global analysis of one-dimensional variational … · introduction the basic variational principle...

GLOBAL ANALYSIS OF ONE-DIMENSIONALVARIATIONAL PROBLEMS

Philip Schrader

This thesis is presented for the degree of

Doctor of Philosophy

of the University of Western Australia

School of Mathematics and Statisitics

2016

iii

Statement of candidate contribution

Chapter 5, published as [73], is work jointly authored by the candidate (majority

contributor and primary author) and Prof. Lyle Noakes. Many other parts of this

thesis were discussed in informal meetings with Prof. Noakes, but not to the extent

of co-authorship.

Prof. Lyle Noakes (supervisor and co-author):

Philip Schrader (candidate):

v

Abstract

From the global analytical point of view a one-dimensional variational problem con-

sists in extremizing a di↵erentiable action/cost function f : X ! R, where X is an

infinite dimensional manifold of paths in a manifold M , over a subset ⌦ ⇢ X of

admissible paths, for example those satisfying some regularity conditions, bound-

ary conditions or other constraints. Thus a solution to the variational problem is a

critical point of the restriction f |⌦.A standard criterion for existence of critical points is the Palais-Smale condition.

If this condition is satisfied then the gradient flow associated with f is well behaved,

and we are guaranteed not only existence of critical points but also existence of a

minimum. Moreover it is then possible to relate the total number of critical points

to topological properties of ⌦.

This thesis is about methods for proving that a one-dimensional variational prob-

lem satisfies the Palais-Smale condition. The methods are demonstrated with ex-

amples motivated by interpolation, approximation, geometric and optimal control

problems in Riemannian manifolds. To begin with we consider conditional extremals :

the critical points of 1

2

RIkx � Ak2dt, where I is the unit interval, x : I ! M is a

path on M , x is the tangent vector along x, A is an arbitrary vector field on M , and

the admissible paths satisfy fixed boundary conditions. Next we treat problems with

higher order covariant derivatives in the action, such as Riemannian cubics in ten-

sion: critical points of 1

2

RIkrtxk2� ⌧ 2kxk2dt with ⌧ 2 R constant. This is followed

by an investigation of curves with minimum total squared curvatureRIk2ds subject

to a fixed length constraint. Such curves are known as elastica and this is the first

example we encounter with a constraint that is not a boundary condition. Finally,

we consider a class of problems known as sub-Riemannian, where the admissible

paths are required to be tangent to a non-integrable distribution on M .

Contents

1 Introduction 1

1.1 Some sources of variational problems in manifolds . . . . . . . . . . . 2

1.1.1 Interpolation and approximation . . . . . . . . . . . . . . . . 2

1.1.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Global analytical methods . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Introduction to Banach manifolds 7

2.1 Preliminary results on Banach spaces . . . . . . . . . . . . . . . . . . 7

2.2 Banach manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Tangent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Submanifolds, immersions, submersions . . . . . . . . . . . . . 10

2.3 Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 The tangent bundle . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 The pullback bundle . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.3 More bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Exact sequences of vector bundles . . . . . . . . . . . . . . . . . . . . 17

2.5 Connections on vector bundles . . . . . . . . . . . . . . . . . . . . . . 18

2.6 Covariant derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.7 The exponential map . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Split derivatives and canonical connections on vector bundles 23

3.1 Split derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Canonical connectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Second order split derivatives . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Geodesics on vector bundles . . . . . . . . . . . . . . . . . . . . . . . 30

4 Geometry of manifolds of maps 33

4.1 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 Banach spaces of sections . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.3 Banach space valued section functors . . . . . . . . . . . . . . . . . . 37

vii

CONTENTS viii

4.4 Banach manifolds of maps . . . . . . . . . . . . . . . . . . . . . . . . 40

4.5 Vector bundles of sections . . . . . . . . . . . . . . . . . . . . . . . . 43

4.6 Local formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.7 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Conditional extremals 49

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.2 Geometry of the path space (after Elıasson) . . . . . . . . . . . . . . 50

5.3 Existence of critical points . . . . . . . . . . . . . . . . . . . . . . . . 51

5.4 Multiplicity of critical points . . . . . . . . . . . . . . . . . . . . . . . 56

5.5 Closed extremals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6 Existence of variationally defined curves with higher order elliptic

Lagrangians 65

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.1.1 Statement of results . . . . . . . . . . . . . . . . . . . . . . . 66

6.1.2 Description of methods . . . . . . . . . . . . . . . . . . . . . . 67

6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.2.1 Method for establishing the Palais-Smale condition . . . . . . 68

6.2.2 Polynomial di↵erential operators . . . . . . . . . . . . . . . . 70

6.2.3 Finsler structures on Hj(Hk(S,M)⇤TM) . . . . . . . . . . . . 72

6.3 Some general results . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.4.1 Higher order conditional extremals . . . . . . . . . . . . . . . 78

6.4.2 Riemannian cubics in tension . . . . . . . . . . . . . . . . . . 80

6.4.3 Closed cubics in tension . . . . . . . . . . . . . . . . . . . . . 82

7 Morse theory for elastica 83

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.1.1 The Palais-Smale condition for total squared curvature . . . . 85

7.2 Lagrange multipliers and elastica . . . . . . . . . . . . . . . . . . . . 87

7.3 Lagrange multipliers and the PS condition . . . . . . . . . . . . . . . 91

7.4 Manifolds of constant speed curves . . . . . . . . . . . . . . . . . . . 95

7.5 The Palais-Smale condition for elastica . . . . . . . . . . . . . . . . . 98

7.6 Morse theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

8 Towards variational problems in linear velocity constraints 109

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.1.1 Sub-Riemannian geometry . . . . . . . . . . . . . . . . . . . . 109

ix CONTENTS

8.2 The path space in sub-Riemannian geometry . . . . . . . . . . . . . . 111

8.2.1 Allotted frames . . . . . . . . . . . . . . . . . . . . . . . . . . 112

8.2.2 Regular points of the endpoint map . . . . . . . . . . . . . . . 113

8.2.3 Rigid curves? . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

9 Conclusion 117

9.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

9.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

A 119

A.1 Proof of Lemma 5.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

A.2 Some comments on the terminology of Eliasson . . . . . . . . . . . . 120

A.3 A symmetry-breaking function for the total squared curvature . . . . 121

A.4 Regularity of weak solutions to ODE . . . . . . . . . . . . . . . . . . 122

Bibliography 125

CONTENTS x

xi CONTENTS

Acknowledgements

In all your ways acknowledge Him, and He will direct your paths. (Prov 3:6)

I am very grateful to my supervisor Prof. Lyle Noakes for sharing his expertise

and ideas, and providing guidance, understanding and kind encouragement during

the course of my studies. He has been precisely the sort of supervisor I needed, and

I consider myself very fortunate.

I also wish to express my gratitude to my lovely wife Nina, who has demonstrated

incredible patience and provided loving support. She has been willing to share the

experience with me as much as possible, despite it being something of an ordeal.

Other family members, especially my parents Keir and Marion, siblings Kristy,

Michael, and Matthew, and parents in-law Richard and Roxanne Morine, have each

made special contributions to making it possible for me to complete this work.

I have also benefited from discussions with Michael Pauley, Shreya Bhattarai,

Brian Corr, Oscar Garay and Paul Baird, and welcome distractions o↵ered in the

company of Dan Hawtin, Mark Ioppolo, David Raithel, Con Savas and members of

the Lone Rangers Sporting Club.

Finally, I am thankful for the financial assistance of the Australian Postgraduate

Award, and the support of UWA sta↵.

CONTENTS xii

CHAPTER 1

Introduction

The basic variational principle underlying what is known as the Lagrangian ap-

proach to mechanics is, somewhat awkwardly, attributed to Hamilton (for example

[51]). According to this principle the equations of motion for a mechanical sys-

tem with configuration space M are obtained as necessary conditions for a path

x : [a, b]!M to extremize the action integral

Z b

a

L(t, x(t), x(t))dt

on the set of all paths with fixed endpoints. Here the Lagrangian L is a real-

valued function, often the kinetic minus potential energy of the system, and x is

the velocity. In this and other variational principles one typically calculates the

variational derivative of the integral by taking an arbitrary variation of a path x(t),

i.e. a smooth map x(t, ✏) : [a, b] ⇥ (��, �) ! M such that x(t, 0) = x(t), x(a, ✏) =

x(a), and x(b, ✏) = x(b), and calculating

d

d✏

Z b

a

L(t, x(t, ✏), x(t, ✏))dt

��✏=0

The paths for which the above quantity is zero are called stationary paths. Neces-

sary conditions for a path to be stationary are obtained, in the form of di↵erential

equations called Euler-Lagrange equations, via the fundamental lemma of the cal-

culus of variations : if f is locally integrable andR b

af(t)h(t)dt = 0 on an interval

(a, b) for all compactly supported smooth functions h then f is identically zero.

For example, the Lagrangian corresponding to a free point-particle is L = 1

2

kxk2.In this case the Euler-Lagrange equations simply require that the acceleration be

zero. The solutions to the Euler-Lagrange equations are called geodesics, and it

turns out that these are also stationary for the length function l =R b

akxkdt. A

natural question to ask is whether, given two points in M , there always exists a

1

1.1. SOME SOURCES OF VARIATIONAL PROBLEMS IN MANIFOLDS 2

minimal length or minimal energy path joining the two points: this is an archetypal

one-dimensional variational problem.

1.1 SOME SOURCES OF VARIATIONAL PROBLEMS IN MANIFOLDS

1.1.1 Interpolation and approximation

The standard example of an interpolation problem in a Riemannian manifold is the

trajectory planning problem for rigid bodies: move a rigid body through a series of

given points with a given orientation at each point. The configuration space for a

rigid body is the special Euclidean group SE(3), which can be given a Riemannian

metric corresponding to the kinetic energy of the rigid body. To interpolate a series of

points in such a manifold one might patch together standard Euclidean interpolants

(eg. cubic polynomials) calculated in coordinate charts, but the result will depend

on the choice of charts and will not take into account the global geometry.

The search for intrinsic (i.e. chart independent) methods of interpolation on

Riemannian manifolds was initiated independently by Gabriel and Kajiya [22] and

Noakes et. al. [60] with the study of stationary paths of the average covariant

acceleration 1

2

Rkrtxk2dt, where x : I ! M is a map from the unit interval to a

Riemannian manifold M and r is the Levi-Civita covariant derivative. In Euclidean

space these paths are cubic polynomials; in Riemannian manifolds they are therefore

known as Riemannian cubics, and are a natural candidate for interpolation when

we require di↵erentiability. For example, in trajectory planning rapid changes of

direction are undesirable so one requires paths to be at least C1. Applications of

this kind have been the primary motivation for studying Riemannian cubics, but

more recently cubics have also been studied by Gay-Balmaz et. al. [24, 25] and

Trouve and Vialard [79] for interpolation on spaces of shapes (images, landmarks,

curves, surfaces or tensors) in computational anatomy, motivated by applications in

medical imaging. Several alternative interpolation schemes have been proposed in

the interim (see for example [50] or [61] and the references therein), most of which

are variational in nature.

1.1.2 Geometry

Geodesics are the solutions of the most basic geometric variational problem: they

are stationary for the length l(x) =R b

akxkdt and also the energy/action E(x) =

1

2

R b

akxk2dt. They are the one-dimensional case of what are known as harmonic

maps, the general case being maps from an n-dimensional manifold N to M which

have stationary energy (we will not need the precise definition of this generalised en-

ergy here). In this context the functional obtained by integrating the norm squared

3 CHAPTER 1. INTRODUCTION

of the left hand side of the Euler-Lagrange equation is known as the bienergy and

its stationary maps are called biharmonic maps. The Euler-Lagrange equation for

E is rtx = 0, so the bienergy coincides with the average covariant acceleration.

Thus Riemannian cubics are precisely the biharmonic curves and have been studied

as such by several authors1 (see eg. [55, 54]).

Another interesting geometric quantity is the total squared curvature along a

curve: F (x) =R `0

k2(s)ds, where s is arclength and k(x) = krTTk, T = xkxk . In

addition to being stationary for the length, geodesics are also absolute minima for

F , i.e. straightest curves. Stationary paths for F are known as free elastica, and

stationary paths subject to the constraint of fixed length are called elastica. This

terminology originates in the classical problem studied by Euler and Bernoulli of

modelling the equilibrium position of a thin inextensible wire with clamped ends

(more details and references can be found in Chapter 7).

1.1.3 Optimal control

Geodesics can also be considered as solutions of an optimal control problem. Suppose

{Xi} is a collection of vector fields onM and consider the ODE x(t) =P

i uiXi(x(t)),

where the ui are real-valued functions called controls. A basic problem in control

theory is to find, given an initial state x(0) = p, controls ui for which the solution

of the ODE satisfies some desired terminal state x(1) = q. In optimal control

one assigns a cost function to the controls and seeks the controls which produce

the desired outcome with minimal cost. For example, the cost function might be1

2

Pi

R1

0

|ui(t)|2dt. If the vector fields Xi happen to form an orthonormal basis then

this cost function coincides with the energy E, and the optimal control problem

is simply a reformulation of the geodesic problem; the trajectories corresponding

to optimal controls are geodesics. However, in a typical control problem the Xi

will not necessarily span the tangent space at each point, and this leads to many

interesting questions about which terminal states are accessible and which kinds

of controls are needed. In particular, suppose the Xi are mutually orthogonal and

span a proper vector subbundle D of the tangent bundle to M , and the cost function

is as above. Then the optimal trajectories are sub-Riemannian geodesics, i.e. the

stationary paths of E restricted to curves which are tangent to D. Sub-Riemannian

geodesics have been widely studied and are an active topic of research (see [56] or

Chapter 8).

Riemmanian cubics and elastica have also been studied from the optimal control

point of view. For example Riemannian cubics, and what might be called sub-

1I am grateful to Paul Baird for bringing this to my attention.

1.2. GLOBAL ANALYTICAL METHODS 4

Riemmanian cubics, appear in [11] and [35], and elastica have been extensively

studied by Jurdjevic [36, 37].

1.2 GLOBAL ANALYTICAL METHODS

When the configuration space M is a Riemannian manifold, the short-time existence

of geodesics satisfying initial conditions is a local question that is settled by the the-

ory of ODEs. However, existence of geodesics satisfying given boundary conditions

is more complicated. There are at least two reasons for looking at this problem: from

a (holonomic) mechanical or optimal control perspective we may wonder whether a

minimal energy trajectory joining two given points always exists; and geometrically

it is interesting to know whether there exists a shortest path between two points.

This question is answered by the Hopf-Rinow theorem, which states that then any

two points can be joined by a length minimizing geodesic i↵ M is connected and

any geodesic segment can be extended to all of R. Another interesting question is

whether multiple such trajectories or paths exist, which leads to the Morse index

theory of geodesics.

The book Morse Theory by Milnor [53] contains an excellent treatment which

proceeds by analogy with the critical point theory of di↵erentiable functions: the

energy E(x) = 1

2

R b

akxk2dt is considered a function on the set of paths joining two

given points on M , which is “. . . something like an ‘infinite dimensional manifold”’,

and geodesics are ‘critical points’ of E.

This analogy was formalised in the work of Eells [13], Palais and Smale [63, 65],

and Elıasson [14], mostly with the aim of producing a Morse theory of harmonic

maps. These authors showed how to construct di↵erentiable structures modelled on

Banach spaces for sets of maps of various degrees of di↵erentiability from a compact

manifold to another manifold. For example, the natural domain for E is the Hilbert

manifold H1(I,M) consisting of continuous paths with square integrable speed, and

E is smooth on this domain. For the geodesic boundary value problem one seeks

critical points of the restriction of E to the submanifold H1(I,M)p,q consisting of

paths with endpoints p, q 2M .

This illustrates the basic structure of a one-dimensional variational problem:

extremize a di↵erentiable action/cost function f : X ! R, where X is an infinite

dimensional manifold of paths in a manifold M , over a subset ⌦ ⇢ X of admissible

paths, eg. those satisfying some boundary conditions or other constraints. For any

variational problem there are questions of existence - do critical points, local/global

maxima/minima exist; multiplicity - how many critical points of each type exist;

and regularity - how di↵erentiable are the critical paths?

5 CHAPTER 1. INTRODUCTION

Existence of minimizers can sometimes be proved by what is known as the direct

method in the calculus of variations : show f is bounded below on ⌦ and take a

minimizing sequence xn such that f(xn)! inf⌦

f , show that there is a topology on ⌦

in which xn has a convergent subsequence, and show that f is lower semi-continuous

with respect to this topology. If X is a Hilbert space and f(xn) is bounded then the

second step follows from the weak compactness of bounded sets in Hilbert spaces.

However, in this thesis the focus will be on a more powerful criterion for existence

of critical points called the Palais-Smale (PS) condition: f : ⌦ ! R is said to

satisfy the PS condition if any sequence (xi) ⇢ ⌦ on which f is bounded, and for

which |df(xi)|! 0, has a convergent subsequence. If this condition is satisfied then

gradient descent will locate critical points, or more precisely, the negative gradient

flow is defined for all positive time and has at least one critical point of f as a limit

point. Moreover, in addition to existence of minimizers, it is also possible to obtain

multiplicity estimates in terms of topological properties of ⌦ using results from

Ljusternik Schnirelman theory and Morse theory. As for regularity, in most of the

variational problems we consider, the solutions satisfy an Euler-Lagrange equation

with a non zero constant coe�cient for the highest derivative. It is therefore possible

to prove by induction that the solutions are smooth (cf. Appendix A.4).

In this thesis the Palais-Smale condition will be verified for the various kinds

of problems discussed in Section 1.1, and the consequent existence and multiplicity

results will be developed.

1.2. GLOBAL ANALYTICAL METHODS 6

CHAPTER 2

Introduction to Banach manifolds

The main novelties in the theory of manifolds modelled on infinite dimensional

Banach spaces, as opposed to finite dimensional manifolds, stem from two basic

facts about Banach spaces. The first of these is that Proposition 2.1.3 below does

not hold in the infinite dimensional case. The second is that a closed subspace

of a Banach space need not split, i.e. it need not have a closed complementary

subspace. This necessitates some slight changes from the standard definitions of

immersions, submersions and vector bundles. In addition there is the total absence of

a ubiquitous element of finite dimensional di↵erential geometry: the tensor product.

This is because even in the case of a separable Hilbert space the natural candidate for

a tensor product fails to satisfy the relevant universal property (see [23] for details).

The references for this chapter are the books [39] and [1], but in Section 2.5 we

depart from both with a di↵erent, but equivalent, approach to the definition of a

connection.

2.1 PRELIMINARY RESULTS ON BANACH SPACES

Many of the results which follow are applicable to what is sometimes called a Ba-

nachable space: a vector space with a topology such that addition and scalar mul-

tiplication are continuous, whose topology can be derived from a complete norm.

We follow Lang [39] in adopting an abuse of language in writing Banach space

when strictly speaking we mean Banachable space. Similarly, by an isomorphismof Banach spaces we will mean a linear homeomorphism (though Lang calls this a

toplinear isomorphism). Moreover, by a direct sum E = E1

� E2

of Banach(able)

spaces we mean a vector space direct sum with the product topology, and so the

projections pri : E ! Ei are continuous.

A closed subspace E1

of a Banach space E, is said to split if there exists a closed

complement E2

, i.e. a direct sum of Banach spaces E = E1

�E2

. As a consequence of

the closed graph theorem (see [1] Corollary 2.2.18) this is equivalent to the existence

7

2.1. PRELIMINARY RESULTS ON BANACH SPACES 8

of a continuous linear projection pr1

: E ! E1

and then E2

= (I�pr1

)(E). Note that

every Banach space is split by every subspace with finite dimension or codimension,

and every closed subspace of a Hilbert space splits via the inner product.

Example 2.1. (cf. [1] p. 52) Let E = `1: the Banach space of bounded sequences

with the sup norm, and consider the subspace consisting of all sequences which

converge to zero. This subspace is closed in E but does not split. Indeed, the com-

plement must contain each sequence xn = (1, 1/n, 1/n . . .), but then the sequence of

sequences {xi, i 2 N} converges to a sequence which converges to zero, and so the

complement cannot be closed.

Let E,F be Banach spaces and U an open subset of E. We say that f, g : U ! F

are tangent at u0

2 U if

limu!u0

kf(u)� g(u)kku� u

0

k = 0

We write L(E,F ) for the Banach space of continuous linear maps E ! F with

the sup norm. The iterated space L(E,L(E,F )) can be identified with the space

L2(E,F ) of multilinear maps E ⇥ E ! F , and inductively L(E,Lk�1(E,F )) =

Lk(E,F ).

Proposition 2.1.1. ([1] p. 75) For f : U ! F and u0

2 U there is at most one

A 2 L(E,F ) such that g(u) := f(u0

) + A(u� u0

) is tangent to f at u0

.

If such an A exists we say f is di�erentiable at u0

and write Df(u0

) := A. If f is

di↵erentiable at every u 2 U and the map Df : U ! L(E,F ) is continuous then we

say f is of class C1. We define Dkf : U ! L(E,Lk�1(E,F ) = Lk(E,F ) inductively,

and if Dkf exists and is continuous then we say f is of class Ck.

Proposition 2.1.2. (cf. [39] p. 11) Let E and F be Banach spaces and U an open

subset of E. If f : U ! L(E,F ) is of class Ck then so is the map U ⇥E ! F given

by (x, v) 7! f(x)v.

If E and F are finite dimensional then the converse holds:

Proposition 2.1.3. (cf. [39] p. 45) Let E and F be finite dimensional vector spaces

and U an open subset of a Banach space. Suppose f : U ⇥E ! F is a Ck map such

that for each x 2 U, fx : E ! F defined by fx(v) = f(x, v) is a linear map. Then

the map U ! L(E,F ) given by x 7! fx is also Ck.

A counterexample showing that the converse to Proposition 2.1.2 does not hold

in the infinite dimensional case is given in [1]. However, according to Prop 3.4.3 of

[1] it is possible to show that the map x 7! fx is of class Ck�1. Thus the converse

to Proposition 2.1.2 does hold in infinite dimensions for C1 maps.

9 CHAPTER 2. INTRODUCTION TO BANACH MANIFOLDS

A map f : U ! V , where U is an open subset of E and V an open subset of

F , is called a Ck di�eomorphism if it is of class Ck and has an inverse of class Ck.

A map f : E ! F is a local Ck di�eomorphism at x0

if there is an open subset U

containing x0

such that f |U is a Ck di↵eomorphism onto an open subset of F .

Theorem 2.1.4. (Inverse function theorem) Let E and F be Banach spaces, U an

open subset of E, and let f : U ! F be a map of class Ck with k � q. If for some

point x0

2 U the derivative Df(x0

) : E ! F is an isomorphism, then f is a local

Ck di↵eomorphism at x0

.

