306 Subject index

– equivalent, 3– up to acceleration, 250

– tangent, 3– of second order, 250

derivative– in Banach spaces, 2– in l.c.s, 81

differentiable map, 1, 81differential, 2, 4, 82, 116

– Maurer-Cartan left, 11– Maurer-Cartan right, 11– left, right, 11– second order, 255

differentiation a la Leslie, 81differentiation in l.c.s, 81direct sum, 20

effective action, 13equation

– with Maurer-Cartan differential,12

– fundamental solution of, 12equivalence

– of second order, 250– up to acceleration, 250

equivalent– curves, 3, 250– flat bundles, 71– pairs, 259– trivializing covers, 15

equivariant– map, 38– morphisms, 53

exact sequence of vector bundles, 24–26exponential map, 9

(f, ϕ, h)-related connections, 59(f, h)-related connections, 34f -related

– connections, 250– vector fields, 7

factors of a projective system, 84fibre

– of a vector bundle, 14– product

– of principal bundles (particularcase), 40

– of vector bundles, 19– type of a vector bundle, 15

flat– bundle, 71– connection, 67

– canonical, 67form

– integrable, 12– left invariant, 10– of a connection, 55– of curvature, 64– right invariant, 10

Frechet space, 77frame bundle, 44

– generalized, 220free action, 13freely transitive action, 13fundamental solution, 12fundamental vector field, 14

G-B-morphism, 40G-morphism, 40G-splitting, 54g-valued forms, 10, 23, 55general linear group, 1, 81generalized

– Christoffel symbols, 232– frame bundle, 220

– of order two, 270– smooth maps, 192

graded space, 80

H0(F)-valued cocycle, 192holonomy

– bundle, 71– homomorphism, 72

holonomy group– of a linear connection, 33, 237– of a principal connection, 70– restricted

– of a linear connection, 33, 237– of a principal connection, 70

homomorphism– holonomy, 72– monodromy, 12

horizontal– component uh, 55– lifting of a curve, 68– subbundle, 27, 55– vector, 28

integrable form, 12integral curve, 7invariant form

– left, 10– right, 10

inverse– function theorem (Nash-Moser),

80– limit

see projective limit, 85isomorphism

Subject index 307

– of principal bundles, 40– of vector bundles, 18– plb, 197

jet– bundle of sections, 24– of (local) sections, 24

k-alternating (antisymmetric) map bun-dle, 22

k-linear map bundle, 22k-Lipschitz, 102Killing vector field, 14

left– canonical form on G, 10– differential, 11– limit, 85

Leibniz condition of a covariant deriva-tion, 32

Leslie Cn, 82Leslie’s differentiation, 81Lie algebra, 9Lie group

– Lie algebra of, 9– canonical form

– left, 10– right, 10

– inversion of, 8– left translation of, 8– multiplication of, 8– product of, 8– right translation of, 8

limit– chart, see projective limit chart– inverse, 85– left, 85– of diagrams, 85

linear connection, 29– Christoffel symbols of, 30– holonomy group of, 33– on a manifold, 249– restricted holonomy group of, 33

linear map bundle, 21local

– component of a connection, 29– connection forms, 56– curvature forms, 65– principal part

– of a vb-morphism, 35– of a vector field, 7

– representation of a smooth map,2

logarithmic derivative, 11loop group, 70

manifold– Banach, 2– modelled on B, 2

map– C1-differentiable in l.c.s, 82– Cn-differentiable in l.c.s, 82– differentiable in l.c.s, 81