Corollary 2.1.5. Let E,F1

, F2

be Banach spaces with U an open subset of E and

f : U ! F1

⇥ F2

a map of class Ck. Let x0

2 U and suppose f(x0

) = (0, 0) and

Df(x0

) induces an isomorphism between E and F1

= F1

⇥ 0. Then there exists a

local di↵eomorphism g of F1

⇥ F2

at (0, 0) such that

g � f : U ! F1

⇥ F2

maps U1

⇢ U into F1

⇥ 0 and induces a di↵eomorphism of U1

at x0

on an open

neighbourhood of 0 in F1

.

2.2 BANACH MANIFOLDS

An atlas of class Ck on a set X is a collection of pairs (�i, Ui) with i in some index

set J such that

• Ui ⇢ X and {Ui : i 2 J} is a cover of X

• Each �i is a bijection Ui ! �i(Ui), where �i(Ui) is an open subset of some

Banach space Ei, and for any i, j, we have �i(Ui \ Uj) open in Ei.

• For each i, j, the map �j��1

i : �i(Ui\Uj)! �j(Ui\Uj) is a Ck-di↵eomorphism.

We give X the unique topology such that the Ui are open and �i are homeomor-

phisms. Each pair (�i, Ui) is called a chart of the atlas. We say (�i, Ui) is a chart atx if x 2 Ui.

If Ui, Uj have non-empty intersection then the derivative of �j � ��1

i gives an

isomorphism Ei ! Ej. Thus for any pair of charts in a connected component of X

we have Ei, Ej isomorphic.

Another homeomorphism � : U ! �(U) ⇢ E from an open subset of X onto an

open set of a Banach space E is said to be compatible with the atlas {(�i, Ui) : i 2 J}if each � � ��1

i is a Ck di↵eomorphism. Two atlases are compatible if their union is

an atlas.

2.2. BANACH MANIFOLDS 10

A Ck-manifold is a set X together with an equivalence class of compatible atlases

of class Ck on X. If all the Banach spaces Ei in some atlas are isomorphic to some

fixed E (eg. if X is connected), then there is an equivalent atlas for which they are

all equal to E. In this case we say that X is modelled on E.

A continuous map f : X ! Y between manifolds is said to be of class Ck if for

all x 2 X there is a chart (�, U) at x and a chart ( , V ) at f(x) such that f(U) ⇢ V

and fUV := � f � ��1 : �(U)! (V ) is of class Ck.

2.2.1 Tangent spaces

Let x be a point in a manifold X and (�, U),( , V ) charts at x. Suppose v, w are

elements of the vector spaces containing �(U) and (V ) respectively. We say that

(�, U, v) and ( , V, w) are equivalent if D( ��1)(�(x))v = w. An equivalence class

of such triples is called a tangent vector of X at x. The set of all tangent vectors at

x is called the tangent space at x, denoted TxX.

Each chart (�, U) at x gives a bijection from TxX to a Banach space; the tan-

gent vector represented by (�, U, v) corresponds to v. We give TxX the linear and

topological structures which make this bijection an isomorphism.

Given di↵erentiable f : X ! Y we define Txf : TxX ! Tf(x)Y as the unique

linear map such that given charts (�, U) at x, ( , V ) at f(x) and a tangent vector

v 2 TxX then Txfv 2 Tf(x)Y is the tangent vector represented by DfUV (�(x))vU ,

where vU is the representative of v with respect to (�, U).

There is an equivalent definition of the tangent space TxX as equivalence classes

of curves on X. A third alternative identifying tangent vectors with derivations on

C1(X,R) is equivalent for finite dimensional X but there are di�culties for infinite

dimensional manifolds because not every derivation is a tangent vector, see [74] p.

105.

2.2.2 Submanifolds, immersions, submersions

Let X be a manifold and Y ⇢ X. Y will be called a submanifold of X if for each

y 2 Y there exists a chart (U,�) for X containing y such that � : U ! U1

⇥ U2

for

U1

, U2

open subsets of the Banach spaces E1

, E2

, and such that

�(Y \ U) = U1

⇥ {a} (2.1)

for some a 2 U2

. The set of all such (Y \ U,�) forms an atlas for Y .

A smooth map f : X ! Y is called an immersion at x 2 X if there is an open

neighbourhood U of x such that f |U is a di↵eomorphism onto a submanifold of Y .


If f is an immersion at every x 2 X then it is called an immersion. If furthermore

f gives a di↵eomorphism onto a submanifold of X then it is called an embedding.A smooth map f : X ! Y is called a submersion at x 2 X if there exist charts

(U,�) at x and (V, ) at f(x) such that � : U ⇠= U1

⇥ U2

⇢ E1

⇥ E2

, and such

that fUV := � f � ��1 : U1

⇥ U2

! (V ) can be factored U1

⇥ U2

! U1

! V

into a projection followed by a di↵eomorphism. f is called a submersion if it is a

submersion at every x 2 X.

Proposition 2.2.1. (cf. [39] p. 29) Let f : X ! Y be a smooth map between

manifolds, then

(i) f is an immersion at x 2 X i↵ Txf is injective and its image splits

(ii) f is a submersion at x 2 X i↵ Txf is surjective and its kernel splits

A point y 2 Y is called a regular value of f : X ! Y if f is a submersion at

each x 2 f�1(y).

Theorem 2.2.2. If y is a regular value of f : X ! Y then f�1(y) is a closed

submanifold of X with tangent space Tx(f�1(y)) = kerTxf .

Proof. Since f is a submersion at any x 2 f�1(y) we have charts (U,�) at x, (V, )

at y such that fUV factors into U1

⇥ U2

! U1

⇠= (V ). Thus there exists a 2 U1

such that f�1

UV ( (y)) = {a} ⇥ U2

and therefore �(U \ f�1(y)) = {a} ⇥ U2

, i.e. �

satisfies the submanifold property (2.1). For the tangent space, suppose x 2 f�1(y)

and v 2 TxX, and we can assume without loss that a = 0. Recall that v can be

identified via charts with the local representative (v1

, v2

) 2 U1

⇥ U2

and then Txfv

with

DfUV (�(x))(v1, v2) =ddt|t=0

fUV (�(x) + t(v1

, v2

)) = ddt|t=0

(0 + tv1

) = v1

Thus if v 2 Tx(f�1(y)) we have v1

= 0 and therefore Txfv = 0. Conversely if

v 2 kerTxf then v1

= 0 and therefore v is in Tx(f�1(y)).

Let f : X ! Y be smooth and W ⇢ Y a submanifold. We say f is transversalover W if for all x 2 X with f(x) 2 W there is a chart (V, ) at f(x) such that

maps V di↵eomorphic to a product V1

⇥ V2

, with (W \ V ) = V1

⇥ {a}, and there

exists a neighbourhood U of x such that the composite

U V V1

⇥ V2

V2

f pr

is a submersion.

2.3. VECTOR BUNDLES 12

2.3 VECTOR BUNDLES

Let X be a manifold and ⇡ a map from some set E to X. Suppose {Ui : i 2 J} is

an open covering for X and for each i we have a Banach space Ei and a bijection

⌧i : ⇡�1(Ui)! Ui ⇥ Ei, called a trivialisation, such that

VB1 ⌧i commutes with the projection onto Ui, i.e.

⇡�1(Ui) Ui ⇥ Ei

Ui e

⌧i

⇡pr

Thus for each x 2 X we have a bijection ⌧ix : ⇡�1(x)! {x}⇥ Ei,

VB2 for each pair i, j and x 2 Ui \ Uj

(⌧j � ⌧�1

i )x : {x}⇥ Ei ! {x}⇥ Ej

is an isomorphism,

VB3 the map ⌧ij : Ui \ Uj ! L(Ei, Ej), x 7! (⌧j � ⌧�1

i )x is Ck.

Then we call {(Ui, ⌧i) : i 2 J} a trivialising covering for ⇡. Two trivialising coverings

are said to be equivalent if their union is also a trivialising covering. When we refer

to a vector bundle ⇡ (or E) over X we mean such a triple ⇡, E , X together with

an equivalence class of trivialising coverings. ⇡ is called the bundle projection, E is

called the total space, X is called the base space and Ex := ⇡�1(x) is called the �breover x 2 X. It will often be convenient to denote an element v 2 E with ⇡(v) = x

as (x; v).

A trivialising covering induces a topology for E by requiring that the ⌧i be home-

omorphisms, and an atlas for E as follows. For any ⇠ 2 E there is a chart (V,�) at

⇡(⇠) and a trivialisation (U, ⌧) with ⇡(⇠) 2 U . Then

(�, Id) � ⌧i : ⇡�1(V \ U)! �(V \ U)⇥ E

gives a chart for E known as a local trivialisation over �. Suppose we have local

trivialisations ⌧1

, ⌧2

over overlapping charts (�1

, V1

), (�2

, V2

). Then using properties

VB2 and VB3 of the trivialising covering, as well as Proposition 2.1.2,

⌧2

� ⌧�1

1

: U1

\ U2

⇥ E1

! U1

\ U2

⇥ E2


is Ck, and therefore the chart transition

(�2

, Id) � ⌧2

� ⌧�1

1

� (�, Id)�1 : �1

(U)⇥ E1

! �2

(U)⇥ E2

where U = U1

\ V1

\U2

\ V2

, is also Ck. Thus the local trivialisations form an atlas

for E with respect to which the projection ⇡ is Ck.

Alternatively, one might begin with a di↵erentiable structure for E and map

⇡ : E ! X of class Ck. Then if we have a trivialising covering {Ui, ⌧i} such that

each ⌧i is of class Ck, then the local trivialisations are compatible with the di↵er-

entiable structure for E . Since we have assumed the ⌧i are Ck, by Proposition 2.1.3

the condition VB3 is automatic in the case where the Ei are finite dimensional. Fur-

thermore, by the comments that follow Proposition 2.1.3, VB3 is also automatic if

the ⌧i are C1.

Notice also the weaker assumption that ⌧j � ⌧�1

i : Ui \ Uj ⇥ Ei ! Ui \ Uj ⇥ Ej

be Ck is actually su�cient for local trivialisations to be Ck. The necessity of VB3as opposed to this weaker condition will be explained in Section 2.3.3.

Suppose now that ⇡ : E ! X, ⇡0 : E 0 ! X 0 are vector bundles, and f0

: X ! X 0

a Ck map. A map f : E ! E 0 is called a VB-morphism (covering f0

) if

(1) ⇡0 � f = f0

� ⇡, and for each x 2 E , the map fx : Ex ! E 0f0(x)

induced by

restriction to fibres is continuous and linear,

(2) for each x0

2 X there exist trivialisations

⌧ : ⇡�1(U)! U ⇥ E, ⌧ 0 : ⇡0�1(U 0)! U0 ⇥ E 0

at x0

, f0

(x0

) respectively such that f0

(U) ⇢ U 0 and the map U ! L(E,E 0)

defined by

x 7! ⌧ 0f0(x) � f � ⌧�1

x

is Ck.

It follows that f is Ck. Indeed suppose we have local trivialisations (U,�, ⌧) and

(U 0,�0, ⌧ 0) with f0

(U) ⇢ U 0, then locally we have

f� : �(U)⇥ E ! �0(U 0)⇥ E 0

f�(�(x), v) = (�0f0

(x), ⌧ 0f0(x) � f � ⌧�1

x v)

which is Ck by Proposition 2.1.2.

Again (2) is automatic if f is C1 or if f is Ck and E and E 0 have finite dimensional

fibres.


2.3.1 The tangent bundle

Let X be a Ck manifold and TX the disjoint union of the tangent spaces TxX,

x 2 X. We will show that TX, with the natural projection TxX 7! x can be given

the structure of a vector bundle of class Ck�1. Indeed given a chart (�i, Ui) for X

with �i(Ui) ⇢ E we have a bijection ⌧i : ⌧�1

i (Ui)! Ui ⇥ E via the identification of

the equivalence class of (�i, Ui, v) with v itself. If (�i, Ui) and (�j, Uj) are overlapping

charts and we denote �ij := �j � ��1

i , then the transition map

⌧ij = ⌧j � ⌧�1

i : Ui \ Uj ⇥ E ! Ui \ Uj ⇥ E

is given by ⌧ij(x, v) = (x,D�ij(�i(x))v) since (�i, Ui, v) and (�j, Uj, D�ij(�i(x))v)

represent the same tangent vector. Now D�ij : Ui \ Uj ! L(E,E) is of class Ck�1

and each D�ij(x) : E ! E is an isomorphism. Thus a Ck atlas for X induces a

Ck�1 trivialisation for TX.

The map Tf : TX ! TY with Txf as defined in Section 2.2.1 is then a VB-

morphism of class Ck�1.

2.3.2 The pullback bundle

Given a vector bundle ⇡ : E ! Y and a Ck map f : X ! Y we define another

vector bundle f ⇤⇡ : f ⇤E ! X, where

f ⇤E := {(x, v) 2 X ⇥ E : ⇡(v) = f(x)}, f ⇤⇡(x, v) = x

called the pullback or induced bundle. Given a trivialising covering {(Ui, ⌧i)} for ⇡ we

have a trivialising covering {(f�1(Ui), ⌧ 0i)} where ⌧ 0i : (f⇤⇡)�1f�1(Ui)! f�1(Ui)⇥Ei

is defined as ⌧ 0i(x, v) := (x, pr2

⌧iv), with pr2

being projection onto Ei. The fibres

(f ⇤E)x = {(x, v), v 2 Ef(x)} are given the linear and topological structures such

that they are isomorphic to Ef(x) via ⌧i � (⌧ 0i)�1, and we give f ⇤E the di↵erentiable

structure which makes these trivialisations Ck isomorphisms.

It follows that the map f⇡ : f ⇤E ! E , f⇡(x, v) := v is a VB-morphism over f

whose restriction to fibres is the identity. Indeed given a trivialisation (f�1(U), ⌧ 0)

the map f�1(U)! L(E,E) given by x 7! ⌧f(x) � f⇡ � (⌧ 0x)�1 = ⌧f(x) � ⌧�1

f(x) is of class

Ck.

Let Z be another manifold and g1

: Z ! X, g2

: Z ! E maps of class Ck such


that the following diagram commutes:

Z

f ⇤E E

X Y

g2

g1

f⇤g2

f⇡

f⇤⇡ ⇡

f

(2.2)

Then the map f ⇤g2

: Z ! f ⇤E defined by f ⇤g2

(z) := (g1

(z), g2

(z)) is Ck and

commutes. Moreover, suppose � : F ! Y is another vector bundle and g : E ! Fa VB morphism. Then as a special case of the above there is an induced morphism

f ⇤(g � f⇡) : f ⇤E ! f ⇤F which we will abbreviate to gf .

The pullback bundle is an example of a more general construction called the

pullback or fiber product (see [39] p. 31). Given Z, g1

, g2

as above the existence of

a unique such map f ⇤g2

is called the universal mapping property of the pullback.

2.3.3 More bundles

Given a vector bundle ⇡ : E ! X there are two kinds of subsets of E which have an

induced vector bundle structure. If Y ⇢ X is a submanifold we define the restrictedbundle ⇡Y : EY ! Y as the induced (pullback) bundle from the inclusion Y ! X.

The restriction of a trivialisation (⌧i, Ui) for E to Ui\Y gives a trivialisation for EY .We say F ⇢ E is a subbundle if for each x 2 X there is a trivialisation (⌧, U) for

E at x, and a split subspace F ⇢ E such that

⌧(EU \ F)! U ⇥ F ⇥ {0}

F is then a vector bundle with fibre Fx = Ex \F , and is a closed submanifold of E .

Let F ⇢ E be a subbundle and define the factor bundle E/F as the disjoint union

over x 2 X of the factor spaces Ex/Fx, with the projection v 2 Ex/Fx 7! x. A

trivialising covering for E/F can be constructed as follows. Suppose {(⌧i, Ui)} is a

trivialising covering for E such that each ⌧i has the subbundle property:

⌧i(EUi

\ F) = Ui ⇥ Fi ⇥ 0.

For each i we let pr2

be the projection Fi ⇥ Ei ! Ei and in2

: Ei ! Fi ⇥ Ei the

inclusion. Then ker(pr2

⌧ix) = Fx, so we have an bijection ⌧ 0ix : Ex/Fx ! Ei, and

then the transition maps ⌧ 0ij : U ! L(Ei, Ej) given by ⌧ 0ij(x) = pr2

�⌧ij(x) � in2

are

of the same class of di↵erentiability as ⌧ij.


Suppose F is a subbundle of E and g : E ! G is a VB morphism with F ⇢ ker g.

Then the induced map g : E/F ! G is also a VB morphism. It is su�cient to show

that this is true locally, that is, for trivial bundles. If F = U⇥F and E = U⇥F⇥F 0

then E/F = U ⇥ F 0 and g = g � inF 0 , which is a VB morphism.

Given vector bundles ⇡ : E ! X, ⇢ : F ! X we define another bundle E�F , called

the Whitney sum, as the disjoint union over x 2 X of direct sums Ex �Fx, with the

projection v 2 Ex�Fx 7! x. Given trivialisations ↵ : EU ! U ⇥E, � : FU ! U ⇥F ,

a trivialisation for E � F can be defined fibrewise by ⌧x : Ex � Fx ! E ⇥ F .

We can also define a vector bundle structure for the union L(E ,F) over all

x 2 X of the vector spaces L(Ex,Fx) of linear maps Ex ! Fx. We will present this

construction in a little more detail, because it explains the need for VB3. Suppose

we have trivialisations:

U ⇥ E1

U ⇥ F1

EU FU

U ⇥ E2

U ⇥ F2

↵1

↵2

�1

�2

Then we can define trivialisations for L(E ,F)

U ⇥ L(E1

, F1

)

L(E ,F)U

U ⇥ L(E2

, F2

)

⌧2�⌧�11

⌧1

⌧2

by ⌧1

(Ax) := (x, �1

� Ax � ↵�1

1

) and similarly ⌧2

(Ax) := (x, �2

� Ax � ↵�1

2

). If E and

F are Ck then we would like L(E ,F) to be also, i.e. we need ⌧2

� ⌧�1

1

to be Ck.

By VB3 we know that ↵21

: U ! L(E2

, E1

) and �12

: U ! L(F1

, F2

) are Ck, so we

factor ⌧2

� ⌧�1

1

as

comp �(Id,↵21

, �12

) :

U ⇥ L(E1

, F1

)! U ⇥ L(E1

, F1

)⇥ L(E2

, E1

)⇥ L(F1

, F2

)! U ⇥ L(E2

, F2

)

(x,A) 7! (x,A,↵21

(x), �12

(x)) 7! (x, �12

(x) � A � ↵21

(x)).

which is Ck because (Id,↵21

, �12

) is Ck and comp defined as

L(E1

, F1

)⇥ L(E2

, E1

)⇥ L(F1

, F2

)! U ⇥ L(E2

, F2

)

(A,B,C) 7! C � A �B


is multilinear. Notice that it is essential here that ↵21

and �12

are Ck. The

weaker assumption that eg. ↵1

� ↵�1

2

: U ⇥ E2

! U ⇥ E1

be Ck is not suf-

ficient in general (cf. Proposition 2.1.3). To verify VB3 for L(E ,F) we factor

⌧12

: U ! L(L(E1

, F1

, L(E2

, F2

)) into

� (↵21

, �12

) : U ! L(E2

, E1

)⇥ L(F1

, F2

)! L(L(E1

, F1

), L(E2

, F2

))

where (A,B)(C) := B � C � A is bilinear.

2.4 EXACT SEQUENCES OF VECTOR BUNDLES

Let f : E ! F be a VB morphism over the identity and define ker f to be the

union over x 2 X of ker fx, im f the union of im fx. A sequence of VB morphisms

over the identity

E F Gf g

will be called split �bre exact at F if im fx, ker gx split and im fx = ker gx for all

x 2 M . We say the sequence is exact at F if it is split fibre exact and im f = ker g

is a subbundle of F .

Theorem 2.4.1. Let f : E ! F be a VB morphism such that im fx and ker fx split

for all x 2M . Then ker f is a subbundle if and only if im f is a subbundle.

Proof. Suppose ker f is a subbundle, then the induced map f : E/ ker f ! F is an

injective VB morphism with im fx split. Then im f is a subbundle by Proposition

3.1 in [39]. Conversely, if im f is a subbundle then f : E ! im f is a surjective VB

morphism with ker fx split, so ker f is a subbundle by Proposition 3.2 (dual to 3.1)

in [39].

Corollary 2.4.2. If im fx and ker fx split and either im f or ker f is a subbundle

then E/ ker f is VB isomorphic to im f .

Corollary 2.4.3. If the sequence

E F Gf g

is split fibre exact at F then to show that it is exact at F it is su�cient to show

that one of ker f or im g is is a subbundle.

In fact, according to Proposition 3.4.20 in [1] it is su�cient to show that ker fx

and im gx split for all x 2 X.

2.5. CONNECTIONS ON VECTOR BUNDLES 18

It follows that the short sequence of VB morphisms over the identity

0 E F G 0f g

is split fibre exact at E ,F ,G i↵ it is exact at E ,F ,G, in which case it is called a shortexact sequence. Here 0 is the trivial bundle X ⇥ {0}. A VB morphism p : F ! Esuch that p � f = IdE is called a le� split for the short exact sequence. A right splitis a VB morphism q : G ! F with g � q = IdG.

Lemma 2.4.4. The exact sequence above has a left split i↵ it has a right split i↵

there is a VB isomorphism � : F ! E � G (Whitney sum) such that the following

diagram commutes

0 E F G 0

E � G

f

in

g

�pr

Proof. Let p : F ! E be a left split, then p is surjective and for all x 2 X we

have im fx \ ker px = 0. Furthermore any v 2 Fx can be decomposed uniquely as

v = (v � fp(v)) + fp(v) and therefore Fx is the algebraic direct sum of im fx and

ker px. But im fx is closed because it splits, and ker px is also closed because px is

continuous. Hence Fx = im fx � ker px (topological direct sum), i.e. ker px splits.

It follows by Theorem 2.4.1 that ker p is a subbundle and then F = im f � ker p.

Now by Corollary 2.4.2 im f is VB isomorphic to E and G is VB isomorphic to

F/ ker g = (im f � ker p)/ im f = ker p, whence F ⇠= E � G.If q is a right split then we claim that im qx is equal to the complement (ker gx)0,

which is closed by assumption. Therefore im qx splits, im q is a subbundle and a

similar argument to the above produces an isomorphism F ⇠= E � G. To see that

im qx = (ker gx)0, first note that gx �qx = IGx

implies im qx ⇢ (ker gx)0. Then suppose

y 2 (ker gx)0, we will show that y 2 im qx. Consider y := qx�gx(y) 2 im qx ⇢ (ker gx)0.

From gx(y) = gxy we have y � y 2 ker gx, but we also have y � y 2 (ker gx)0 since it

is closed. Thus y = y, i.e. y 2 im qx.

Conversely, if � is given then prE �� is a left split, and ��1�inG is a right split.

2.5 CONNECTIONS ON VECTOR BUNDLES

Let ⇡ : E ! X be a vector bundle and consider the tangent space TeE with e 2 Ex.In Section 2.2.1 a tangent vector w 2 TeE is defined as an equivalence class of charts

and elements of the model space. In the case of a vector bundle we can use local

trivialisations for charts and so w is an equivalence class [�, ⇡�1(U), (v, e)] where

(v, e) is an element of the model space B ⇥ E. Thus a local trivialisation gives an


isomorphism TeE ⇠= B⇥E ⇠= TxX⇥Ex. Note however that this isomorphism depends

on the local trivialisation. Part of our motivation for defining a connection is that

it will be useful to have an identification TeE ⇠= TxX ⇥ Ex which is independent of

coordinates.

Given a submersion f : X ! Y , by the universal mapping property (2.2) we have

a VB morphism over the identity

f ⇤Tf : TX ! f ⇤TY

Since Tfx is surjective and has split kernel the same is true of f ⇤Tfx, so ker f ⇤Tf

is a subbundle of TX called the vertical subbundle. In the case of a vector bundle

⇡ : E ! X this morphism can be extended to a short exact sequence

0! ⇡⇤E TE ⇡⇤TX ! 0vE ⇡⇤T⇡ (2.3)

with vE defined as follows. The inclusion of the fibre Ex ! E gives an injection

TeEx ! TeE for all e 2 Ex, then since TeEx = Ex = (⇡⇤E)e we have a linear injection

ve : (⇡⇤E)e ! TeE . It remains to check that the map vE : ⇡⇤E ! TE thus formed is

a VB morphism and that im vE = ker(⇡⇤T⇡). For this observe that a local trivial-

isation (�, ⌧) : EU ! �U ⇥ E for E induces local trivialisations for ⇡⇤E , TE , ⇡⇤TX

such that the following diagram commutes:

⇡⇤EEU

TEEU

(⇡⇤TX)EU

�U ⇥ E ⇥ E �U ⇥ E ⇥ B ⇥ E �U ⇥ E ⇥ B

vE ⇡⇤T⇡

pr

We let lE : E�E ! ⇡⇤E be the di↵eomorphism given by lE(x; e0, e1) = (x, e0

; e1

),

and define the vertical li� vlE := vE � lE

Following [84] we define a connection on E as a smooth splitting of the exact

sequence (2.3), i.e. a VB isomorphism TE ⇠= ⇡⇤TX � ⇡⇤E . It can be characterised

by a VB morphism KE : TE ! E over ⇡ : E ! X, called a connector, such that

⇡⇤KE is the left splitting map of the connection.

Lemma 2.5.1. Given a VB morphism KE : TE ! E over ⇡ : E ! X, ⇡⇤KE will be

a left split of (2.3) i↵ KE � vlE = pr2

: E � E ! E .

Proof. First note that pr1

�l�1

E = ⇡⇤⇡ and pr2

�l�1

E = ⇡⇡. Therefore

pr1

�l�1

E � ⇡⇤KE = ⇡⇤⇡ � ⇡⇤KE = ⌧E

2.6. COVARIANT DERIVATIVES 20

and pr2

�l�1

E � ⇡⇤KE = KE . The condition for a left split is that ⇡⇤KE � vE = Id⇡⇤E ,

which is equivalent to l�1

E � ⇡⇤KE � vE � lE = IdE�E , i.e. (⌧E , KE) � vlE = IdE�E . But

from ⌧E � v = ⇡⇤⇡ = pr1

�l�1

E we have ⌧E � vlE = pr1

, and therefore KE � vlE = pr2

is

necessary and su�cient.

IfKE is also a VB morphism over ⌧ : TX ! X (i.e. with respect to the secondary

vector bundle structure T⇡ : TE ! TX) then the corresponding connection is called

a linear connection.

Remark. From now on all connections will be taken to be linear.

It follows that in a local trivialisation TE ! �U ⇥ E ⇥ B ⇥ E, KE must have

the form

KE(x, e0; y, e1) = (x; e1

+ �E(x)(e0, y)) (2.4)

where �E : U ! L(E,B;E) is known as the local connector. The (linear) connectionis then given by

kE = (⇡⇤T⇡, ⇡⇤KE) : TE ⇠= ⇡⇤TX � ⇡⇤E (2.5)

with local expression

kE(x, e0; v1, e1) = (x, e0

; v1

, e1

+ �E(x)(e1, v1)) (2.6)

2.6 COVARIANT DERIVATIVES

Let N be another smooth manifold and f : N ! E a smooth map. Following [38]

the covariant derivative of f is the VB morphism defined by

rf := KE � Tf : TN ! TE ! E (2.7)

As a special case, for a section s : X ! E we have rs : TX ! E . It is natural

to identify rs with the corresponding section rs : X ! L(TX, E), and we usually

write rvs(x) for rs(x)(v), where v 2 TxX. Given a connection also for TX,

Elıasson has shown that there is an induced connection KL on L(TX, E) (and more

generally for L(E ,F), see [14] Proposition 1.2 for a detailed construction, or the

local formula (2.10) below). Thus, by treating rs as a section, one defines r2s by

(2.7): r2s := KL � Trs : TX ! L(TX, E). This is identified with a section of

L(TX,L(TX, E)) = L(TX � TX, E) and satisfies

rv(rws) = r2(v, w) +rrv

ws (2.8)

where v, w are vector fields on X. Iterating this procedure gives higher order deriva-

tives rks : X ! Lk(TX, E).


More generally, given a VB morphism over the identity f : E ! F with connec-

tors KE , KF , we can identify f with the section f : X ! L(E ,F), give L(E ,F) the

induced connection KL (again we refer to [14] Proposition 1.2 for details), define

rf by (2.7)

rf := KL � Tf : TX ! L(E ,F) (2.9)

and identify with the section X ! L(TX � E ,F). In a local trivialisation we write

f(x, e) = (x, f2

(x)e), and the local formula1 for KL from [14] gives

rf(x)(v, e) = (x, (Df2

(x)v)e+ �F(x)(f2(x)e, v)� f2

(x)�E(x)(e, v)) (2.10)

Moreover according to Lemma 2.1 of [14], if s : X ! E is a section then rf satisfies

r(f � s) = rf(·, s) + f � rs (2.11)

This is the familiar product rule which characterises covariant di↵erentiation of

di↵erential forms, eg: ! : X ! L(TX,R).However, if we apply (2.7) directly to f : E ! F instead of f : X ! L(E ,F),

the result: rf = KF � Tf , is quite di↵erent to (2.9). So it is not clear at this point

how to define rf when f : E ! F is not a VB morphism, because such a map can

not be identified with a section of L(E ,F), and a sensible definition should reduce

to the standard one when f is a VB morphism (i.e. it should satisfy (2.11)). We

will take up this task in Section 3.1.

2.7 THE EXPONENTIAL MAP

Consider the case N = I, the unit interval. In this case we write ddt

to mean the

section (t, 1) of TI = I ⇥ R. For � : I ! X, we write � := T� � ( ddt) : I ! TX and

then rt� := r�( ddt) = KTX � T � � ( d

dt).

Solutions to the equation rt� = 0 are called geodesics. In local coordinates this

equation has the form � + �X(�)(�, �) = 0 which is a linear ODE. For the basic

existence and uniqueness results for solutions of di↵erential equations defined on

Banach spaces, and flows of vector fields on Banach manifolds, we refer to Chapter

IV of [39]. Just as in the finite dimensional case, it follows from these results that

there is a neighbourhood Vx ⇢ TxX containing 0 such that there exists a unique

geodesic �v(t) with �v(0) = x, �v(0) = v which is extendible at least over the interval

[0, 1]. Moreover, dependence on the initial conditions is smooth. Thus there exists

a neighbourhood U of the zero section in TX such that exp(x; v) := �v(1) is a well

1with a di↵erent convention for the ordering of the arguments of the local connector.

2.7. THE EXPONENTIAL MAP 22

defined smooth map U ! X. Furthermore, from the homogeneity of the geodesic

equation, �sv(t) = �v(st), we have exp(x; tv) = �v(t) for t in a neighbourhood of 0,

whence

D2

exp(x; 0)v = ddt|t=0

exp(x; tv) = v

and so exp(x; ·) : TxX ! X is a local di↵eomorphism by the inverse function

theorem.

CHAPTER 3

Split derivatives and canonical connections onvector bundles

This chapter develops some more specialised results on connections and covariant

derivatives which play an important role in the geometry of manifolds of maps in

Chapter 4. Despite the preparatory nature, there is actually some new content.

To the best of my knowledge the definition of a split derivative (Definition 3.1)

does not appear elsewhere, though it has precursors in [14] and [39]. Nor does the

construction of a connection on the tangent bundle to a vector bundle in Section 3.2

and the description of its geodesics (Section 3.4)1.

3.1 SPLIT DERIVATIVES

The aim of this section is to provide a suitable definition of the covariant derivative

of a smooth map between vector bundles (which is not necessarily a VB morphism).

Elıasson has already defined covariant derivatives of exp : U ⇢ TM ! M in [14].

The main reason for doing this is that covariant derivatives of exp play a major role

in the geometry of manifolds of maps, particularly in Section 4.6. Lang has also given

a definition of a closely related object which he terms the tensorial tangent map or

tensorial derivative ([39] p. 288). What follows is something of a combination of the

two, but extended to general maps between vector bundles. We have two reasons

for requiring this generality: we will need to iterate the procedure in Section 3.3,

and the definitions will also be used in Chapter 8.

Suppose ⇡ : E ! M and ⇢ : F ! N are vector bundles with connections kE , kF

respectively and let f : E ! F be a smooth map. Then we define the split derivativeof f as the morphism T f given by

T f := kF � Tf � k�1

E : ⇡⇤TM � ⇡⇤E ! ⇢⇤TN � ⇢⇤F

1It turns out there is a similar connection in [30], but less information about its geodesics.

23

3.1. SPLIT DERIVATIVES 24

where Tf is the standard tangent map. In a local trivialisation we write f(x, ⇠) =

(f1

(x, ⇠), f2

(x, ⇠)), (x, ⇠) 2 U ⇥ E, and we have the following local formula for T f :

T f(x, ⇠, y, ⌘) = (f1

, f2

, @1

f1

y + @2

f1

(⌘ � �E(x)(⇠, y)),

@1

f2

y + @2

f2

(⌘ � �E(x)(⇠, y))

+ �F(f1, f2, @1f1y + @2

f1

(⌘ � �E(x, ⇠, y)))) (3.1)

where �F ,�E are local connectors for kF , kE , and each fi and its partial derivatives

is to be evaluated at (x, ⇠). It will be convenient to write this map in two parts:

T1

f : ⇡⇤TM ! ⇢⇤TM � ⇢⇤F ,

T2

f : ⇡⇤E ! ⇢⇤TM � ⇢⇤F

so that T f = T1

f + T2

f (fibrewise addition). From (3.1) the local principal parts

(fibre components) are

T1

f(x, ⇠, y) = (@1

f1

y � @2

f1

�E(x)(⇠, y),

@1

f2

y � @2

f2

�E(x)(⇠, y) + �F(f1)(f2, @1f1y � @2

f1

�E(x)(⇠, y)))

T2

f(x, ⇠, ⌘) = (@2

f1

⌘, @2

f2

⌘ + �F(f1)(f2, @2f1⌘))

where once again each fi and its partial derivatives is to be evaluated at (x, ⇠).

In the case of a smooth map g : M ! F we can identify M = M ⇥ {0}, whichwe give the trivial connector, and then

T g = kF � Tg : TM ! TF ! ⇢⇤TN � ⇢⇤F (3.2)

Similarly, for h : E ! N we have T h = Th � k�1

E : ⇡⇤TM � ⇡⇤E ! TE ! TN .

Comparing equation (3.2) with (2.7), and also using (2.5), we observe that in

this case rg = ⇢⇢ � pr2

T g. Indeed for a section s : N ! F we have T s = kE � Tfand supposing that in a local trivialisation s(x) = (x, s

2

(x)), the principal parts of

T s(x, y, z) are (y, ds2

(x)y + �F(x)(s2(x), y)).

Furthermore, for a VB morphism f : E ! F over the identity (i.e. assume

N = M), locally we write f(x, e) = (x, f2

(x)e) and

T1

f(x, e, v) = (x, f2

(x)e, v, (Df2

(x)v)e� f2

(x)�E(x)(e, v) + �F(x)(f2(x)e, v))

(3.3)

T2

f(x, e, w) = (x, f2

(x)e, 0, f2

(x)w) (3.4)

25 CHAPTER 3. SPLIT DERIVATIVES AND CANONICAL CONNECTIONS

so ⇢⇢ � pr2

�(T2

f) = f and

⇢⇢ � pr2

�T1

f(x, e, v) = (x, (Df2

(x)v)e� f2

(x)�E(x)(e, v) + �F(x)(f2(x)e, v))

If we compare the above expression with (2.10) we see that ⇢⇢�pr2

�T1

f : ⇡⇤TM ! Fcoincides, up to identification with a section, with rf . This leads us to the following

definition:

Definition. If f : E ! F is a smooth map between vector bundles with connections,

then the covariant derivative rf : ⇡⇤TM � ⇡⇤E ! F is defined by rf := ⇢⇢ �pr

2

�T f , and splits into r1

f : ⇡⇤TM ! F , r2

f : ⇡⇤E ! F .

Remark. Following Elıasson’s ([14] p. 179) definition of r exp one might instead

define rf : E � TM � E ! F by

rf := KF � Tf � (⌧E , ⇡, KE)�1 (3.5)

where (⌧E , ⇡, KE) : TE ! E � TM � E is the so-called Dombrowski splitting. As a

map between manifolds this definition is equivalent to the previous one by the di↵eo-

morphism E �TM �E ! ⇡⇤TM �⇡⇤E , however when one makes this identification

the bundle structure is lost and rf is not a VB morphism.

Geometrically, r1

f is the vertical (fibre) component of the derivative of f in the

horizontal (base) direction, and r2

f is the vertical component of the derivative in

the vertical direction. If f is a VB morphism we expect the vertical component of

the derivative in the vertical direction to be f itself. Indeed, from (3.4) we observe

that r2

f = f � ⇡⇤⇡ when f is a VB morphism.

Suppose � : N ! E is a lift of h = ⇡ �� and we want to calculate r(f ��). Firstcalculate

T � = kE � T� = (⇡⇤T⇡ � T�, ⇡⇤KE � T�) = (�;Th,r�)

then

r(f � �) = ⇢⇢ � pr2

�T f � T � = ⇢⇢ � pr2

(T2

f(�;Th) + T2

f(�;r�))

= r1

f(�;Th) +r2

f(�;r�) (3.6)

In particular if v : I ! E is a lift of x = ⇡ � v, then for T v : TI ! ⇡⇤TM � ⇡⇤Ewe have

T v( ddt) = (v; x,rtv) (3.7)

and therefore

rt(f � v) = r1

f(v; x) +r2

f(v;rtv)

3.2. CANONICAL CONNECTORS 26

Moreover, if f is a VB morphism then r2

f(v;rtv) = f(x;rtv), and we recover the

Leibniz rule.

For the exponential map exp : U !M we have T exp = T exp �k�1

TM because the

codomain is not a vector bundle. Thus r1

exp = T1

exp and r2

exp = T2

exp. As

mentioned above these maps coincide with those defined by Elıasson [14] modulo

the di↵eomorphism U � TM ⇠= ⇡⇤TM |U . In a local trivialisation:

r1

exp(x, v; u) = (exp(x, v), @1

exp(x, v)u+ @2

exp(x, v)(�TM(x)(v, u)))

r2

exp(x, v;w) = (exp(x, v), @2

exp(x, v)w)

so we have r2

exp = D2

exp.

3.2 CANONICAL CONNECTORS

Lemma 3.2.1. Let ⇡ : E ! M be a vector bundle with connector KE , and h :

N !M a smooth map. Then there is an induced connector for the pullback bundle

h⇤⇡ : h⇤E ! N given by Kh⇤⇡ := h⇤(KE � Th⇡), which we will call the pullbackconnector.

Proof. By Lemma 2.5.1 it is su�cient to prove that Kh⇤⇡ � vlh⇤⇡ = pr2

. Note that

vlE �(h⇡, h⇡) = Th⇡ � vlh⇤⇡ (in fact this is true for any VB morphism, not just h⇡),

and therefore

h⇤(KE � vlE �(h⇡, h⇡) = h⇤(KE � Th⇡ � vlh⇤⇡).

Since KE is a connector the left hand side is equal to pr2

, and by definition the right

hand side is Kh⇤⇡ � vlh⇤⇡.

Lemma 3.2.2. (Vilms [84]) Let � : E ! E 0 be a smooth isomorphism of vector

bundles. Then � �KE � T� is a connector for E 0.

Suppose ⇢ : F ! M is another vector bundle with connector KF and consider

the Whitney sum E�F !M . It is proved in [14] that there is a bundle equivalence

T (E � F) ⇠= TE � TF which is given locally by the identity map, and with this

identification (KE , KF) gives the direct sum connector for E � F .

We recall that the total space TE of the tangent bundle ⌧E : TE ! E has a

secondary vector bundle structure: T⇡ : TE ! TM . A connection for the secondary

is described in [84] and attributed to Kobayashi. The corresponding connector is

KT⇡ := TKE � E , where E : T 2E ! T 2E is the ‘canonical flip’ (for a manifold

M , the canonical flip T 2M ! T 2M can be defined locally by M(x, v0

; v1

v2

) =

(x, v1

; v0

, v2

)). We will call this the �bre connector.


It will be useful to also have a connection for ⌧E : TE ! E . In the special case

where E = TM the canonical flip M : T 2M ! T 2M gives an isomorphism between

the two bundles T ⌧ and ⌧TM , so using Lemma 3.2.2 the fibre connector induces a

connector for ⌧TM by the formula:

K⌧TM

:= M � TK⌧M

� TM � T�1

M : T 3M ! T 2M

This is the connection investigated by Vilms [84] and is equivalent to the one con-

structed by Elıasson [14]. Both authors proved that the corresponding geodesics

in TM are Jacobi lifts of geodesics on M , meaning that the fibre component of a

geodesic in TM is a Jacobi field along its projection onto M .

Returning now to a general vector bundle E , there is no isomorphism between

T⇡ and ⌧E , so we can not mimic the construction above. Instead we will use the

connection on E , i.e. the splitting k⇡ : TE ⇠= ⇡⇤TM � ⇡⇤E , and take the direct sum

of the pullback connectors. These connectors are

K⇡⇤⌧ = ⇡⇤(KTM � T⇡⌧ ) (3.8)

K⇡⇤⇡ = ⇡⇤(KE � T⇡⇡) (3.9)

for ⇡⇤TM and ⇡⇤E and then the direct sum connector is (K⇡⇤⌧ , K⇡⇤⇡). Now by

Lemma 3.2.2 the induced connector for ⌧E : TE ! E is

K⌧E = k�1

⇡ � (K⇡⇤⌧ , K⇡⇤⇡) � Tk⇡ (3.10)

Using k⇡ = (⇡⇤T⇡, ⇡⇤KE) (equation (2.5)) gives

K⌧E = k�1

⇡ � (⇡⇤K⌧M

� T⇡⌧ � T (⇡⇤T⇡), ⇡⇤K⇡ � T⇡⇡ � T (⇡⇤K⇡))

= k�1

⇡ � (⇡⇤K⌧M

� T 2⇡, ⇡⇤K⇡ � TK⇡)

It will be useful to have the local expression for K⌧E . Let E be the model space

for the fibres of E and B the model space for M . A local trivialisation EU ! �U⇥E

over a chart (U,�) for M induces tangent trivialisations

TEEU

! (�U ⇥ E)⇥ (B ⇥ E)

T 2ETEU

! (�U ⇥ E ⇥ B ⇥ E)⇥ (B ⇥ E ⇥ B ⇥ E)

3.3. SECOND ORDER SPLIT DERIVATIVES 28

for the bundles (TE , ⌧E , E) and (T 2E , ⌧ 2E , TE). In these coordinates we have

kE(x, e0; v1, e1) = (x, e0

; v1

, e1

+ �E(x)(e0, v1))

k�1

E (x, e0

; v1

, e1

) = (x, e0

; v1

, e1

� �E(x)(e0, v1))

KTM � T 2⇡(x, e0

, v1

, e1

; v2

, e2

, v3

, e3

) = (x; v3

+ �TM(x)(v1

, v2

))

KE � TKE(x, e0, v1, e1; v2, e2, v3, e3) = (x; e3

+ @1

�E(x)(e0, v1, v2)

+ �E(x)(e3, v1) + �E(x)(e1, v3)

+ �E(x)(e2 + �E(x)(e1, v1), v2))

and therefore

KTE(x, e0, v1, e1; v2, e2, v3, e3)

= k�1

E (⇡⇤, ⇡⇤)(x; v3

+ �TM(x)(v1

, v2

),

e3

+ @1

�E(x)(e0, v1, v2) + �E(x)(e2, v1) + �E(x)(e0, v3)

+ �E(x)(e1 + �E(x)(e0, v1), v2))

= (x, e0

; v3

+ �TM(x)(v1

, v2

),

e3

+ @1

�E(x)(e0, v1, v2) + �E(x)(e2, v1) + �E(x)(e1 + �E(x)(e0, v1), v2))

� �E(x)(e0,�TM(x)(v1

, v2

))).

3.3 SECOND ORDER SPLIT DERIVATIVES

Suppose f : E ! F as in Section 3.1. We would like to calculate the split derivatives

of r1

f : ⇡⇤TM ! F and r2

f : ⇡⇤E ! F . The pullback connection k⇡⇤⌧ (3.8) for

⇡⇤TM combined with the connection for E gives a splitting

(k⇡, Id) � k⇡⇤⌧ : T (⇡⇤TM) ⇠= (⇡⇤⌧)⇤(TE � ⇡⇤TM) ⇠= (⇡ � ⇡⇤⌧)⇤(TM � E � TM)

and similarly

(k⇡, id) � k⇡⇤⇡ : T (⇡⇤E) ⇠= ⇡⇤⇤(TM � E � E)

where we have used ⇡⇤⇤ to abbreviate (⇡ � ⇡⇤⇡)⇤. Define

T r1

f := kF � Tr1

f � k�1

⇡⇤⌧ (k�1

E , Id)

T r2

f := kF � Tr2

f � k�1

⇡⇤⇡(k�1

E , Id)


Each of these has three components, so we define

rirjf := ⇢⇢ � pr2

�Tirjf

for i = 1, 2, 3, j = 1, 2. For the exponential map

rri exp : ⌧ ⇤⇤(TM � TM � TM)! TM

Corollary 3.3.1. (Corollary 3.2 [14])

(a) ri exp(0, v) = v

(b) r3

ri exp(u, v, w) = ri exp(u, w)

(c) r1

r2

exp(u, v, w) = r2

r1

exp(u, w, v) is bilinear in (v, w)

(d) r2

r2

exp(u, v, w) is bilinear and symmetric in (v, w)

(e) rjri exp(0, ⇠, ⌘) = 0, i, j = 1, 2.