Maurer-Cartan– equations, 10– form

– left , 10– right, 10

– left differential, 11– right differential, 11

model of chart, 2monodromy homomorphism, 12morphism

– of principal bundles, 39– B, 40– G, 40– G-B, 40

– of vector bundles, 17, 195– plb, 197

multiplicative differentials, 11

Nash-Moser inverse function theorem, 80natural sections of a principal bundle, 40

parallel displacement– in Frechet vb-bundles, 236– in plb-principal bundles, 163

parallel displacement/translation– in principal bundles, 70– in vector bundles, 33

parallel translation– in Frchet vb-bundles, 236– in plb-principal bundles, 163

pb-isomorphism, 40pb-morphism, 39plb

– Christoffel symbols, 232– connection, 161– connection (on a vector bundle),

229– group, 124– isomorphism, 197– manifold, 108

– tangent bundle of, 114– tangent space of, 113

– morphism, 197– principal bundle, 143– transition maps, 192– trivializations, 192– vector bundle, 189

pls-maps, 112

308 Subject index

polynomials of degree k, 23principal bundle, 37

– associated, 47– base of, 38– connection of, 54– natural sections of, 40– projection of, 38– pull-back of, 43– structure group of, 38– total space of, 38– transition functions of, 41– trivializing cover of, 38

principal bundles– isomorphism of, 40

principal connections– projective limit of, 154

principal part (local)– of a vb-section, 23– of a vector field, 7

projection– of a principal bundle, 38– of a vector bundle, 15– of tangent bundle, 4

projective limit– canonical projections of, 84– chart, 108– of Banach manifold, 108– of maps, 86– of principal bundles, 143– of principal connections, 154– of smooth maps, 112

projective system, 84– connecting morphisms of, 84– factors of, 84– of connections, 154, 228– of maps, 86– of principal bundles, 142– of vector bundles, 187– of vector fields, 121

pull-back– of a principal bundle, 43– of a vector bundle, 20– universal property of, 21, 43

related connections– on principal bundles, 59– on vector bundles, 34

restricted holonomy group, 33, 70, 237right canonical form on G, 10right differential, 11

second order– differential, 255– frame bundle

– of a Banach manifold, 264

– tangent bundle, 251– tangent space, 251

section– along a curve, 33

– parallel, 33– of a vector bundle, 23– of the tangent bundle, 6

sections– jet of, 24– natural of a principal bundle, 40

seminorm, 76similar homomorphisms, 72smooth

– curve, 3– map in a Banach manifold, 2– map in a Banach space, 2– maps generalized, 192

space– ambient, 2– graded, 80– tame, 80

structure– equation, 64

– local, 65– of a plb-connection, 174

– group of a principal bundle, 38subbundle

– of a vector bundle, 26– vertical, 26, 52

system– of connections, 275– space, 276

tame– linear map, 80– space, 80

tangent– bundle, 4

– of a plb-manifold, 114– of order two, 251– section of, 6

– curves, 3– map, 4, 6– space, 3

– of a plb-manifold, 113– of order two, 251

– vector, 3– of a curve, 5

tensorial form, 64topology of seminorms, 76total

– differential, 6– left/right differential, 11– space

– of a principal bundle, 38– of a vector bundle, 15– of the tangent bundle, 4

Subject index 309

transition functions– (maps) of a vector bundle, 16– of a principal bundle, 41

transitive action, 13translation

– left of a Lie group, 8– right

– of a Lie group, 8– of an action, 13

trivialization– of a principal bundle, 38– of a vector bundle, 15

trivializing– map of a vector bundle, 15– cover

– of a principal bundle, 38– of a vector bundle, 15

universal property of the pull-back, 21,43

vb– chart, 15– connection, 27

– local component of, 29– isomorphism, 18– morphism, 17

vector bundle– Banach, 15– associated to a principal bundle,

48– base (space) of, 15– chart, 15– connection of, 27– fibre type of, 15– jet of sections of, 24– morphism, 17

– local principal part of, 35– plb, 189– projection of, 15– section

– along a curve, 33– local principal part of, 23

– section of, 23– subbundle of, 26– total space of, 15– transition functions of, 16– trivialization of, 15– trivializing cover of, 15– trivializing map of, 15

vector bundles– direct/Whitney sum of, 20– exact sequence of, 24–26– fibre product of, 19– plb-morphism of, 195

vector field, 6

– Killing, 14– basic, 33– fundamental, 14– left invariant , 8

vector fields f -related, 7velocity vector, 5vertical

– component uv, 56– subbundle, 26, 52

Whitney sum, 20

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