Proof. Eliasson gives a nice intrinsic proof [14]. Alternatively, these properties can

also be observed from local formulas. For example (d) is clear from the local expres-

sion for r2

r2

exp(x, u, v, w):

(exp(x, u), @22

exp(x, u, v, w) + �(exp(x, u), @2

exp(x, u, v), @2

exp(x, u, w))) (3.11)

From (a), applying the implicit function theorem to

r1

exp�r2

exp : ⌧ ⇤(TM � TM)|U ! TM

gives the existence of a smooth fibre preserving map ✓ : U ! L(TM, TM) such that

r2

exp(v, ✓(v)w) = r1

exp(v, w) (3.12)

Similarly, there is a smooth fibre preserving map ⇤ : U ! L2(TM, TM) defined

implicitly by

r2

r2

exp(u, v, w) = r2

exp(u,⇤(u)(v, w)) (3.13)

Lemma 3.3.2. (Lemma 3.1 [14])

r1

r2

exp(u, v, w) = r2

exp(u,D2

✓(u)(v, w) + ⇤(u)(✓(u)w, v))

3.4. GEODESICS ON VECTOR BUNDLES 30

3.4 GEODESICS ON VECTOR BUNDLES

Next we deduce some properties of the corresponding geodesics on E . Denoter(TE)the covariant derivative on E . A curve ↵ : I ! E is a geodesic i↵ r(TE)↵ =

KTE � T 2↵ = 0↵. Here ↵ is the tangent lift and 0↵ the zero section of TE over ↵.

Using the VB isomorphism kE we have that ↵ is a geodesic i↵

kE �KTE � T 2↵ = (⇡�1

⌧ �KTM � T 2(⇡ � ↵), ⇡�1

⇡ �KE � T (KE � T↵)) = 0↵

i.e. i↵ ⇡ �↵ is a geodesic on M and r(E)2↵ = 0⇡↵. In other words, ↵ is a 2-parallel

lift of a geodesic on M . Working in a local trivialisation for E we let ↵ = (�,!) and

we can write the geodesic equation KTE(�,!, �, !, �, !, �, !) = 0 as the system

� + �TM(�)(�, �) = 0 (3.14a)

! + @1

�E(�)(!, �, �) + �E(!, �) + �E(�)(! + �E(�)(!, �), �) + �E(!, �) = 0

(3.14b)

Equation (3.14b) can be obtained either by writing KTE � T 2↵ = 0 in local coor-

dinates and substituting (3.14a), or as the local expression of kE �KTE � T 2↵ = 0.

Given a solution of (3.14a), i.e. a geodesic on M , (3.14b) is a second order linear

ODE and therefore solvable for any initial conditions !(0) = e0

, !(0) = e1

.

We will denote the corresponding exponential map by expE : U ⇢ TE ! E .Working in the tangent trivialisation for TE induced by a local trivialisation for E ,we have the local expression expE : �(U)⇥ E ⇥ UB ⇥ E ! �(U)⇥ E given by

expE(x, e0

, v1

, e1

) = (expM(x, v), expE2

(x, e0

, v, e1

))

Here expM(x, v) := �(1), where �(t) is the solution of (3.14a) with initial conditions

�(0) = x, �(0) = v, and expE2

(x, e0

, v, e1

) := !(1), where !(t) is the solution of

(3.14b) with x = �,!(0) = e0

and !(0) = e1

.

Since the exponential map is a local di↵eomorphism (Section 2.7) there is a

neighbourhood UE of the zero section of TE ! E such that F : UE ! E ⇥ E defined

by F (w) := (⌧Ew, expE w) is a di↵eomorphism onto a neighbourhood of the diagonal

in E ⇥ E .

Theorem 3.4.1. There is a neighbourhood UM of the zero section in TM such thateF := F � �1

E : ⇡⇤UM � ⇡⇤E ! E ⇥ E is a di↵eomorphism onto a neighbourhood of

the diagonal.

Proof. In our local coordinates F : �(U)⇥E ⇥ UB ⇥E ! �(U)⇥E ⇥ �(U)⇥E is


given by

F (x, e0

, v, e1

)! (x, e0

, expMx v, expE

2

(x, e0

, v, e1

)) (3.15)

We know that there exist neighbourhoods VB ⇢ B, VE ⇢ E each containing zero

such that F |�U⇥E⇥VB⇥VE is a di↵eomorphism onto its image. But since equation

(3.14b) is linear in e we have that expE2

(x,�e0

, v,�e1

) = � expE2

(x, e0

, v, e1

) for any

scalar � 2 R. Thus if we define �E : B ⇥ E ⇥ B ⇥ E ! B ⇥ E ⇥ B ⇥ E by

�E(x, e0, v, e1) = (x,�e0

, v,�e1

) then F (x, e0

, v,�e1

) = �EF (x, e0�, v, e

1

), and we can

take VE = E. Now since kE is a VB isomorphism we have that eF |�U ⇥E ⇥ VB ⇥E

is a di↵eomorphism and the result follows.

Corollary 3.4.2. For any v 2 UM the restriction eFx,0,v : Ex ! Eexp

M

(x,v) is a linear

homeomorphism.

Proof. In a local trivialisation, using the expression for kE and (3.15)

eF (x, e0

, v, e1

) = (x, e0

, expM(x, v), expE2

(x, e0

, v, e1

� �E(x)(v, e0)))

Thus for e 2 E we have eFx,0,v(e) = expE2

(x, 0, v, e) which is linear in e by the linearity

of (3.14b). Moreover since eF is a di↵eomorphism we have

eF (x, 0, v, w) = (x, 0, expM(x, v), 0) () w = 0

so eFx,0,v has trivial kernel.

3.4. GEODESICS ON VECTOR BUNDLES 32

CHAPTER 4

Geometry of manifolds of maps

In this chapter the di↵erentiable structures for manifolds of maps N ! M are

constructed, when N is a compact Riemannian manifold and M a manifold with

connection. We follow the treatment of Elıasson [14], from which the title is bor-

rowed, but provide more detail in some parts. In particular, the charts are modelled

on Banach spaces of sections of vector bundles. So we begin with a description of

the most useful examples: the Ck sections and the Sobolev Lpk sections. We also give

a proof of the Sobolev imbedding theorem for spaces of sections.

Once again there is some new content: in Section 4.5 we use the connection for

TE from Section 3.2 and the properties of expE to prove that a manifold of maps

N ! E has the structure of a vector bundle over the manifold of maps N ! M .

This was previously only proved for E = TM in [14].

4.1 FUNCTION SPACES

Let ⌦ be an open subset of Rn and denote C1(⌦,Rm) the space of infinitely con-

tinuously di↵erentiable functions ⌦ ! Rm. Thus for each u 2 C1(⌦,Rm), and

for any k we have the continuous (total) derivative Dku : ⌦ ! Lk(Rn;Rm), where

Lk(Rn;Rm) is the space of k-multilinear maps from (Rn)k ! Rm. This space can

be given an inner product

A · B := Tr(B⇤A) =nkX

i=1

A(ei) · B(ei)

where {ei} is an orthonormal basis for (Rn)k, and B⇤ is the adjoint of B considered

as a linear map Rnk ! Rm. The corresponding norm is called the Frobenius or

Hilbert-Schmidt norm:

kAk := Tr(A⇤A)

33

4.1. FUNCTION SPACES 34

Suppose u 2 Ck(⌦,Rm) and define, for positive integers k, p,

|u|k :=kX

i=0

supx2⌦

kDiu(x)k (4.1a)

kukk,p :=

kX

i=0

Z

⌦

kDiu(x)kpdx!

1/p

(4.1b)

The subspace Ckb (⌦,Rm) := {u 2 Ck(⌦,Rm) : |u|k < 1} is a Banach space with

respect to the norm (4.1a). We will denote by Lpk(⌦,Rm) the completion of {u 2Ck(⌦,Rm) : kukk,p <1} in the norm (4.1b). These Lpk spaces are collectively known

as Sobolev spaces. They have an equivalent formulation as spaces of p-integrable

functions with p-integrable distributional (or weak) derivatives up to order k (see

eg: [2]). In the case p = 2 we have an inner product on L2k(⌦,Rm)

hu, vik :=kX

i=0

Z

⌦

Diu(x) ·Div(x)dx

making it into a Hilbert space which is often abbreviated (slightly!) to Hk(⌦,Rm).

Remarks

- In most expositions (a standard example being [2]), D represents a partial

derivative instead of the total derivative we have used here, and the sums

in (4.1) are taken over all partial derivatives. The norms thus obtained are

equivalent to those defined above. Moreover, usually only maps ⌦ ! R are

considered, but all results extend easily to maps ⌦! Rm.

- Sometimes the k kk,p norm is defined with the pth root taken inside the sum-

mation. But then when p = 2 it does not coincide with the norm induced from

the inner product.

- Note that strictly speaking the elements of Lpk are not functions but equivalence

classes of functions which are equal almost everywhere (i.e. except on a set of

measure zero). However, we will follow the common practice of ignoring this

distinction most of the time.

Given Banach spaces X, Y , we say that X is continuously imbedded in Y , denoted

X ,! Y , if X ⇢ Y and there is a constant C such that kukY CkukX for all u 2 X.

We say the imbedding is compact if every bounded sequence in X has a subsequence

which converges in Y .

35 CHAPTER 4. GEOMETRY OF MANIFOLDS OF MAPS

For j k continuity of the imbeddings Lpk(⌦,Rm) ,! Lpj(⌦,Rm) and Ck(⌦,Rm) ,!Cj(⌦,Rm) follow immediately from the definitions above. When ⌦ has finite volume

and 1 p q < 1 then Lq(⌦,Rm) ,! Lp(⌦,Rm) by the Holder inequality, and

therefore Lqk(⌦,Rm) ,! Lpj(⌦,Rm).

Theorem 4.1.1. (Sobolev imbedding theorem) If k � j > npthen Lpk(Rn,Rm) ,!

Cjb (Rn,Rm).

Proof. See [6] Theorem 2.10 or [2] Theorem 5.4 for a more general statement.

There are, as Fraenkel writes [21], “a bewildering variety of conditions” on an

open set ⌦ ⇢ Rn which ensure that Sobolev imbeddings also hold for Lpk(⌦,Rm).

Here we will be content with the observation that if there is a continuous linear map

ext : Lpk(⌦,Rm) ! Lpk(Rn,Rm) such that ext(u)|⌦ = u for all u 2 Lpk(⌦,Rm) then

the composition of ext with the Sobolev imbedding and restriction to ⌦

Lpk(⌦,Rm)! Lpk(Rn,Rm)! Cjb (Rn,Rm)! Cj

b (⌦,Rm)

is an imbedding. If such a map exists it is called a (p, k)-extension operator for ⌦.

Moreover if ext is a linear map from functions (defined a.e) on ⌦ to functions on Rn

whose restriction to Lpk(⌦,Rm) is a (p, k)-extension operator for every 1 � p < 1,

k � 0 then ext is called a total extension operator for ⌦.

Theorem 4.1.2. If ⌦ is either a half space in Rn or a bounded domain with smooth

boundary, then there exists a total extension operator for ⌦.

Proof. [2] Theorem 4.28.

Theorem 4.1.3. (Rellich-Kondrakov theorem) If ⌦ is bounded and has su�ciently

regular boundary (C1 or Lipschitz continuous), and k � j > npthen the Sobolev

imbedding Lpk(⌦,Rm) ,! Cjb (⌦,Rm) = Cj(⌦,Rm) is compact.

Proof. [6] Theorem 2.33 or [2] Theorem 6.2.

Remarks Most statements of Sobolev imbedding and Rellich-Kondrakov theorems

include several other imbeddings. We have only stated those which will be used later.

Theorem 4.1.4. Let � : ⌦! ⌃ be a di↵eomorphism between bounded open sets of

Rn, then the induced map �⇤ : Lpk(⌦,Rm) ! Lpk(⌃,Rm), u 7! ��1 � u is an isomor-

phism.


4.2. BANACH SPACES OF SECTIONS 36

4.2 BANACH SPACES OF SECTIONS

Let N be a compact Riemannian manifold and ⇡ : E ! N a vector bundle with

metric h iE and compatible connection r. We will denote the space of smooth

sections of E by C1(E). For each ⇠ 2 C1(E) the covariant derivative r⇠ is a smooth

section N ! L(TN, E). The vector bundle L(TN, E) has a connection induced by

the Levi-Civita connection on N and the connection on E which we also denote byr.

By induction we have an induced connection on Lk(TN, E) = L(TN,Lk�1(TN, E))and the iterated covariant derivative rk⇠ 2 C1(Lk(TN, E)).

From the metrics on TN and E we have an induced metric on Lk(TN, E) definedfibrewise by hAx, Bxi = Tr(B⇤

xAx) for Ax, Bx 2 Lk(TxN, Ex), which in turn induces

a Finsler structure which we denote by k k.Let dV denote the volume form on N . Then for ⇠ 2 Ck(E) define

|⇠|k :=kX

i=0

supx2N

kri⇠(x)k (4.2a)

k⇠kk,p :=

kX

i=0

Z

N

kri⇠(x)kpdV!

1/p

(4.2b)

Since N is compact each of these norms exists for all u 2 Ck(E) and in fact Ck(E)is a Banach space with respect to (4.2a). The completion of Ck(E) in (4.2b) will be

denoted by Lpk(E). In the case p = 2 we again have an inner product

h⇠, ⌘ik :=kX

i=0

Z

N

hri⇠,ri⌘idV

making Hk(E) := L2k(E) a Hilbert space.

Proposition 4.2.1. The spaces Ck(E) and Lpk(E) are independent (as Banachable

spaces) of the choice of Riemannian metric on N and metric connection on E .

Proof. For Ck(E) the definition does not depend on the metric on N , so we only

need to prove this part of the proposition for Lpk(E). Suppose g and g are metrics on

N with corresponding volume forms dV and dV . Let (Ui,�i) be a finite collection

of charts covering N , and ↵i a subordinate partition of unity. Then writing x1

i . . . xni

for the coordinates in �i(Ui), and gi for �i⇤g, we have the local expression

dV =X

i

↵i

pdet gidx

1

i ^ . . . ^ dxni

and similarly for dV . Since each ↵i has compact support and g, g are positive def-


inite, there exist constants ci, Ci such that ci↵i

pdet gi ↵i

pdet gi Ci↵i

pdet gi.

Now for ⇠ 2 Lpk(E) it follows that ck⇠kp,k,g k⇠kk,p,g Ck⇠kk,p,g, where cp = mini ci

and Cp = maxi Ci.

Suppose h and h are di↵erent metrics on E , and given v 2 Ex write kvk =ph(x)(v, v), and v = v

kvk . Then there is a constant c1

such that

kvk2 = h(x)(v, v) = kvk2h(x)(v, v) kvk2|h(x)| c1

kvk2

because |h(x)| depends continuously on x and N is compact. By a symmetric

argument there is a constant c2

such that kvk c2

kvk2. Now if KE and KE are

connectors which are compatible with h, h respectively, and ⇠ 2 Ck(E), then r⇠ �r⇠ = (KE � KE)T ⇠ : TN ! E . In a local trivialisation (recall equation (2.4))

KE � KE : TE ! E has the expression (KE � KE)(x, e0, v, e1) = (x,�x(e0, v) ��(x)(e

0

, v)), so there is a corresponding section1 A : N ! L(TN � E , E) such that

r⇠ � r⇠ = A(·, ⇠). Thus, using the compactness of N again, kr⇠k kr⇠k +|A(⇡⇠)|k⇠k kr⇠k+ c

3

k⇠k, therefore kr⇠kp 2p�1(kr⇠)kp + cp3

k⇠kp) and finally

k⇠kp1,p =

Z

N

k⇠kp + kr⇠kpdV Z

N

(1 + 2p�1cp3

)c2

k⇠kp+ 2p�1c

2

kr⇠kpdV

ck⇠kp

1,p

for some constant c. By a symmetric argument there is a constant C such that

k⇠kp

1,p Ck⇠kp1,p, and so h,KE and h, KE produce equivalent norms on Lp

1

(E). More-

over, since rr⇠ 2 C1(Lr(TN, E) for all r < k and k⇠kpr+1,p = k⇠kpr�1,p + krr⇠kp1,p it

follows by induction that the respective norms on Lpk(E) are equivalent. The same

is true for Ck(E) by a very similar proof.

It seems to be widely known that the Sobolev imbedding and Rellich-Kondrakov

theorems extend to these spaces of sections, but it is surprisingly di�cult to find

detailed proofs. We will take up this task in the next section.

4.3 BANACH SPACE VALUED SECTION FUNCTORS

Let N be compact manifold and V B(N) the category consisting of smooth vector

bundles over N with Banach space fibres and smooth VB morphisms. Denote by

� the functor from V B(N) to vector spaces and linear maps which assigns to each

E 2 V B(N) the vector space �(E) of sections of E . Given vector bundles E ,F over

1A global expression is A := (KE�KE)�kE � in1 : TN�E = ⇡⇤TN ! ⇡⇤TN�⇡⇤E ! TE ! E ,this does not depend on which connection is used for the splitting kE .

4.3. BANACH SPACE VALUED SECTION FUNCTORS 38

N and a VB-morphism f 2 hom(E ,F) the induced map f⇤ : �(E)! �(F), f⇤(⇠) :=

f � ⇠ is linear, so � is functorial.

A Banach space section functor S on V B(N) is a covariant functor S which

assigns to each E 2 V B(N) a Banach space S(E) ⇢ �(E). Here functoriality means

that f⇤ : S(E)! S(F), is a continuous linear map, i.e. f⇤ 2 hom(S(E),S(F)). This

is Palais’ first axiom ([65] p. 9). Elıasson [14] makes the equivalent requirement that

S⇤ : C1(L(E ,F)) ! L(S(E),S(F)), S⇤(A)(⇠) := A ⇠, where denotes fibrewise

composition, should be a continuous linear inclusion.

Note that S is an additive functor: S(E �N F) ⇠= S(E)�S(F), and also has the

following localisation property.

Lemma 4.3.1. A section ⇠ 2 �(⇠) is in S(E) i↵ for any p 2 N there is a neighbour-

hood Up of p in N and a section ⇠p 2 S(E) such that ⇠|Up = ⇠p|Up.

Proof. If ⇠ is in S(E) then let ⇠p = ⇠. Conversely given ⇠ 2 �(⇠) let Ui be a cover

of N and ⇠i 2 S(E) a corresponding collection with the property ⇠|Ui = ⇠i|Ui. If i

is a partition of unity subordinate to Ui then fibrewise multiplication by gives a

VB-morphism i : E ! E , and then i⇤ : S(E)! S(E) is linear. SinceP

i (p) = 1

for any p 2 N we have ⇠ =P

i i � ⇠ =P

i i � (⇠|Ui) =P

i i⇤(⇠i|Ui) 2 S(E).

We will say S has the restriction property if when N1

is a compact submanifold

of N then restriction ⇠ 7! ⇠|N1

is a continuous linear map S(E)! S(E|N1

).

Theorem 4.3.2. Suppose S is a section functor with the restriction property, E !N a vector bundle, and N

1

. . . Nr compact submanifolds of N whose interiors cover

N . Define

S(E) := {(⇠1

, . . . , ⇠r) 2 �ri=1

S(E|Ni) : ⇠i|Nj = ⇠j|Ni}

and F : S(E)! S(E), ⇠ 7! (⇠|N1

, . . . , ⇠|Nr). Then F is an isomorphism of Banach

spaces.

Proof. (cf. Palais [65] p. 10) Continuity of F follows from the restriction prop-

erty because S(E) is a subspace of �ri=1

S(E|Ni). Let i be a partition of unity

subordinate to the interiors of Ni. Given (⇠1

, . . . , ⇠r) 2 S(E) we have as in the

proof of Lemma 4.3.1 that fibrewise multiplication i⇠i gives an element of S(E|Ni).

Moreover by Lemma 4.3.1

⌘i(p) :=

8<

: i(p)⇠i(p) p 2 Ni

0 p /2 Ni

is an element of S(E), and therefore ⌘ :=P

i ⌘i 2 S(E). To show that ⌘ is the inverse

of (⇠1

, . . . ⇠r) we need ⌘|Ni = ⇠i. Indeed if p 2 Ni \Nj then by definition of S(E) we


have ⇠i(p) = ⇠j(p) and so for any p 2 Ni, ⌘(p) =P

j ⌘j(p) =P

j j(p)⇠i(p) = ⇠i(p).

So we have constructed an inverse for F , which also happens to be continuous.

Corollary 4.3.3. If we define instead

S := {(⇠1

, . . . , ⇠r) 2 �ri=1

S(E|Ni) : ⇠i|Nj = ⇠j|Ni}

where Ni is the interior of Ni, and F (⇠) := (⇠|N1

, . . . , ⇠|Nr) : S(E) ! S(E) then F

is still an isomorphism.

Proof. The proof is the same, because the each function i has compact support

contained in the interior Ni.

Lemma 4.3.4. Both Lpk and Ck are section functors with the restriction property.

Proof. Suppose f : E ! F is a VB morphism and ⇠ 2 C1(E). We identify f 2C1(L(E ,F)), and then rf 2 C1(L(TN,L(E ,F))) satisfies r(f ⇠) = rf(·, ⇠) +f r⇠ and therefore for each x 2 N

kr(f � ⇠)xk krfxkk⇠xk+ kfxkkr⇠xk

For C1 and Lp1

the functoriality is proved by taking the sup over and integrating the

above inequality respectively, and then Ck and Lpk follow by induction. Checking

the restriction property is straightforward.

Theorem 4.3.5. (Sobolev imbedding/Rellich-Kondrakov for sections) If E ! N is

a vector bundle with compact n-dimensional base and k�j > npthen Lpk(E) ,! Cj(E)

compactly.

Proof. Since N is compact there is a finite cover of E by trivialisations �i : E|Ni !Bi ⇥ Rm where each Bi ⇢ Rn is a closed ball. Since �i is a composition of a VB

isomorphism E|Ni ! Ni ⇥ Rm and a di↵eomorphism Ni ! Bi, by the functoriality

of Lpk and di↵eomorphism invariance (Theorem 4.1.4), it induces a continuous linear

injection �i⇤ : Lpk(E|Ni)! Lpk(Bi ⇥ Rm). Moreover by Proposition 4.2.1 we can use

the Euclidean metric and derivative on Bi. Similarly, ��1

i⇤ : Cj(Bi⇥Rm)! Cj(E|Ni)

is a continuous linear injection. Therefore, defining

Lpk(B ⇥ Rm) := {(�1

⇠1

, . . . ,�r⇠r) 2 �ri=1

Lpk(Bi ⇥ Rm) : (⇠1

, . . . ⇠r) 2 Lpk(E)}

Cj(B ⇥ Rm) := {(�1

⇠1

, . . . ,�r⇠r) 2 �ri=1

Cj(Bi ⇥ Rm) : (⇠1

, . . . ⇠r) 2 Cj(E)}

4.4. BANACH MANIFOLDS OF MAPS 40

the following composition is an imbedding

Lpk(E) Lpk(E) Lpk(B ⇥ Rm) Cj(B ⇥ Rm) Cj(E|Ni) Cj(E)F �⇤ �

�1⇤ F�1

where the central map is the Sobolev imbedding 4.1.1 applied to each Lpk(Bi,Rm). As

for compactness of the imbedding, if (⇠`) is a bounded sequence in Lpk(E) then each

�i(⇠`i) has a subsequence which converges in Cj(Bi ⇥ Rm). Taking the diagonal

subsequence over i and mapping through to Cj(E) gives a subsequence ⇠` which

converges in Cj(E).

Suppose now that W is a compact manifold with boundary, and assume W

is imbedded as a submanifold of a compact manifold N with the same dimension

(Taylor [78] p. 285 explains how this can always be arranged). Furthermore let

F ! W be a vector bundle which is equal to the restriction E|W of a vector bundle

E ! N . We define Lpk(F) := {⇠|W : ⇠ 2 Lpk(E)} and then Lpk(E) ,! Cj(E) compactly

implies Lpk(F) ,! Cj(F) compactly.

4.4 BANACH MANIFOLDS OF MAPS

This section will follow Elıasson [14], except that it is su�cient for our purposes to

assume that the manifolds N,M, E ,F are smooth (C1). Without this assumption

one needs to be very careful with losses in di↵erentiability in Lemma 4.4.2 and

Theorem 4.4.3 etc.

A section functor M which satisfies the following three conditions will be called

a manifold model.

(M1) For any E 2 V B(N) we have M(E) ,! C0(E).

(M2) For any E ,F 2 V B(N) there is a continuous linear imbedding M(L(E ,F)) ,!L(M(E),M(F)).

(M3) Let U ⇢ E be an open subset which projects onto N and f : U ! F a smooth

fibre map, then, setting M(U) := {⇠ 2M(E) : ⇠(N) ⇢ U}, for any ⇠ 2M(U)

we have f �⇠ 2M(F) and the induced mapM(f) : M(U)!M(F), ⇠ 7! f �⇠is continuous.

Note that since N is compact, C0(U) is open in C0(E) with the compact-open

topology, and then by (M1) M(U) is open in M(E).

Theorem 4.4.1. Lpk and Ck are manifold models.


Proof. For Lpk this is [15] Theorem 6.

With f as in (M3) define the fibre derivative Di2

f : U ! Li(E ,F) by (Di2

f)|Up

=

Di(f |Up

), where Up denotes the intersection of U with the fibre over p 2 N .

The following lemma will be used extensively, in particular to prove the di↵er-

entiability of charts, and is essentially a consequence of the fact that addition of

sections is carried out fibrewise.

Lemma 4.4.2. Let M be a manifold model and f : U ! F as in (M3), then

M(f) : M(U)!M(F) is smooth and

DiM(f) = M(Di2

f) (4.3)

Proof. M(Di2

f) is continuous by (iii) so it su�ces to show (4.3) holds for i = 1, i � 1

then follows inductively by replacing f by Di2

f and F by Li(E ,F). Now M(D2

f) :

M(U) ! M(L(E ,F)) but by property (ii) M(L(E ,F)) ⇢ L(M(E),M(F)). Let

⇠ 2M(U) and ⌫ 2M(E), then

(M(D2

f)(⇠)⌫)(p) = (D2

f(⇠)⌫)(p)

= ddtf(⇠(p) + t⌫(p))

��t=0

= ddtf(⇠ + t⌫)

��t=0

(p)

= ddtM(f)(⇠ + t⌫)

��t=0

(p)

= (DM(f)(⇠)⌫) (p)

Let M be a smooth manifold with a connection, and denote by C0(N,M) the

set of continuous maps N ! M . For any smooth map h : N ! M the pullback

bundle h⇤TM is in V B(N).

Theorem 4.4.3. Let N be a compact Riemannian manifold, M a smooth man-

ifold with a connection, and M a manifold model on V B(N). Then there is a

well defined subset M(N,M) of C0(N,M) such that M(N,M) can be given the

structure of a smooth Banach manifold modelled on the Banach spaces M(h⇤TM),

h 2 C1(N,M).

Proof. Since exp is a local di↵eomorphism (Section 2.7) there is a neighbourhood

D ⇢ TM of the zero section in TM such that (⌧, exp)|D !M ⇥M is a di↵eomor-

phism onto a neighbourhood of the diagonal. Writing h⇤ exp = exp �h⌧ : h⇤TM !TM ! M , it follows that there is a corresponding neighbourhood h⇤D of the zero

section in h⇤TM such that �h := h⇤(⌧, exp)|h⇤D ! N ⇥M is a di↵eomorphism

4.4. BANACH MANIFOLDS OF MAPS 42

onto an open neighbourhood of the graph of h. We define Uh to be the set of all

g 2 C(N,M) such that graph(g) ⇢ �h(h⇤D) and ��1

h � (Id, g) : N ! N ⇥M ! h⇤Dis in M(h⇤D). Then the map �h : Uh ! �h(Uh) ⇢ M(h⇤D) defined by �h(g) :=

��1

h � (Id, g) is injective with inverse

��1

h (⇠) = pr2

��h � ⇠ = h⇤ exp �⇠ : N !M.

We define

M(N,M) =[

h2C1(N,M)

Uh.

We will prove that the �h are charts. Suppose h, f 2 C1(N,M) are such that

Uh \Uf is non-empty. Then V := �h(h⇤D)\�f (f ⇤D) ⇢ N ⇥M) is non-empty and

��1

h (V ) ⇢ h⇤D is an open subset which projects onto N . Now �hf := ��1

f � �h :

��1

h (V ) ! f ⇤D is smooth and fibre preserving so M(�hf ) is smooth by Lemma

4.4.2. For any ⇠ 2 �h(Uh \ Uf ) we have

�f � ��1

h (⇠) = �f (h⇤ exp �⇠)

= ��1

f � (Id, h⇤ exp �⇠)

= ��1

f � (h⇤⌧, h⇤ exp) � ⇠

= ��1

f �h � ⇠ = M(�hf )(⇠).

Thus �f � ��1

h is smooth and {(Uh,�h) : h 2 C1(N,M)} is an atlas.

For each h 2 C1(N,M) the chart �h constructed above is called the naturalchart centred at h.

Theorem 4.4.4. The induced map ⌧� : M(N, TM) !M(N,M) can be given the

structure of a vector bundle which is naturally isomorphic to the tangent bundle

TM(N,M).


Lemma 4.4.5. LetM,M 0 be smooth manifolds and g : M !M 0 a smooth function.

Then g� : M(N,M)!M(N,M 0) defined by g � (x) := g � x is also smooth and we

have T (g�) = (Tg)�


Theorem 4.4.6. Let M be a manifold model and S a section functor on V B(N)

such that for any E ,F 2 V B(N) we have M(L(E ,F)) ,! L(S(E),S(F)). Then S


can be uniquely extended over the vector bundles h⇤TM for h 2M(N,M) and

S(M(N,M)⇤TM) :=[

h2M(N,M)

S(h⇤TM)

is a smooth vector bundle over M(N,M).


4.5 VECTOR BUNDLES OF SECTIONS

Theorem 4.5.1. Let N be a compact Riemannian manifold, ⇡ : E ! M a smooth

finite dimensional vector bundle with connections on E and M , and M a manifold

model on V B(N). Then ⇡� : M(N, E)!M(N,M), (⇡�)(↵) := ⇡ � ↵ can be given

the structure of a vector bundle.

Proof. The di↵erentiable structure onM(N, E) is that given by Theorem 4.4.3 using

the connection 3.10. It will be convenient to omit the subscript from kE for the

remainder of the proof. Let ↵ 2 C1(N, E) be such that ⇡ � ↵ = h, and denote

⇢ := (⇡⇤⌧M , ⇡⇤⇡) the projection for ⇡⇤TM �E ⇡⇤E . The following diagram will be

helpful, noting that h⇤TM �N h⇤E = ↵⇤⇡⇤(TM � E),

h⇤(TM � E) ⇡⇤TM �E ⇡⇤E

↵⇤TE TE E ⇥ E

N E

↵⇢

eFk↵

↵⌧E

↵⇤⌧E

k

F

⌧Epr1

↵

(4.4)

Let �↵ be the natural chart for M(N, E) centred at ↵, i.e. �↵ : U↵ ⇢ M(N, E) !M(↵⇤DE) where DE ⇢ TE is a neighbourhood of the zero section such that

F := (⌧E , expE) : DE ! E ⇥ E

is a di↵eomorphism. Let VE ⇢ DE be such that k(VE) = ⇡⇤VM � ⇡⇤E with VM

as in Theorem 3.4.1, and then restrict �↵ to the set U 0↵ := ��1

↵ (M(↵⇤VE)). Then

(cf. (4.4)) we have a di↵eomorphism k↵ : ↵⇤VE ! h⇤VM � h⇤E which induces a

di↵eomorphism k↵� : M(↵⇤VE)!M(h⇤VM)⇥M(h⇤E) with (k↵�)�1 = k�1

↵ �. Thenthe charts defined by ✓↵ := k↵ � �↵ are locally trivial. To show that they give a

local trivialisation for ⇡� : M(N, E) ! M(N,M) we first need to check that the

4.5. VECTOR BUNDLES OF SECTIONS 44

following diagram commutes

M(h⇤VM)⇥M(h⇤E) U 0↵

M(h⇤VM) Uh

✓�1↵

pr !⇡

��1h

(4.5)

where �h is the natural chart for M(N,M) centred at h. We know from the proof

of Theorem 4.4.3 that ��1

↵ = expE �↵⌧E� and ��1

h = expM �h⌧M

�, and therefore

⇡ � ✓�1

↵ (⇠, ⌘) = ⇡ � expE �↵⌧E � k�1

↵ (⇠, ⌘)

= ⇡ � expE �↵⌧E � ↵�1

⌧E(k�1 � ↵⇢)(⇠, ⌘)

= ⇡ � expE �k�1 � ↵⇢ � (⇠, ⌘)

We have previously noted that the induced connection on TE is such that if ! is a

geodesic on E then ⇡ � ! is a geodesic on M . The initial values are (⇡ � !)(0) =

⇡(!(0)), (⇡ � !)0(0) = T⇡(!(0)), therefore ⇡ � expE = expM �T⇡ and

⇡ � ✓�1

↵ (⇠, ⌘) = expM �T⇡ � k�1 � ↵⇢ � (⇠, ⌘) (4.6)

Now since, by definition, k = (⇡�1

⌧M

T⇡, ⇡�1

⇡ KE), we have ⇡⌧M

� pr1

�k = T⇡ and

therefore T⇡ � k�1 = ⇡⌧M

� pr1

. Substituting into (4.6), recalling ⇢ = (⇡⇤⌧M , ⇡⇤⇡),

we have

⇡ � ✓�1

↵ (⇠, ⌘) = expM �⇡⌧M

� pr1

�↵⇢ � (⇠, ⌘) = expM �⇡⌧M

� ↵⇡⇤⌧M

� pr1

�(⇠, ⌘)

= expM �h⌧M

� ⇠ = ��1

h (⇠)

which shows that (4.5) commutes. It follows that (⇡�)�1(Uh) = U 0↵ = U 0

� for any

liftings ↵, � of h. In particular, we can take the zero lift 0h and then

{U 00

h

, h 2 C1(N,M)}

covers M(N, E). To show that the ✓0

h

, U 00

h

give a trivialising covering we need

to check that overlapping trivialisations give linear isomorphisms between fibres.

Suppose the natural charts centred at g, h 2 C1(N,M) overlap, i.e. Uh \Uf is non

empty. Fix x 2 Uh \ Uf with �h(x) = ⇠, and define ✓hf,⇠ : M(h⇤E) ! M(f ⇤E) by✓hf,⇠(⌘) := ✓

0

f

�✓�1

0

h

(⇠, ⌘). We will show ✓hf,⇠ is an isomorphism. For any ⌘ 2M(h⇤E)we have

✓hf,⇠(⌘) = ✓0

f

� ✓�1

0

h

(⇠, ⌘) = k0

f

� �0

f

� ��1

0

h

� k�1

0

h

� (⇠, ⌘)


and from the proof of Theorem 4.4.3 we know �0

f

� ��1

0

h

= F�1

0

f

� F0

h

so

✓hf,⇠(⌘) = k0

f

� F�1

0

f

� F0

h

� k�1

0

h

� (⇠, ⌘) = eF�1

0

f

� eF0

h

� (⇠, ⌘) (4.7)

It follows from Corollary 3.4.2 that eF�1

0

f

� eF0

h

� (⇠, ·) is an isomorphism. Moreover,

if we define eF0

h

0

f

: Vhf ⇢ h⇤VM ! L(h⇤E , f ⇤E) by eF0

h

0

f

(⇠)⌘ = pr2

eF�1

0

f

� eF0

h

(⇠, ⌘)

then eF0

h

0

f

is C1 and therefore

eF0

h

0

f

� : M(Vhf )!M(L(h⇤E , f ⇤E)) ⇢ L(M(h⇤E),M(f ⇤E))

is also C1 by Lemma 4.4.2. Now from (4.7) we have that ✓hf : �h(Uh \ Uf ) !L(M(h⇤E ,M(f ⇤E)) satisfies ✓hf (⇠)⌘ = eF

0

h

0

f

� (⇠)⌘. Thus ✓hf is also C1 and

M(N, E) is a vector bundle.

Suppose now that S is a section functor which satisfies

M(L(E ,F) ,! L(S(E),S(F))

for any E ,F in V B(N). Let x 2 M(N,M) and define S(x⇤E) as the set of all

⌘ 2 C0(N, E) such that ⇡ � ⌘ = x and there is a natural chart (Uh,�h) with eF�1

0

h

�(Id, ⌘) 2 {�h(x)}⇥ S(h⇤E).

Proposition 4.5.2. S(M(N,M)⇤E) :=S

x2M(N,M)

S(x⇤E) can be given the struc-

ture of a vector bundle over M(N,M).

Proof. Local trivialisations over the natural charts on M(N,M) are given by

(⇡�)�1(Uh)!M(h⇤VM)⇥ S(h⇤E)

(x, ⌘) 7! eF�1

0

h

� (Id, ⌘) (4.8)

and then the transition functions are again given by composition with eF0

h

0

f

. We saw

in the proof of the previous theorem that eF0

h

0

f

� : M(Vhf ) ! M(L(h⇤E , f ⇤E)) is

C1, and we have assumed that there is a continuous imbedding M(L(h⇤E , f ⇤E)) ,!L(S(h⇤E),S(f ⇤E)), so the transition functions for S(M(N,M)⇤E) are also smooth.

Corollary 4.5.3. M(N, E) and M(M(N,M)⇤E) are isomorphic vector bundles.

4.6 LOCAL FORMULAS

We mentioned in Section 3.2 that Elıasson has constructed a connector for TM

whose geodesics are Jacobi lifts of geodesics in M . This connection also has the

4.6. LOCAL FORMULAS 46

property that the corresponding exponential map expT : T 2M ! TM satisfies

expT = T expM �TM (see Cor. 3.1 in [14]). It then follows that

gexpT = T expM �k�1

TM = T expM �kTM � TM � k�1

TM

Using the notation from Section 3.1 we therefore have

gexpT (x, 0; v, w) = T expM(x, v; 0, w)

= r exp(x, v; 0) +r2

exp(x, v;w)

= r2

exp(x, v;w)

So in this case, for the zero lift 0h : I ! TM , the induced map eF0

h

: h⇤VM�h⇤TM !N ⇥ TM has the formula

eF0

h

(p; v, w) = (p,r2

exp(h⌧ (p, v;w))) (4.9)

Now for simplicity we only consider the tangent lift of a curve, i.e. assume N is

either the unit circle S1 or the unit interval I, and define @(x) := x

Hk(N,M) Hk�1(Hk(N,M)⇤TM)

Hk(h⇤VM) Hk(h⇤VM)⇥Hk�1(h⇤TM)

@

�h

�0h

@h

From (4.8) and (4.9) we have, cf. proof of Theorem 4.4.3, that the natural chart

centred at 0h is ��1

0

h

= r2

exp � and therefore

��1

0

h

@h(⇠) = r2

exp �@h(⇠) = T (exp �⇠)( ddt)

= r1

exp �(⇠, h) +r2

exp(⇠,rt⇠)

= r2

exp �(⇠, ✓(⇠)h) +r2

exp �(⇠,rt⇠)

where we have also used (3.7) and (3.12). Thus

@h(⇠) = (⇠,rt⇠ + ✓(⇠)h) (4.10)

and @ is a smooth section because rt is linear on Hk(h⇤TM) and ✓� is smooth by

Lemma 4.4.2. There is also a VB morphism

rt : Hk(Hk(I,M)

⇤TM)! Hk�1(Hk(I,M)

⇤TM)


defined, of course, by rtv = K � Tv( ddt). To see that it is smooth we calculate the

local expression:

r2

exp �(rt)h(⇠, ⌘) = K � T (r2

exp(⇠, ⌘))( ddt)

= r1

r2

(⇠, ⌘, h) +r2

r2

(⇠, ⌘,rt⇠) +r3

r2

exp(⇠, ⌘,rt⌘)

= r2

exp(D2

✓(⇠)(⌘, h) + ⇤(⇠)(✓(⇠)h, ⌘)

+r2

exp(⇠,⇤(⇠)(⌘,rt⇠) +rt⌘)

= r2

exp(⇠,rt⌘ +D2

✓(⇠)(⌘, h) + ⇤(⇠)(@h⇠, ⌘))

where the penultimate step uses Lemma 3.3.2. So

(rt)h(⇠, ⌘) = rt⌘ +D2

✓(⇠)(⌘, h) + ⇤(⇠)(@h⇠, ⌘) (4.11)

which is smooth for the same reasons as @h.

4.7 METRICS

For the sake of clarity in this section we will write g(x)( , ) for the Riemannian

metric on TxM . Define G : VM ! L(TM, TM) implicitly by

g(x)(G(u)v, w) = g(exp(u))(r2

exp(u, v),r2

exp(u, w)). (4.12)

Then G is a smooth fibre preserving map and G(u) is self adjoint and positive

definite. As usual we will denote the pullback by Gh : h⇤TM ! L(h⇤TM, h⇤TM).

Theorem 4.7.1. The bundle H0(H1(N,M)⇤TM) can be given a Riemannian metric

G which is characterised by the property that for h 2 C1(N,M), G(h) coincides withh , i

0

on H0(h⇤TM).

Proof. Suppose V,W are in the fibre over x 2 H1(N,M). Then by the definition

of the fibre there exists a natural trivialisation centred at h and ⌘, ⇣ 2 H0(h⇤TM)

such that x = exp �⇠, V = r2

exp �(⇠, ⌘) and W = r2

exp �(⇠, ⇣). So we define

G(x)(V,W ) :=

Z

N

g(x)(r2

exp(⇠, ⌘),r2

exp(⇠, ⇣))

=

Z

N

g(h)(Gh(⇠)⌘, ⇠) = hGh(⇠)⌘, ⇠i0

Then since Gh� is smooth by Lemma 4.4.2 it follows that G is smooth. Also, if x

is smooth then we can use the natural chart centred at x and then since r2

exp(0, ·)and Gx(0) are both identity maps, G(x)(V,W ) = hV,W i

0

.

4.7. METRICS 48

Remark. From now on we will write h , i0

instead of G.

Corollary 4.7.2. The tangent bundle TH1(N,M) = H1(H1(N,M)⇤TM) has a

Riemannian metric defined by hV,W i1

:= hV,W i0

+ hrtV,rtW i0

Proof. Smoothness follows from Theorem 4.7.1 and the smoothness of rt.

CHAPTER 5

Conditional extremals

The contents of this chapter have been published as [73]. Some minor changes have

been made to make the notation consistent with previous chapters and to make

references internal to the thesis where possible, and a preparatory section has been

removed to avoid duplication of Chapter 4. The exposition which follows is otherwise

unchanged from the published version.

Abstract

Conditional extremal curves in a complete Riemannian manifold M are

defined as the critical points of the squared L2 distance between the

tangent vector field of a curve and a so-called prior vector field. We

prove that this L2 distance satisfies the Palais-Smale condition on the

space of absolutely continuous curves joining two submanifolds of M ,

and thus establish the existence of critical points. We also prove a Morse

index theorem in the case where the two submanifolds are single points,

and use the Morse inequalities to place lower bounds on the number of

critical points of each index.

5.1 INTRODUCTION

We consider the problem of interpolating Riemannian manifold data obtained from

an integral curve of an unknown vector field which is assumed to be in some collection

A of vector fields. Suppose x : I ! M where I is the unit interval and M a

Riemannian manifold. The pair (x,A) is considered optimal if x interpolates the

data, A 2 A, and the squared L2 distance between the tangent vector field x and A

is minimised, as in [59] . We are therefore interested in finding critical points of

S(x) :=1

2

Z

I

hx� A, x� Ai dt, (5.1)

49

5.2. GEOMETRY OF THE PATH SPACE (AFTER ELIASSON) 50

which we refer to as conditional extremals1.

Necessary conditions for extremality of curves joining any two given points on

M are derived in [59] using standard variational methods. Various properties of

solutions are examined in [59], in particular when A is a potential field or a left

invariant field on a semisimple Lie group.

Here we restrict attention to the case where A contains a single vector field and

find su�cient conditions for the existence of critical points of S. As usual the key

result is the Palais-Smale condition; existence of critical points and a lower bound

for the total number of critical points are consequences of this condition [67]. More

detailed multiplicity results are obtained using Morse theory.

5.2 GEOMETRY OF THE PATH SPACE (AFTER ELIASSON)

The natural domain for S is the Hilbert manifoldH1(I,M) of continuous curves with

square integrable tangent vector fields. Thus we require the critical point theory of

functions defined on Hilbert manifolds (see eg. [67, 52]). For instance, this theory

is used to prove existence of Riemannian cubics [27] and elastic curves [42]. Our

particular approach is closely aligned with that used by Elıasson to show existence

of closed geodesics in [17].

With ⇤ : D ! L2(TM, TM) := L(TM�TM, TM) as defined by equation (3.13)

we have an induced map

⇤ : Hk(h⇤D)! Hk(L2(h⇤TM, h⇤TM)) ⇢ L2(Hk(h⇤TM), Hk(h⇤TM))

where the inclusion is the second manifold model property (see Section 4.4). In

particular this gives a smooth map

⇤ : H1(h⇤D)! L(H1(h⇤TM), H0(h⇤TM);H0(h⇤TM))

which is the local connector of a symmetric connector for H0(H1(I,M)⇤TM). As

described in [17], by di↵erentiating each side of (4.12) it can be shown that this is a

metric connector, i.e. it is compatible with the metric. Furthermore, the connector

constructed in the same way for the tangent bundle H1(H1(I,M)⇤TM) corresponds

to the induced map H1(K) : A ! K � A (Theorem 5.4 in [14]), where K is the

connector for TM . We denote the corresponding covariant derivative in both cases

by r.

1this is actually a slight misnomer, since they may be saddle points.

51 CHAPTER 5. CONDITIONAL EXTREMALS

5.3 EXISTENCE OF CRITICAL POINTS

Let A be a smooth vector field on a complete Riemannian manifold M . Note that

A : M ! TM induces a smooth vector field A� : H1(I,M) ! H1(H1(I,M)⇤TM)

by composition. Similarly, a smooth section Y of L(TM, TM) induces a section

Y � : H1(I,M)! H1(L(H1(I,M)⇤TM,H1(I,M)⇤TM))

⇢ L(H1(H1(I,M)⇤TM), H1(H1(I,M)⇤TM)),

where once again the inclusion is a consequence of the second manifold model prop-

erty proved for Hk in [15]. If V is a vector field on H1(I,M) and x, y 2 H1(I,M) are

such that x(t1

) = y(t2

) = p 2 M then it is not necessary that V (x(t1

)) = V (y(t2

)).

Thus it only makes sense to compare r and r for vector fields of the form A�, sincein this case A � (x(t

1

)) = A � (y(t2

)) = A(p).

Lemma 5.3.1. (i) If A is a smooth vector field on M then r(A�) = (rA)�.

(ii) The same assertion applies to sections of L(TM, TM) with the connection

induced from that on TM : suppose Y is a section of L(TM, TM), then

r(Y �) = (rY )�.

(iii) If A is a vector field on M then r2(A�) = (r2A)�

(iv) Let R denote the curvature tensor of r, then

R(⇠, ⌘, ⌫)(x) = R(⇠(x), ⌘(x), ⌫(x))

for any vector fields ⇠, ⌘, ⌫ on H1(I,M).

(v) Let ⇠ be a smooth vector field on H1(I,M) and suppose x 2 H1(I,M) satis-

fies @(x) 2 H1(H1(I,M)⇤TM) so that r@(x)⇠ is well defined, then r@(x)⇠ =

r⇠@(x).

Proof. See A.1.

Remark. We appear to be saying in (v) that [⇠, @] = 0. However it is not clear how

[⇠, @] should be defined because @ is not a vector field on H1(I,M) (it is a section

of H0(H1(I,M)⇤TM) not H1(H1(I,M)⇤TM)).

We can now rewrite (5.1) as

S = 1

2

k@ � A � k20

(5.2)

5.3. EXISTENCE OF CRITICAL POINTS 52

Since @, A� and the metric are smooth, S is also smooth. We can calculate the

derivative using the metric compatibility of r: dS(⇠) = hr⇠@ � r⇠A�, @ � A�i0

where ⇠ is a vector field on H1(I,M). As in the proof of Lemma 5.3.1(v) we have

r⇠@(x) = rt⇠(x) and by Lemma 5.3.1(i) r⇠(A�)(x) = r⇠(x)A(x). Therefore for

each x 2 H1(I,M)

dS(⇠(x)) = hrt⇠ �r⇠(x)A(x), x� A(x)i0

(5.3)

We view rA(x) as a smooth section of L(x⇤TM, x⇤TM) and taking the adjoint

pointwise with respect to the Riemannian metric for M gives another section which

we denote by rA⇤(x).

Proposition 5.3.2. Let X be s submanifold of H1(I,M) such that for any x 2 X,

TxX contains all ⇠ 2 TxH1(I,M) which satisfy ⇠(0) = 0 and ⇠(1) = 0. If x is a

critical point of S|X then x is smooth and satisfies

(rt +rA⇤(x))(x� A(x)) = 0 (5.4)

Proof. From (5.3) x is a critical point of S|X i↵

dS(⇠(x)) = hrt⇠ �r⇠(x)A, x� A(x)i0

= 0

for all ⇠ 2 TxX. In particular this must hold for all ⇠ 2 C10

(x⇤TM) and using

integration by parts x must be a weak solution of (5.4). But the weak solutions are

in fact smooth (see Appendix A.4).

We are interested in critical points of the restriction of S to the space of curves

whose endpoints are contained in closed submanifolds N0

, N1

⇢ M . The map

P : H1(I,M) ! M ⇥ M , x 7! (x(0), x(1)) is a submersion. So the pre-image

of N := N0

⇥ N1

, which we will denote by ⌦N(M), is a closed and therefore

complete submanifold of H1(I,M). The tangent space Tx⌦N(M) consists of all

⇠ 2 H1(x⇤TM) such that ⇠(0) 2 Tx(0)N0

, and ⇠(1) 2 Tx(1)N1

.

Proposition 5.3.3. x 2 ⌦N(M) is a critical point of S|⌦

N

(M)

if and only if x is a

smooth solution of (5.4) and x(0) � A(x(0)) and x(1) � A(x(1)) are orthogonal to

Tx(0)N0

and Tx(1)N1

respectively.

Proof. Note that ⌦N(M) satisfies the conditions of Proposition 5.3.2 so the critical

points are smooth and satisfy (5.4). We may therefore integrate by parts in (5.3) to

obtain h⇠, x� A(x)i|10

= 0, which must hold for any ⇠ 2 Tx⌦N(M).


Lemma 5.3.4. Suppose S and also kA � k0

are bounded on some subset U ⇢H1(I,M), then U is an equicontinuous family of curves with uniformly bounded

length.

Proof. If S(x) = 1

2

kx� A(x)k20

and kA(x)k0

are bounded for x 2 U then kxk0

kx� A(x)k

0

+ kA(x)k0

k for some constant k. Furthermore, by the Cauchy-

Schwarz inequality

d(x(t1

), x(t2

)) Z t2

t1

kxk dt |t1

� t2

|1

2k

Hence U is equicontinuous and any x 2 U has length less than or equal to k.

Assumption We shall henceforth assume that kA � k0

is bounded on ⌦N(M). For

example, this will be satisfied when kAk is bounded on M (as assumed in [59] to

prove extendibility of the solutions of (5.4)).

Corollary 5.3.5. Suppose at least one of N0

or N1

is compact. Then any sequence

(xi) in ⌦N(M) on which S is bounded has a uniformly convergent subsequence.

Proof. From Lemma 5.3.4, (xi) is an equicontinuous family of curves with uniformly

bounded length. Then since at least one of N0

, N1

is compact there exists a closed

and bounded K ⇢M such that xi(I) ⇢ K for all i. By the Hopf-Rinow theorem K

is compact and then by the Arzela-Ascoli theorem (xi) is compact in C(I,M) and

has a uniformly convergent subsequence.

Definition. (cf. [16]) A C1 function f : H1(I,M) ! R is called locally coercive if

there exist constants � > 0 and C such that in a natural chart centred at h we have

(Dfh(⇠)�Dfh(⌘))(⇠ � ⌘) � �k⇠ � ⌘k21

� C|⇠ � ⌘|20

(5.5)

whenever ⇠, ⌘ 2 H1(h⇤D) with k⇠k1

, k⌘k1

bounded. Here we have used the abbrevi-

ation fh for f � exp. When f is of class C2 the following condition is equivalent to

(5.5)

D2fh(⇠)(⌘, ⌘) � �k⌘k21

� C|⌘|20

(5.6)

for ⇠ 2 H1(h⇤D) such that k⇠k1

is bounded and all ⌘ 2 H1(h⇤TM).

Proposition 5.3.6. The function S : H1(I,M)! R is locally coercive.

Proof. We will show that (5.6) holds. Using the local expression for the metric from

Theorem 4.7.1, the local expression for S(x) is

Sh(⇠) =1

2

hG(⇠)(@h(⇠)� Ah(⇠)), @h(⇠)� Ah(⇠)i0

5.3. EXISTENCE OF CRITICAL POINTS 54

where x = exp �⇠ and Ah is defined by A(x) = D2

exp(⇠, Ah(⇠)). Note that the inner

product in the above expression is defined in terms of the metric along h so it is

independent of ⇠. Hence from (4.10)

DSh(⇠)⌘ = hG(⇠)(rt⌘ +D2

✓(⇠, ⌘)�D2

Ah(⇠, ⌘)), @h(⇠)� Ah(⇠)i0

+ 1

2

hD2

G(⇠)(@h(⇠)� Ah(⇠), ⌘), @h(⇠)� Ah(⇠)i0

(5.7)

and furthermore

D2Sh(⇠)(⌘, ⌘)

= 2hD2

G(⇠)(@h(⇠)� Ah(⇠), ⌘),rt⌘ +D2

✓(⇠, ⌘)�D2

Ah(⇠, ⌘)i0

+ hG(⇠)(D2

2

✓(⇠, ⌘, ⌘)�D2

2

Ah(⇠, ⌘, ⌘)), @h(⇠)� Ah(⇠)i0

+ hG(⇠)(rt⌘ +D2

✓(⇠, ⌘)�D2

Ah(⇠, ⌘)),rt⌘ +D2

✓(⇠, ⌘)�D2

Ah(⇠, ⌘)i0

+ 1

2

hD2

2

G(⇠)(@h(⇠)� Ah(⇠), ⌘, ⌘), @h(⇠)� Ah(⇠)i0

Recall that the embedding H1(h⇤TM) ,! C0(h⇤TM) is compact, i.e. a bounded

subset U ofH1(h⇤TM) is relatively compact in C0(h⇤TM). It follows that a function

defined on U which is continuous with respect to the C0(h⇤TM) topology is bounded.

The functions G, ✓, Ah and their derivatives are all continuous on C0(h⇤TM), so

using the assumption that k⇠k1

(and therefore also k@h(⇠)k0

) is bounded we have

D2Sh(⇠)(⌘, ⌘) � �krt⌘ +D2

✓(⇠, ⌘)�D2

Ah(⇠, ⌘)k20

� const k⌘k0

k@h(⇠)� Ah(⇠)k0

krt⌘ +D2

✓(⇠, ⌘)�D2

Ah(⇠, ⌘)k0

� const k@h(⇠)� Ah(⇠)k20

k⌘k20

� �krt⌘k20

� const k⌘k0

krt⌘k0

� const k⌘k20

with � > 0. It follows from k⌘k0

|⌘|0

that krt⌘k20

� k⌘k21

� |⌘|20

. Therefore using

2ab "a2 + 1

"b2 and choosing " su�ciently small

D2Sh(⇠)(⌘, ⌘) � �k⌘k21

� �|⌘|20

� const("krt⌘k20

+ 1

"k⌘k2

0

)� const k⌘k20

� �k⌘k21

� C|⌘|20

with � > 0.

As in [32] it will be useful to define the orthogonal decomposition H1(h⇤TM) =

H1(h⇤TM)0

+ V, where H1(h⇤TM)0

is the space of sections which are zero at the

endpoints and

V := {⇠ 2 H1(h⇤TM) : h⇠, ⌘i1

= 0, 8⌘ 2 H1(h⇤TM)0

}


Now since dP (h) : H1(h⇤TM) ! Th(0)M � Th(1)M is surjective and H1(h⇤TM)0

=

ker dP (h) we have V = Th(0)M ⇥ Th(1)M which has dimension 2m. If we define

another subspace

V 0 := {⇠ 2 H1(h⇤TM) : r2

t ⇠ = ⇠}

then for any ⇠ 2 V 0 and ⌘ 2 H1(h⇤TM)0

, using integration by parts,

h⇠, ⌘i1

= h⇠, ⌘i0

+ hrt⇠, ⌘i|10

� hr2

t ⇠, ⌘i0

= 0

which shows that V 0 ✓ V . Each solution of r2

t ⇠ = ⇠ must be smooth because r2

t ⇠

has the same order of di↵erentiability as ⇠. Furthermore there are 2m linearly

independent solutions of this equation so V 0 = V because they have the same

dimension.

Lemma 5.3.7. If ⇠ 2 V then k⇠k1

k|⇠|0

Proof. Since r2

t ⇠ = ⇠ we have k⇠k22

= k⇠k21

+ k⇠k20

2k⇠k21

, and then the continuous

linear inclusion H2(h⇤TM) ,! C1(h⇤TM) gives |rt⇠|0

|⇠|1

k4

k⇠k2

k2

k⇠k1

.

Now using integration by parts and the Cauchy-Schwarz inequality

k⇠k21

= h⇠, ⇠i0

+ hrt⇠, ⇠i|10

� hr2

t ⇠, ⇠i0

= hrt⇠, ⇠i|10

2|rt⇠|0

|⇠|0

kk⇠k1

|⇠|0

and the result follows.

Theorem 5.3.8. If at least one of the closed submanifolds N0

, N1

⇢M is compact

and kAk0

is bounded then S satisfies the Palais-Smale condition on ⌦N(M).

Proof. Let (xi) be a sequence in ⌦N(M) such that |dS(xi)| ! 0 and S(xi) is

bounded. By Corollary 5.3.5 we can assume (xi) is uniformly convergent and choose

h 2 C1N (I,M) uniformly close to the limit curve x so that, for su�ciently large i,

xi is contained in the natural chart centred at h, i.e. xi 2 exp(h⇤D).

We define a submanifold ⌦N(h⇤D) ⇢ H1(h⇤D) by ⌦N(h⇤D) := (P � exp)�1(N),

where as before P : H1(I,M)!M⇥M is defined by P (x) := (x(0), x(1)). From the

orthogonal decomposition H1(h⇤TM) = H1(h⇤TM)0

+V and P � exp(H1(h⇤D)0

) ⇢N we have ⌦N(h⇤D) = H1(h⇤D)

0

+VN where H1(h⇤D)0

is used to denote H1(h⇤D)\H1(h⇤TM)

0

and VN := {⇠ 2 V \H1(h⇤D) : P � exp ⇠ 2 N} is a submanifold of V .

Let (⇠i) be the sequence corresponding to (xi) by xi = exp ⇠i. Of course (⇠i) must

be uniformly convergent; we show that it converges in H1. From (4.10) we obtain

krt⇠ik0

k@h⇠ik0

+ k✓(⇠i)k0

but kxik0

is bounded (by the proof of Lemma 5.3.4) and ⇠i is uniformly convergent

so k@h⇠ik0

and k⇠ik0

are bounded, and the above inequality shows that k⇠ik1

is

5.4. MULTIPLICITY OF CRITICAL POINTS 56

bounded. We write ⇠i = ⇠0i + ⇠vi where ⇠0i 2 H1(h⇤D)0

and ⇠vi 2 VN , and abbreviate

Sh|⌦N

(h⇤D)

to S⌦

h . Using the local coercivity of S on H1(I,M) (Proposition 5.3.6),

we have constants � > 0, C such that

�k⇠i � ⇠jk21

(DSh(⇠i)�DSh(⇠j))(⇠i � ⇠j) + C|⇠i � ⇠j|20

= (DS⌦

h (⇠i)�DS⌦

h (⇠j))(⇠0

i � ⇠0j )

+ (DSh(⇠i)�DSh(⇠j))(⇠vi � ⇠vj ) + C|⇠i � ⇠j|2

0

here we have used the fact that ⇠0i � ⇠0j 2 H1(h⇤TM)0

⇢ Th⌦N(M) (it is not

necessarily true that ⇠vi 2 Th⌦N(M)). Thus

�k⇠i � ⇠jk21

C|⇠i � ⇠j|20

+ (|DS⌦

h (⇠i)|+ |DS⌦

h (⇠j)|)k⇠0i � ⇠0j k1

+ (|DSh(⇠i)|+ |DSh(⇠j)|)k⇠vi � ⇠vj k1

We know |⇠i � ⇠j|0

! 0, k⇠ik1

is bounded and |DS⌦

h (⇠i)| ! 0 by assumption. Fur-

thermore ⇠vi � ⇠vj 2 V so k⇠vi � ⇠vj k1

k|⇠vi � ⇠vj |0

! 0 and from the local expression

(5.7) |DSh(⇠j)| is bounded so the above inequality shows that (⇠i) is Cauchy. By

the completeness of H1(h⇤TM) and the fact that ⌦N(M) is closed in H1(I,M), (xi)

converges in ⌦N(M).

Corollary 5.3.9. In any homotopy class of curves joining N0

and N1

there exists a

critical point of S which minimises S with respect to the given class. Furthermore,

S attains its infimum on ⌦N(M).

Proof. Both statements follow from the Palais-Smale condition [67].

5.4 MULTIPLICITY OF CRITICAL POINTS

The category catX(Q) of a subset Q of a topological space X is defined as the

minimal number of closed contractible subsets of X which cover Q. We abbreviate

the homotopy-type invariant catX(X) to cat(X). By the Lusternik-Schnirelman

multiplicity theorem (eg. Theorem 7.2 in [64]), since the restriction of S to ⌦N(M)

satisfies the Palais-Smale condition it has at least cat(⌦N(M)) critical points. We

denote by CN(M) the space of continuous curves joining N0

and N1

. The inclusion

⌦N(M) ! CN(M) is a homotopy equivalence (Theorem 1.3 in [32]) and therefore

we have the following corollary.

Corollary 5.4.1. The restriction of S to ⌦N(M) has at least cat(CN(M)) critical

points.

In order to find conditions under which the category of CN(M) is infinite we will

use the following theorem:


Theorem 5.4.2. (Fadell and Husseini [19]) Let F ,! E ! B be a fibration with

F,E,B path connected which admits a section. If Q ⇢ F then catF Q catE Q.

Proposition 5.4.3. Suppose M is 1-connected and not contractible, N0

, N1

are

path connected, and the fibration P : CN(M) ! N has a section (see examples

below). Then CN(M) has infinite category.

Proof. The fibre Cp,q(M) has the same homotopy type as the based loop space

⌦(M) and since M is 1-connected ⌦(M) is path connected. A fibration with path

connected fibre and base space must have path connected total space. Therefore

by Theorem 5.4.2 cat(Cp,q(M)) cat(CN(M)). But by Corollary 3.1 in [20] ⌦(M),

and therefore Cp,q(M), has infinite category.

Proposition 5.4.4. Let M be path connected, finite dimensional and not con-

tractible. Suppose N0

, N1

are path connected and that the fibration P : CN(M)!N has a section. If the fundamental group ⇡

1

(M) is not an infinite group with

finitely many conjugacy classes then cat(CN(M)) is infinite.

Proof. Fix (a, b) 2 N and a section s : N ! CN(M). Let C 0N(M) denote the path

component of s(a, b), we then have a sub-fibration C 0p,q(M) ,! C 0

N(M) ! N with

path connected base and total space. To see that the fibre is also path connected let

x 2 C 0a,b(M) and suppose H : I ! C 0

NM is a path from x to s(a, b). Let P0

denote

the composition of P with projection onto N0

, and similarly P1

the composition of

P with projection onto N1

. Define ↵it : I ! Ni by ↵i

t(u) := Pi �H(ut), i = 0, 1 and

similarly �it(u) := Pi �H(1� ut). Then F : I ! C 0

a,b(M)

F (t) :=

(↵0

2t ⇤H(2t) ⇤ ↵1

2t 0 t 1

2

�0

2t�1

⇤ s � P �H(2t� 1) ⇤ �1

2t�1

1

2

t 1

where ↵1

t (u) := ↵1

t (1 � u) is path reversal and ⇤ denotes concatenation of paths.

Then F (0) is homotopic to x relative to the endpoints, and similarly F (1) to s(a, b).

Therefore there exists a path in C 0a,b(M) joining x and s(a, b). Now by Theorem

5.4.2 cat(C 0p,q(M)) cat(C 0

N(M)), and of course cat(C 0N(M)) cat(CN(M)). But

C 0p,q(M) has the same homotopy type as the space ⌦

0

(M) of loops which represent

the identity element of ⇡1

(M): a homotopy equivalence is given by the map which

takes y 2 C 0p,q(M) to y⇤s(p, q). Furthermore ⌦

0

(M) can be identified with the based

loop space ⌦(M) where M is the universal covering of M , and with the assumption

on ⇡1

(M), cat(⌦(M)) is infinite (as in the proof of Corollary 3.4 in [20]).

We now give one example where the fibration CN(M)! N does have a section,

and one where it does not.


Example 5.1. IfN0

, N1

are contractible then we can construct a section of CN(M)!N as follows. Suppose H : N

0

⇥ I ! N0

and G : N1

⇥ I ! N1

are deformation

retractions to the points p 2 N0

, q 2 N1

respectively, and let ↵ : I ! M be con-

tinuous with ↵(0) = p,↵(1) = q. Then we define a section s : N ! CN(M) as the

concatenation: s(a, b) := G(b, 1� t) ⇤ ↵ ⇤H(a, t).

Example 5.2. Suppose M is not contractible, N0

is a single point p 2M and N1

=

M . Then CN(M) is contractible. If there exists a section s : N = {p}⇥M ! CN(M)

of P : CN(M) ! N then in homotopy groups we have induced homomorphisms

s#

: ⇡q(M) ! ⇡q(CN(M)) = 0, P#

: ⇡q(CN(M)) ! ⇡q(M) with P#

� s#

= 1. But

then s is a weak homotopy equivalence from M to a contractible space. Since M

has the homotopy type of a CW-complex, s is a homotopy equivalence [77, p. 405].

The remainder of this section will be devoted to applying the Morse inequalities

to give lower bounds for the number of critical points of each Morse index. For

simplicity we restrict attention to the case where N0

, N1

are single points. Note

that the Morse inequalities do not render the category lower bound redundant,

since they only apply when S is a Morse function, i.e. when the Hessian Hess(S) is

represented by an invertible self-adjoint operator at all critical points of S.

Let ⇠, ⌘ be smooth vector fields on H1(I,M), then using (2.8)

Hess(S)(⇠, ⌘) : = ⇠(⌘S)� (r⇠⌘)S

= hr2(@ � A)(⇠, ⌘), @ � Ai0

+ hr⇠(@ � A), r⌘(@ � A)i0

If @x 2 H1(x⇤TM), in particular if x is a critical point of S (Proposition 5.3.2),

we have from (2.8) and Lemma 5.3.1(v) r2@(⇠, ⌘) = r⇠r@⌘ � r@r⇠⌘ = R(⇠, @)⌘.

Combining this with the other parts of Lemma 5.3.1 we obtain

Hessx S(⇠, ⌘) = hR(⇠, x)⌘ �r2A(⇠, ⌘), x� Ai0

+ hrt⇠ �r⇠A,rt⌘ �r⌘Ai0

(5.8)

Proposition 5.4.5. If x is a critical point of S then Hessx S is represented by a self-

adjoint operator hessx S : H1(x⇤TM)! H1(x⇤TM) with finite dimensional kernel.

In particular it follows that hessx S is Fredholm with index zero.

Proof. Let x be a critical point of S. Hessx S is continuous and symmetric and

therefore represented by a bounded self-adjoint operator characterised by

Hessx S(⇠, ⌘) = hhessx S(⇠), ⌘i1

Suppose ⇠ 2 ker(hessx S), then in particular Hessx S(⇠, ⌘) = 0 for all ⌘ 2 C10

(x⇤TM)


and (5.8) shows that ⇠ must be a weak solution of

�r2

t ⇠+rA⇤(rt⇠)+rtr⇠A�rA⇤r⇠A�r2A⇤⇤(⇠, x�A)+R(x�A, ⇠)x = 0 (5.9)

Again the weak solutions are in fact smooth (Appendix A.4) and the space of solu-

tions has dimension 2m.

Now ⇠ 2 ker(hessx S) i↵ h⇠, hessx S(⌘)i1

= 0 i↵ ⇠ 2 im(hessx S)?. Furthermore,

self adjoint operators are closed so we have

ker(hessx S) = im(hessx S)? = H1(x⇤TM)/ im(hessx S) = coker(hessx S)

Then since ker(hessx S) is finite dimensional hessx S is Fredholm with index zero.

Remark. We shall refer to (5.9) as the Jacobi equation for S by analogy with the

equation obtained in the same way for the energy function E = 1

2

h@, @i0

, i.e. when

A = 0 (see e.g. [53]). Solutions will be referred to as Jacobi fields for S.

Recall that the Morse index (not to be confused with Fredholm index) of a

critical point x is the dimension of the maximal subspace on which Hessx is negative

definite. We will show that the following general Morse index theorem is applicable

to S.

Theorem 5.4.6. (Uhlenbeck [81]) Let B be a bilinear form on a Hilbert space H,

and H0

⇢ Ht ⇢ H1

= H, 0 t 1 an increasing family of closed subspaces. Denote

B|Ht

⌦Ht

by Bt and let Nt be the nullspace of Bt. If

(i) the dimension of the maximal subspace on which B is non-positive is finite

(ii) Nt \Nk = 0 for t 6= k

(iii) B is Fredholm of finite index

(iv) [t<kHt = Hk = \t>kHt

then there are only finitely many conjugate points, i.e. t 2 [0, 1] such that n(t) :=

dimNt is non zero. Furthermore indexB � indexB0

=P

0t<1

n(t) where indexBt

is the dimension of the maximal subspace on which Bt is negative definite.

We let H1

= H1(x⇤TM), Ht := {⇠ 2 H1

: supp ⇠ ⇢ (0, t)} and B = Hessx S.

Then (iv) is satisfied and (iii) has been proved already. As for (ii), suppose there

exists ⇠ 2 Nt \Nk, k > t, then ⇠(⌧) = 0 for all t < ⌧ < k and ⇠ satisfies the Jacobi

equation, therefore ⇠ is identically zero.


In order to prove that (i) is satisfied we first show that Hessx S is positive definite

on Ht for small t. Using the fact that R,rA,r2A restricted to x are bounded

maps, as well as the Cauchy-Schwarz inequality, we have from (5.8) Hessx S(⇠, ⇠) �krt⇠k2

0

� ↵krt⇠k0

k⇠k0

� �k⇠k20

for some positive constants ↵, �. Then using 2ab a2 + b2, with a = krt⇠k

0

and b = ↵k⇠k0

we have Hessx S(⇠, ⇠) � 1

2

krt⇠k20

�Nk⇠k20

for some N . Now if ⇠ has support in (0, ⌧) then we may rescale ⌘(t) := ⇠(⌧ t) and

k⌘k20

= 1

⌧k⇠k2

0

. Furthermore, in an orthonormal parallel frame along x we have

rt⌘ = ddt⌘ and k d

dt⌘k2

0

= ⌧R ⌧0

k ddu⇠(u)k2 du = ⌧krt⇠k2

0

. Therefore Hessx S(⇠, ⇠) �1

2⌧krt⌘k2

0

� N⌧k⌘k20

and choosing ⌧ su�ciently small we have Hessx S(⇠, ⇠) > 0 on

H⌧ .

We now prove (i) by adapting a technique used in [53] and splitting H1(x⇤TM)0

into two subspaces with Hessx S being positive definite on one of the spaces. Choose

ti, i = 0, . . . , n such that Hessx S is positive definite for all ⇠ 2 H1(x⇤TM)0

with

supp ⇠ ⇢ (ti, ti+1

). Let V be the vector space consisting of all ⇠ 2 H1(x⇤TM)0

such that ⇠(ti) = 0 for all i and V 0 the finite dimensional space consisting of ⇠ 2H1(x⇤TM)

0

such that ⇠ satisfies the Jacobi equation on each [ti, ti+1

]. Given any

⌘ 2 H1(x⇤TM)0

we let ⌘0 be the unique element of V 0 such that ⌘0(ti) = ⌘(ti). Then

⌘ � ⌘0 2 V and therefore H1(x⇤TM)0

= V + V 0. Since Hessx S is positive definite

on V and V 0 has finite dimension, (i) is satisfied and we have the following theorem.

Theorem 5.4.7. Each critical point of S in ⌦p,q(I,M) has a finite number of con-

jugate points ti and finite Morse index equal toP

i n(ti).

A consequence of Proposition 5.4.5 is that hessx S is an isomorphism if and only

if its kernel is trivial. Thus S is a Morse function on ⌦p,q(M) if and only if 0, 1 are

not conjugate along any critical point of S in ⌦p,q(M), in which case we say that

p, q are non-conjugate.

Theorem 5.4.8. Let p, q 2 M be non-conjugate points. If �i denotes the ith Betti

number of Cp,q(M) over a field F, then there are at least �i criticial points of S|⌦p,q

(M)

with Morse index i. Furthermore the total number of critical points of S|⌦

p,q

(M)

is

either infinite or odd.

Proof. The first statement follows from the homotopy equivalence of ⌦N(M) and

CN(M) [32], Theorem 5.3.8, Proposition 5.4.5 and the weak Morse inequalities [63].

If i denotes the number of critical points having Morse index i then the Morse

inequalities are equivalent to the statement

1X

i=0

izi =

1X

i=0

�izi + (1 + z)

1X

i=0

qizi (5.10)


for some coe�cients qi 2 N [ {1}. If M is not contractible the based loop space

has infinite category [20] and therefore S|⌦

p,q

(M)

has infinitely many critical points

(Corollary 5.4.1). If M is contractible then Cp,q(M) is also contractible and the sum

of the Betti numbers of Cp,q(M) is 1. Then setting z = 1 in (5.10) the total number

of critical points is equal to 1 + 2P1

i=0

qi which is either infinite or odd (cf. [27]).

It remains to show that non-conjugate points exist. In fact we will show that,

just as in the case of geodesics, given p 2 M the points which are not conjugate to

p are generic in M . It is shown in [59] that if M is complete and kAk is bounded

the solutions of (5.4) are uniquely extendible over all of R. Therefore we may define

the map cexp : TM ! M which takes (p, v) 2 TM to x(1) where x is the solution

of (5.4) with initial conditions x(0) = p, x(0) = v.

Proposition 5.4.9. Suppose kAk is bounded. Then the point cexpp(v) is conjugate

to p 2M along a conditional extremal if and only if cexpp : TpM !M is critical at

v.

Remark. The proof given in [53] of the analogous result for exp uses the fact that

exp(tv) is a geodesic. We use the same proof for Proposition 5.4.9 except that

cexp(tv) is not necessarily an extremal, so we require the map ↵ (defined below)

instead.

Proof. If cexpp is critical at v 2 TpM then there exists a non zero X 2 Tv(TpM) such

that T cexpp(v)X = 0. Let v(s) be a path in TpM with v(0) = v and dvds(0) = X. For

each s we denote by ↵(s, t) the solution of (5.4) with initial conditions ↵(s, 0) = p

and ddt↵(s, 0) = v(s), i.e. ↵(s, 1) = cexpp(v(s)). Furthermore ↵ depends smoothly

on s because the solutions of (5.4) depend smoothly on initial conditions. Thus

W (t) = @@s↵(0, t) is well defined and must be a Jacobi field for S along ↵(0, t).

Now W (0) = 0 and W (1) = @@s

cexpp(v(s))|s=0

= T cexpp(v)X = 0 and W is not

identically zero because

rtW (0) = rt@@s↵(0, 0) = rs

@@t↵(0, 0) = d

dsv(0) = X

and X is non-zero, therefore cexpp(v) is conjugate to p.

Now suppose T cexpp is non-singular at v, so if we choose a basis Xi for Tv(TpM)

then the T cexpXi are linearly independent. We can carry out the above construc-

tion to obtain a variation through extremals ↵i(s, t) for each Xi. We then obtain n

independent Jacobi fields Wi along ↵i(0, t) each of which vanishes at p, i.e. a basis

for the space of Jacobi fields along ↵i(0, t) which vanish at p. However the Wi(1)

5.5. CLOSED EXTREMALS 62

are linearly independent, so no Jacobi field along ↵i(0, t) can vanish at both p and

cexpp(v).

Corollary 5.4.10. If p 2M the set of points which are not conjugate to p is open

and dense in M .

Proof. By Sard’s theorem the set of critical values of cexpp has measure zero in M .

5.5 CLOSED EXTREMALS

An absolutely continuous map from S1 to M can be identified with a map x 2H1(I,M) such that x(0) = x(1). Thus we identify H1(S1,M) with the submanifold

P�1(diagM ⇥ M). The tangent space TxH1(S1,M) consists of all liftings ⇠ 2TxH1(I,M) such that ⇠(0) = ⇠(1). We may therefore use Proposition 5.3.2 to show

that a critical point x of S|H1(S1,M)

is smooth on the interval (0, 1). Then integrating

(5.3) by parts for ⇠ 2 TxH1(S1,M) we have h⇠, x� A(x)i|10

= h⇠(0), x(0)�x(1)i = 0,

since ⇠(0) = ⇠(1) and x(0) = x(1). This shows that x is a C1 mapping of S1, and

then since x satisfies (5.4) it must in fact be smooth.

We may now use local coercivity (Proposition 5.3.6) to prove the Palais-Smale

condition for S restricted to H1(S1,M), provided we can show that if S is bounded

on a sequence inH1(S1,M) then the sequence is uniformly convergent. In the case of

⌦N(M) this requires only the assumption that either N0

or N1

is compact (Corollary

5.3.5), but if x 2 H1(S1,M) then x(0) is not constrained, so we must assume that

M is compact.

Theorem 5.5.1. If M is compact then S satisfies the Palais-Smale condition on

H1(S1,M).

Proof. The proof is quite similar to that of Theorem 5.3.8. Let (xi) ⇢ H1(S1,M)

be a sequence such that S(xi) is bounded and |dS(xi)| ! 0. Since M is compact

kA(p)k, p 2 M is bounded. Therefore by Lemma 5.3.4 (xi) is equicontinuous and

has a uniformly convergent subsequence by the Arzela-Ascoli theorem. We replace

(xi) by this subsequence and choose h 2 C1(S1,M) such that for su�ciently large

i, xi is contained in the natural chart centred at h. Thus if (⇠i) is the sequence in

H1(h⇤TM) defined by exp(⇠i) = xi we have that k⇠ik1

is bounded just as in the proof

of Theorem 5.3.8. Now we abbreviate S|H1(S1,M)\U

h

to SS and from Proposition 5.3.6

(DSSh (⇠i)�DSS

h (⇠j))(⇠i � ⇠j) + C|⇠i � ⇠j|20

� �k⇠i � ⇠jk21


Since (xi) is uniformly convergent and |DSSh (⇠i)|! 0 by assumption, the inequality

above shows that (⇠i) converges in H1 and therefore (xi) converges in H1(S1,M).

Corollary 5.5.2. In any homotopy class of closed curves onM there exists a critical

point which minimises S with respect to the given class. Furthermore S attains its

infimum in H1(S1,M).

The following example, used in [67] for the energy functional (A = 0), shows

that the condition that M be compact is necessary.

Example 5.3. Consider the surface of revolution in R3 obtained by rotating y = ex

about the x-axis. Then the homotopy class of circles of rotation has no element of

minimal length or energy.

Lemma 5.5.3. The inclusion of H1(S1,M) in C(S1,M) is a homotopy equivalence.

Proof. The proof is the same as that of Theorem 1.3 in [32], except that in place

of the fibration Cp,q(I,M) ! CN(I,M) ! N we have Cp,p(I,M) ! C(S1,M) !diag(M ⇥M), and similarly for H1(S1,M).

Once again we may use the Ljusternik-Schnirelman multiplicity theorem [64] to

obtain a lower bound for the number of critical points:

Corollary 5.5.4. There are at least cat(C(S1,M)) critical points of S restricted to

H1(S1,M). If M is not contractible then there are infinitely many critical points.

It follows from Corollary 5.5.2 that non-constant closed extremals exist when-

ever M is compact with non-trivial fundamental group. The following proposition

addresses the simply connected case with an additional assumption on A.

Proposition 5.5.5. If M is simply connected and compact, and if kAk has a finite

number of critical points, then there exist non-constant critical points of S restricted

to H1(S1,M).

Proof. Let p be the constant curve t 7! p 2 M . From the Euler-Lagrange equa-

tion (5.4) p is a critical point i↵ rA⇤(p)(A(p)) = 0. For any V 2 TpM we have

hrA⇤(A), V i = hA,rVAi = 1

2

V kAk2 and therefore rA⇤(A) = 1

2

grad kAk2. Hence

p is a critical point of S i↵ it is a critical point of kAk and therefore there are only

finitely many constant critical points of S. However, according to [19] the free loop

space C(S1,M) has infinite category when M is simply connected and compact.

Then by Corollary 5.5.4 there must be infinitely many non-constant critical points.

5.5. CLOSED EXTREMALS 64

CHAPTER 6

Existence of variationally de�ned curves withhigher order elliptic Lagrangians

This chapter has been published as [71]. Once again, minimal changes have been

made to ensure continuity of notation with previous chapters.

Abstract

We present a method for proving the existence of solutions to a class

of one dimensional variational problems, namely those where the La-

grangian is strongly elliptic. The method is demonstrated by some ex-

amples of optimal interpolation problems which are motivated by appli-

cations to the mechanics and control of rigid bodies. In each case the key

step is to show that the variational problem satisfies the Palais-Smale

condition. We do so in a general setting, showing that the number of

initial conditions required depends on the higher order energy bounding

properties of the Lagrangian.

6.1 INTRODUCTION

Recently there has been increased interest in variationally defined curves with higher

order derivatives in the Lagrangian for intrinsic methods of interpolation in mani-

folds. This was initiated independently by Gabriel and Kajiya [22] and Noakes et.

al. [60] with the study of stationary paths of the average covariant acceleration1

2

Rkrtxk2dt, where x : I ! M is a map from the unit interval to a Riemannian

manifold M . In Euclidean space these curves are cubic polynomials; they are there-

fore known as Riemannian cubics, and are a natural candidate for interpolation

when we require di↵erentiability at knot points. For example, in trajectory plan-

ning rapid changes of direction are ine�cient so it is desirable for paths to be at

least C1. Applications of this kind for oriented rigid bodies require interpolation in

65

6.1. INTRODUCTION 66

the Lie group SO(3) of 3D orientations (or SE(3), the group of rigid body motions),

and have been the primary motivation for studying Riemannian cubics. More re-

cently Riemannian cubics have also been studied by Gay-Balmaz et. al. [24, 25] and

Trouve and Vialard [79] for interpolation on spaces of shapes (images, landmarks,

curves, surfaces or tensors) in computational anatomy, motivated by applications in

medical imaging.

Several alternative interpolation schemes have been proposed in the interim (see

for example [50] or [61] and the references therein), most of which are variational in

nature. Of particular interest for this chapter are the so called Riemannian cubics

in tension and conditional extremals. Cubics in tension are the stationary paths

ofRkrtxk2 + ⌧ 2kxk2dt, where ⌧ is a constant known as the tension parameter.

A detailed qualitative analysis of a special case of these curves can be found in

[61]. They have been studied from an optimal control point of view in [75, 76]

where they are called elastic curves, and also [34, 35] where they are called ⌧ -elastic

curves and applied to control of spacecraft for interferometric imaging. In this

particular application acceleration requires fuel expenditure and image quality is

inversely related to speed, so it makes sense to penalise both. Conditional extremals

are the stationary paths ofRkx� A(x)k2dt where A is a fixed vector field [62, 73].

These curves are motivated by a problem in which x = A is an estimate for the

equations of motion of a system, but is not compatible with the observed data.

Conditional extremals are then an estimate of the actual trajectories.

So far the only existence results for higher order interpolants are those of Giambo

et. al. for Riemannian cubics and higher order geodesics [27, 29]. In this paper we

prove existence of cubics in tension and conditional extremals in a general setting

which also includes the prior existence results.

6.1.1 Statement of results

Let S denote either the unit interval or the circle I/{0, 1} and M a complete Rie-

mannian manifold. We prove existence of solutions to variational problems of the

form: minimiseRSP among all curves satisfying initial conditions up to order j,

where P is strongly elliptic of order k and such that boundingRSP bounds the L2

normRSkri

txk2 for each j < i k. Using this result we show existence of solutions

to three specific variational problems:

(i) (higher order conditional extremals) given a non-autonomous vector field A

on M with kA(t, p)k bounded, minimizeRIkrk�1

t x � A(t, x)k among curves

I !M satisfying boundary conditions up to order k � 1,

(ii) (interpolation by cubics in tension) given a parameter ⌧ , fixed times (ti)n�1

0

⇢ I

67 CHAPTER 6. VARIATIONAL CURVES WITH ELLIPTIC LAGRANGIAN

and (pi)n�1

0

⇢M , minimizeRI(krtxk2 + ⌧ 2kxk2) among all curves x : I !M

such that x(ti) = pi,

(iii) (closed cubics in tension) minimizeRI(krtxk2 + ⌧ 2kxk2) over all curves S1 !

M , assuming M is compact.

6.1.2 Description of methods

From the global analytical point of view a variational problem of the kind discussed

above consists of an action functional f and a Hilbert manifoldX of curves satisfying

some boundary conditions, which is a submanifold of the natural domain of the

action. The standard criterion for existence of critical points is the Palais-Smale

condition: f : X ! R is said to satisfy the Palais-Smale condition if any sequence

(xi) ⇢ X on which f is bounded, and for which |df(xi)| ! 0, has a convergent

subsequence. If this condition is satisfied and the function is bounded below then

we are guaranteed not only existence of critical points of f but also existence of a

minimum (see eg. [67]). Furthermore we obtain lower bounds for the total number

of critical points via Ljusternik-Schnirelman theory, and possibly also Morse theory.

Elıasson has in fact verified the Palais-Smale condition for a large class of varia-

tional problems [15, 18]. The results of Giambo et. al. and the variational problems

we consider in this paper do not fit into the class of problems treated therein. Nev-

ertheless we approach the problem using Elıasson’s observation that verification of

the Palais-Smale condition can be conveniently separated into three parts [16]. One

verifies that the action is weakly proper on X with respect to some larger manifold

X0

containing X as a weak submanifold, and this allows the problem to be treated

in local coordinates for X. It remains to confirm that the action functional is lo-

cally bounding and locally coercive on X with respect to X0

and the Palais-Smale

condition follows (see Section 6.2.1 for precise definitions and a proof).

It is proved in [18] that the action functionals with so-called strongly elliptic

Lagrangians are locally bounding and locally coercive, with respect to the Banach

manifold of continuous maps C0(I,M), on the domain consisting of curves satisfying

boundary conditions of su�ciently high order. This assumption on the domain is

not compatible with, for example, the problem of C1 optimal interpolation of n

points by cubics in tension, because the order of the boundary conditions is too low.

We therefore prove in Section 6.3 that action functionals of this kind are locally

coercive and locally bounding on their natural domain with respect to Ck�1(I,M).

Moreover we give some conditions under which these properties are inherited by the

restriction of the action functional to a submanifold of the natural domain. This

leads to the central result mentioned above, which roughly states that the Palais-

6.2. PRELIMINARIES 68

Smale condition will hold if lower order boundary conditions are o↵set by terms in

the Lagrangian which bound the higher order energies kritxk2.

6.2 PRELIMINARIES

In this section we review the relevant material from [14, 15, 16, 18], including the

method for establishing the Palais-Smale condition and the definition of strongly

elliptic polynomial di↵erential operators.

6.2.1 Method for establishing the Palais-Smale condition

Let X,X0

be Banach manifolds modelled on B,B0

respectively, and suppose X ⇢X

0

, B ⇢ B0

with the latter a continuous linear inclusion. Then X is a weak subman-

ifold of X0

if for any x0

in the closure of X there is a chart (�0

, U0

) for X0

containing

x0

, such that, setting U = U0

\X, we have �0

(U) ⇢ B and the restriction of �0

to

U is a chart � : U ! �(U) for X. Any chart for X which arises in this way will be

called a weak chart1 at x0

.

Note that this definition allows the topology of the weak submanifold to be finer

than the relative topology.

A Finsler structure on a Banach manifold X is a continuous function v 7! kvkon the tangent bundle ⌧ : TX ! X such that the restriction to each fibre TxX is a

norm, and such that in any local trivialisation � : ⌧�1U ! �(U) ⇥ B, and for any

constant k > 1, we have

1

kk��1(⇠, ⌘)k k��1(⇠

0

, ⌘)k kk��1(⇠, ⌘)k (6.1)

with ⌘ 2 B and ⇠ su�ciently close to ⇠0

= �(x). That is, the fibre norms are locally

equivalent.

Suppose X is a weak submanifold of X0

and let k kB denote the norm for B and

| |0

the norm for B0

. We call a Finsler structure on X locally bounded with respect

to X0

if for any x0

in the closure X and any constant L, there is a local trivialisation

� over a weak chart � at x0

and a constant c such that

k��1(⇠, ⌘)k ck⌘kB

for all ⇠ 2 �(U) with k⇠kB < L, and all ⌘ 2 B.

A function f : X ! R is called

1When reading these definitions it is helpful to keep in mind the typical example X = Hk(I,M)and X0 = C0(I,M), where the weak charts are given by exph �⇠ 7! ⇠, where h 2 C1(I,M) and⇠ 2 C0(h⇤TM).


(i) weakly proper with respect to X0

if any subset A ⇢ X on which f is bounded

is relatively compact in X0

.

(ii) locally bounding with respect to X0

if for any constants K,L and x0

2 X there

is a weak chart (U,�) at x0

and a constant ↵ such that for all ⇠ 2 �(U) with

|⇠|0

< K and f�(⇠) := f(��1(⇠)) < L, we have k⇠kB < ↵.

(iii) locally coercive with respect to X0

if it is C1 and for any x0

2 X and any

constant K, there is a weak chart (U,�) at x0

and there exist constants c1

>

0, c2

such that

(Df�(⇠)�Df�(⌘))(⇠ � ⌘) � c1

k⇠ � ⌘k2B � c2

|⇠ � ⌘|20

(6.2)

for all ⇠, ⌘ 2 �(U) with k⇠kB < K, k⌘kB < K. If f is of class C2 we have an

equivalent condition:

D2f�(⇠)(⌘, ⌘) � c1

k⌘k2B � c2

|⌘|20

(6.3)

for all ⇠ 2 �(U) with k⇠kB < K and all ⌘ 2 B.

The assumption of an upper bound for |⇠|0

is not included in the original defini-

tion of locally bounding [16] because it does not need to be assumed if f is weakly

proper. However we will find it useful to be able to prove that f is locally bounding

independently. See Appendix A.2 for further comments on the terminology above.

Proof of the equivalence of equations (6.2) and (6.3): If (6.2) holds then a change

of variables gives

(df⇠ � df⇠�⌘)(⌘) � c1

k⌘k2 � c2

|⌘|20

By Taylor’s theorem we have

(df⇠ � df⇠�⌘)(⌘) = d2f⇠(⌘, ⌘) + o(⌘)⌘

where lim⌘!0

o(⌘)/k⌘k = 0 and therefore for all " > 0 there is a � such that k⌘k <� =)

d2f⇠(⌘, ⌘) � (c1

� ")k⌘k2 � c2

|⌘|2

and we choose " su�ciently small. The proof of the converse is similar.

Theorem 6.2.1. (Elıasson [18]) Let X be a regular Banach manifold and a weak

submanifold of X0

as above, with a locally bounded Finsler structure. If f : X ! Ris of class C1 and weakly proper, locally bounding and locally coercive each with

respect to X0

, then f satisfies the Palais-Smale condition.


Proof. The assumption that X be (topologically) regular is required so that the

manifold topology coincides with the Finsler metric topology (see [66]). Let (xi) ⇢ X

be a sequence for which f(xi) is bounded and |df(xi)|! 0. Since f is weakly proper

we can find a subsequence converging in X0

to some x0

. Then we may choose a

weak chart � at x0

and a subsequence (⇠i) := (��1xi) contained in the domain of �

which is bounded in B because f is locally bounding, and for which the inequality

(6.2) therefore holds. Now since the Finsler structure for X is locally bounded

|Df�(⇠i)(⇠i � ⇠j)| |df(xi)|ck⇠i � ⇠jk

and then from (6.2) we have

k⇠i � ⇠jk2 c1c2|⇠i � ⇠j|2

0

+ 1

c2(Df�(⇠i)�Df�(⇠j))(⇠i � ⇠j)

c1c2|⇠i � ⇠j|2

0

+ 1

c2(|df(xi)|+ |df(xj)|)ck⇠i � ⇠jk

Using the assumption |df(xi)|! 0 and the convergence of (⇠i) in B0

, it follows that

(⇠i) is Cauchy and converges in B, and the corresponding subsequence (xi) converges

in X.

Throughout this chapter we let S be either the unit interval I or the circle

I/{0, 1} and M a complete Riemannian manifold of class C1 and with finite di-

mension. Many of the results that follow hold for any compact S (see [14, 16])

but for now we are only interested in applications where S is one-dimensional, and

making this assumption simplifies the exposition.

6.2.2 Polynomial di�erential operators

Let E ! S be a vector bundle with a Riemannian metric connection and V ⇢ E an

open subset which projects onto S. Let F ! S be another vector bundle and denote

by Lr(E,F ) the bundle with fibre over s 2 S being the vector space of r-multilinear

maps Ers ! Fs. Suppose we have a smooth fibre preserving map A↵ : V ! Lr(E,F )

for each multi-index ↵ = (↵1

, . . . ,↵r) with r � 0, 1 ↵i k,P↵i w. Then the

map P : C1(V )! C1(F ) defined by

P (⇠) =X

↵

A↵(⇠)(r↵1t ⇠, . . . ,r↵

r

t ⇠)

is called a local polynomial di↵erential operator(PDO) from V to F of order k and

weight w, denoted P 2 PDwk (V, F ). From Corollaries 7.1,7.2 in [15] we have:

(i) If P 2 PDjj(V, F ) and k � j then P can be extended to a smooth map

P : Hk(V )! Hk�j(F ).


(ii) P 2 PD2kk (V, F ) can be extended to a smooth map P : Hk(V )! L1

0

(F )

These results allow us to make global extensions of maps defined on C1(S,M)

which can be represented locally by PDOs. For example, the map @ : x 7! x

which takes a smooth curve to its tangent lift is a smooth map @ : Hk(S,M) !Hk�1(Hk(S,M)⇤TM) (cf. (4.10)), and j-times covariant di↵erentiation of vector

fields along curves is a smooth map

rjt : H

k(S,M)! L(Hr(Hk(S,M)⇤TM), Hr�j(Hk(S,M)⇤TM))

with local principal part ([15] Theorem 12)

(rjt)h(⇠)⌘ = rj

t⌘ +j�1X

i=0

Pij(⇠)rit⌘ (6.4)

where Pij 2 PDj�ij�i(h

⇤D, L(h⇤TM, h⇤TM)).

If we take the contraction of @ with rj�1

t we obtain a smooth section rj�1

t @ :

Hk(S,M) ! Hk�j(Hk(S,M)⇤TM), x 7! rj�1

t x, with local principal part (cf. [15]

Theorem 11)

(rj�1

t @)h(⇠) = rjt⇠ +Qj(⇠) (6.5)

where Qj 2 PDjj�1

(h⇤D, h⇤TM). This is the basic operator for (global) polynomial

di↵erential operators, which are defined as follows.

Let E ! S ⇥ M be a vector bundle and suppose we have a smooth section

B↵ : S ⇥M ! Lr(S ⇥ TM, E) for each multi-index ↵ = (↵1

, . . .↵r) with r � 0, 1 ↵i k and

Pi ↵i w. Then given x 2 C1(S,M) we define a smooth section

P (x) : S ! (Id, x)⇤E by

P (x)(t) :=X

↵

B↵(t, x(t))(r↵1�1

t x, . . . ,r↵r

�1

t x)

Thus P is a map C1(S,M)! C1(C1(S,M)⇤E) which we call a polynomial di↵er-

ential operator on E of order k and weight w, denoted P 2 PDwk (E). These operators

can also be extended to smooth sections. In particular P 2 PD2kk (E) extends to a

smooth map Hk(S,M) ! L1

0

(Hk(S,M)⇤E). In a local trivialisation �h we have,

using (6.5),

Ph(⇠) =X

↵

A↵(⇠)(r↵1t ⇠, . . . ,r↵

r

t ⇠)

for some A↵ : h⇤D ! Lr(h⇤TM, h⇤E) so Ph 2 PDwk (h

⇤D, h⇤E) is a local PDO.

We say P 2 PD2kk (S⇥M ⇥R) is strongly elliptic if there exists a constant � > 0

such that B(k,k)(t, x)(v, v) � �kvk2 for all t 2 S, x 2M and v 2 TxM , i.e. B

(k,k)(t, x)


is a coercive bilinear form (this notion of coercivity for bilinear forms is di↵erent

to the definition of locally coercive functions in Section 6.2.1, see Appendix A.2 for

further explanation). It follows that in the local expression A(k,k) is also coercive,

since A(k,k)(t, ⇠)(⌘, ⌘) = B

(k,k)(t, exph ⇠)(D2

exp(⇠)⌘, D2

exp(⇠)⌘).

6.2.3 Finsler structures on Hj(Hk(S,M)⇤TM)

On each fibre Hj(x⇤TM), x 2 C1(S,M) we have an inner product (cf. Section 4.7):

h⇠, ⌘ij :=jX

i=0

Z

S

g(x)(rit⇠,ri

t⌘) (6.6)

where g is the Riemannian metric on M . These inner products can be extended to a

smooth Riemannian metric for Hj(Hk(S,M)⇤TM), which we also denote by h , ij.Working in a local trivialisation, suppose x = exph(⇠), and ⌘, ⌫ 2 Hj(h⇤TM) are

the local representatives of V,W 2 Hj(x⇤TM). Then

hV,W ij =jX

i=0

Z

S

g(exph(⇠))(ritD2

exp(⇠)⌘,ritD2

exp(⇠)⌫)

=X

i

Z

S

g(h)(G(⇠)(rit)h(⇠)⌘, (r

it)h(⇠)⌫)

=X

i

hG(⇠)(rit)h(⇠)⌘, (r

it)h(⇠)⌫i

0

(6.7)

Here G : D ! L(TM, TM) is defined by

g(p)(v,G(u)w) = g(exp u)(D2

exp(u)v,D2

exp(u)w)

In particular the case j = k gives a Riemannian metric for Hk(S,M). It is proved

in [15] that the associated norm generates a Finsler structure, i.e. it satisfies (6.1),

and that each connected component is a complete metric space.

Lemma 6.2.2. The Finsler structure on Hk(S,M) defined above is locally bounded

with respect to Cj(S,M) for j < k.

Proof. First note that in the definition of locally bounding (Section 6.2.1) the topol-

ogy of X0

is involved only in the choice of weak chart: it must come from a chart

for X0

and it must contain x0

2 X where the closure is taken in X0

. Note also that

Hk(S,M) is a weak submanifold of Cj(S,M) for each j < k, and in each case by

restriction of the natural charts for C0(S,M). Thus we will use natural charts, and

this will prove the result for every j < k. Let ��1

h = (exph, D2

exp) be a local trivi-

alisation for THk(S,M) = Hk(Hk(S,M)⇤TM), ⇠ 2 Hk(h⇤D), and ⌘ 2 Hk(h⇤TM).


In order to distinguish the norm on TxHk(S,M) from that on Hk(h⇤TM) we will

temporarily write the former as kx kk, i.e.

kxvk2k =kX

i=0

Z

S

g(x)(ritv,ri

tv)

and

k⌘k2k =kX

i=0

Z

S

g(h)(rit⌘,ri

t⌘) =kX

i=0

Z

S

krit⌘k2

where we continue to denote the norm derived from the metric on M by k k. From(6.7) and (6.4)

kx��1

h (⇠, ⌘)k2k =kX

i=0

Z

S

g(h)(G(⇠)(rit)h(⇠)⌘, (r

it)h(⇠)⌘)

X

i

Z

S

|G(⇠)|k(rit)h(⇠)⌘k

2

constX

i

Z

S

|G(⇠)|(krit⌘k2 +

i�1X

j=0

|Pij(⇠)|krjt⌘k2)

Now k⇠(t)k |⇠|0

const k⇠kk by the Sobolev imbedding theorem, so if k⇠kk < L

then ⇠(t) is contained in a compact subset of D, and since G,Pij are continuous we

have that |G(⇠)|, |Pij(⇠)| are both bounded. Thus

kx��1

h (⇠, ⌘)k2k const k⌘k2k

for all k⇠kk L and ⌘ 2 Hk(h⇤TM).

6.3 SOME GENERAL RESULTS

Let P 2 PD2kk (S ⇥ M ⇥ R) be strongly elliptic and define J(x) :=

RSP (x). It

is proved in [18] (Theorem 7) that if we restrict J to a submanifold of Hk(S,M)

consisting of curves satisfying boundary conditions on the derivatives up to order

k � 1, then this restriction is locally bounded and locally coercive with respect to

C0(S,M). In the following theorem we prove instead that J is locally bounding and

locally coercive on its natural domain Hk(S,M). This is in order to accomodate a

larger variety of variational problems, such as those considered in sections 6.4.2 and

6.4.3 where the boundary conditions are of lower order than P . The cost of this

generality is that we must weaken the statement to locally bounding and locally

coercive with respect to Ck�1 instead of C0.

6.3. SOME GENERAL RESULTS 74

Theorem 6.3.1. Let P be a strongly elliptic PDO on S⇥M⇥R of order k. Then the

smooth function J : Hk(S,M) ! R defined by J(x) :=RSP (x) is locally bounding

and locally coercive with respect to Ck�1(S,M).

Proof. Let �h be a natural chart centred at h and Ph(⇠) the local expression for P ,

so

Jh(⇠) =

Z

S

Ph(⇠) =X

↵

Z

S

A↵(⇠)(r↵1t ⇠, . . . ,r↵

r

t ⇠)

For ↵ 6= (k, k) let X↵ =RSA↵(⇠)(r↵1

t ⇠, . . . ,r↵r

t ⇠), then since |⇠|0

is bounded so

is |A↵(⇠)|, and using the Holder inequality |X↵| const kr↵1t ⇠k

0,p1. . . kr↵

r

t ⇠k0,p

r

whereP

i 1/pi = 1. Let us assume that ↵1

is the largest index and p1

= 2. From

the linear inclusion C0(h⇤TM) ⇢ Lp0

(h⇤TM) we have kr↵i

t ⇠k0,p const |r↵

i

t ⇠|0

and

therefore |X↵| const k⇠kk|⇠|rk�1

. Now using the estimate 2ab "a2 + 1

"b2 for all

" > 0, we obtain

|X↵| const�"k⇠k2k + 1

"|⇠|2rk�1

�

For ↵ = (k, k), since P is strongly elliptic there exists �0 > 0 such that

Z

S

Ak,k(⇠)(rkt ⇠,rk

t ⇠) � �0krkt ⇠k

2

0

Hence Jh(⇠) � �0krkt ⇠k

2

0

� const�"k⇠k2k + 1

"|⇠|2rk�1

�, and choosing " su�ciently small

there exists � > 0 such that

Jh(⇠) + �0k⇠k2k�1

� �k⇠k2k � const |⇠|2rk�1

It follows that if Jh(⇠) and |⇠|k�1

are bounded then so is k⇠kk, i.e. J is locally

bounding with respect to Ck�1.

We now prove that J is also locally coercive. For this we require the derivatives

of Jh:

DJh(⇠)⌘ =X

↵

Z

S

D2

A↵(⇠)(⌘,r↵1t ⇠, . . . ,r↵

r

t ⇠)

+X

↵

rX

i=1

Z

S

A↵(⇠)(r↵1t ⇠, . . . ,r↵

i

t ⌘, . . . ,r↵r

t ⇠)

D2Jh(⇠)(⌘, ⌘) =

Z

S

Ak,k(⇠)(rkt ⌘,rk

t ⌘) +X

i+j<2k

Z

S

Cij(⇠)(rit⌘,r

jt⌘)

where the Cij are sums of derivatives of A↵,↵ 6= (k, k). It will not be necessary

to write down their precise expressions here; we merely note that that they are

continuous functions of ⇠ and its derivatives up to order k. Hence if k⇠kk is bounded


then using the Cauchy-Schwarz inequality and strong ellipticity there exists � > 0

such that

D2Jh(⇠)(⌘, ⌘) � �krkt ⌘k

2

0

�X

i+j<2k

const krit⌘k

0

krjt⌘k

0

� �krkt ⌘k

2

0

� const k⌘kkk⌘kk�1

� �krkt ⌘k

2

0

� const("k⌘k2k + 1

"k⌘k2k�1

)

for all " > 0. Choosing " su�ciently small gives

D2Jh(⇠)(⌘, ⌘) � c1

k⌘k2k � c2

k⌘k2k�1

(6.8)

where c1

> 0 and c2

are constant. The result now follows from the imbedding

Ck�1 ⇢ Hk�1.

We now give some conditions under which the locally coercive and locally bound-

ing properties persist upon restriction to a submanifold.

Lemma 6.3.2. Let Y be a submanifold of Hk(S,M) such that for any h 2 Y \C1(S,M), the natural chart (�h, Uh) centred at h satisfies the submanifold property:

�h(Uh\Y ) = �h(Uh)\E, where E ⇢ Hk(h⇤TM) is the model space for Y . Then Y

is a weak submanifold of Ck�1(S,M), and if J : Hk(S,M)! R is locally bounding

and locally coercive with respect to Ck�1(S,M) then so is J |Y .

Proof. It will be convenient to denote J := J |Y . We let (�h, Uh) be a natural chart

centred at h 2 C1(S,M), then �h|(Uh \ Y ) is a weak chart for Y . From the proof

of Theorem 6.3.1 we have

Jh(⇠) + const |⇠|2k�1

� �k⇠k2k

for any ⇠ 2 �h(Uh) \ E. But k⇠kE const k⇠kk because E splits, so J is locally

bounding with respect to Ck�1. As for local coercivity, from Theorem 6.3.1 we also

have

(DJh(⇠)�DJh(⌘))(⇠ � ⌘) + c2

k⇠ � ⌘k2k�1

� c1

k⇠ � ⌘k2k

for any ⇠, ⌘ 2 �h(Uh) \ E. But DJh(⇠) = DJh(⇠)|E and (⇠ � ⌘) 2 E, therefore

(DJh(⇠)�DJh(⌘))(⇠ � ⌘) + c2

|⇠ � ⌘|2k�1

� const k⇠ � ⌘k2E

i.e. J is locally coercive with respect to Ck�1.

6.3. SOME GENERAL RESULTS 76

Lemma 6.3.3. Let U be a subset of Hk(S,M) such that for all x 2 U , kxk0

K

for some constant K. Then U is an equicontinuous family of curves of bounded

length.

Proof. For any t1

, t2

2 I, x 2 U the Holder inequality gives

d(x(t1

), x(t2

)) Z t2

t1

kxk dt |t1

� t2

|1

2K

Thus U is equicontinuous and length(x) =R

1

0

kxk dt K.

Lemma 6.3.4. The energy function Ek : Hk(S,M)! R, Ek(x) := kxk2k�1

is locally

bounding with respect to C0(S,M).

Proof. By induction: note that in Ek =RS

Pihri

tx,ritxi the integrand is strongly

elliptic. Therefore Ek is locally bounding with respect to Ck�1 by Theorem 6.3.1

and in particular E1 is locally bounding with respect to C0. Suppose Ej is locally

bounding with respect to C0, i.e. for any x 2 Hj(S,M) there exists a weak chart

(Uh,�h) at x such that for all ⇠ 2 Uh, if |⇠|0

and Ejh(⇠) are bounded then so is k⇠kj.

Then since Ej+1 � Ej it is su�cient to prove that if Ej+1

h is bounded then so is

krj+1

t ⇠k0

.

Now if Ej+1(��1

h ⇠) = k@(��1

h ⇠)k2j L for some constant L then

krjt@(�

�1

h ⇠)k0

= k��1

h (⇠, (rjt@)h⇠)k

0

L

and since k k0

satisfies (6.1) it follows that k(rjt@)h⇠k

0

= k�h(0, (rjt@)h⇠)k

0

is also

bounded. From (6.5) we have

k(rjt@)h⇠k

0

= krj+1

t ⇠ +Qj+1

(⇠)k0

� krj+1

t ⇠k0

� kQj+1

(⇠)k0

where Qj+1

2 PDj+1

j , so now it su�ces to prove kQj+1

(⇠)k0

bounded. Let Qj+1

(⇠) =P

↵ q↵(⇠)(r↵1t ⇠, . . . ,r↵

r

t ⇠), where ↵i j andP

i ↵i j + 1, and let

Y↵ =

Z

S

kq↵(⇠)(r↵1t ⇠, . . . ,r↵

r

t ⇠)k2

so that kQj+1

(⇠)k20

P

↵ Y↵. For each Y↵ we have, by the Holder inequality,

Y↵ const kr↵1t ⇠k

0,p1. . . kr↵

r

t ⇠k0,p

r

whereP

i1

pi

= 1. Let us assume that for all i we have ↵1

� ↵i, and choose p1

= 2,


so kr↵1t ⇠k

0,p1 k⇠kj. Then for i 6= 1 we have ↵i < j, and

kr↵i

t ⇠k0,p

i

const |r↵i

t ⇠|0

const |⇠|j�1

via the imbedding C0 ⇢ Lp0

. Then we have Y↵ const k⇠kj|⇠|r�1

j�1

const, since k⇠kjis bounded (by assumption), and kQj+1

(⇠)k0

is bounded.

Corollary 6.3.5. Suppose U ⇢ Hk(S,M) is relatively compact in C0(S,M) and

kxkk�1

const for all x 2 U , then U is relatively compact in Ck�1(S,M).

Proof. Consider a sequence (xi) ⇢ U which converges in C0(S,M) to x 2 U and

choose h 2 C1(S,M) C0-close to x such that for all i su�ciently large, xi is con-

tained in the domain Uh of the natural chart centred at h. Define the local sequence

(⇠i) by xi := exph ⇠i. Then (⇠i) is bounded in C0(h⇤TM), and by the previous lemma

kxkk�1

is locally bounding with respect to C0. Thus we have k⇠ikk bounded, and

since the Sobolev imbedding Hk(h⇤TM) ,! Ck�1(h⇤TM) is compact, (⇠i) converges

in Ck�1(h⇤TM).

Let�jTM denote the Whitney sum of j copies of TM , or simplyM in case j = 0.

Define Ij : Hk(S,M) ! �jTM by Ij(x) = (x(0), x(0), . . .rj�1

t x(0)) for 0 j k.

It can be shown that Ij is a submersion and therefore, given v 2 �jTM , the set

Hk(S,M)v := I�1

j (v) of curves with initial conditions (x(0), x(0), . . .rj�1

t x(0)) = v

is a submanifold of Hk(S,M) with tangent space

TxHk(S,M)v = Hk(x⇤TM)j := {w 2 Hk(x⇤TM) : w(0) = . . . = rj

tw(0) = 0}

Theorem 6.3.6. Fix v 2 �jTM with j < k and let P be a strongly elliptic PDO of

order k on S⇥M ⇥R such that if J(x) :=RSP (x) is bounded then (Ek(x)�Ej(x))

is bounded. Then J |Hk(S,M)v satisfies the Palais-Smale condition.

Proof. By Theorem 6.3.1 J is locally bounding and locally coercive with respect

to Ck�1(S,M). Given h 2 C1(S,M) \ Hk(S,M)v, the natural chart (�h, Uh)

for Hk(S,M) has the submanifold property: �h(Uh \ Hk(S,M)v) = �h(Uh) \Hk(h⇤TM)j. Thus by Lemma 6.3.2 we have that Hk(S,M)v is a weak subman-

ifold of Ck�1(S,M) and J |Hk(S,M)v is also locally bounding and locally coercive

with respect to Ck�1(S,M). It remains to prove that J |Hk(S,M)v is weakly proper

with respect to Ck�1. For each 0 < i < j we have, by the Cauchy-Schwarz and

6.4. EXAMPLES 78

Holder inequalities,

|ritx|

2

0

= kritx(s)k

2

= kritx(0)k

2

+

Z s

0

ddthri

tx,ritxi dt

kvi+1k2 + 2

Z1

0

|hri+1

t x,ritxi| dt

kvi+1k2 + 2kri+1

t xk0

kritxk

0

Dividing the above inequality by |ritx|

0

and using kvi+1k, kritxk

0

|ritx|

0

we obtain

kritxk

0

kvik+ 2kri+1

t xk0

and then

kxkj�1

constjX

i=1

kvik+ const krjt xk

0

Thus if U ⇢ Hk(S,M)v is such that J is bounded on U , then kxkk�1

is bounded and

in particular kxk0

K. By Lemma 6.3.3 U is equicontinuous and each x 2 U has

length at most K, and then since each x 2 U has the same initial point, U(I) ⇢M

is bounded. Thus by the Arzela-Ascoli theorem U is relatively compact in C0(S,M),

and by Corollary 6.3.5 U is relatively compact in Ck�1(S,M).

Suppose now that ⇤ is a submanifold of Hk(S,M)v and J satisfies the conditions

of the previous theorem so that Jv := J |Hk(S,M)v satisfies the Palais-Smale con-

dition. Note that it does not immediately follow that J := J |⇤ satisfies the Palais-

Smale condition, because |dJ(xi)|! 0 does not necessarily imply that |dJv(xi)|! 0.

However we do have the following.

Corollary 6.3.7. If ⇤ is a submanifold of Hk(S,M)v which satisfies the conditions

in Lemma 6.3.2, and J : Hk(S,M) ! R satisfies the conditions of Theorem 6.3.6,

then J |⇤ satisfies the Palais-Smale condition.

Proof. J |⇤ is locally bounding and locally coercive with respect to Ck�1(S,M) by

Lemma 6.3.2. It is shown in the proof of Theorem 5.3.8 that J |Hk(S,M)v is weakly

proper with respect to Ck�1(S,M), and it follows that J |⇤ is also weakly proper.

6.4 EXAMPLES

6.4.1 Higher order conditional extremals

As our first application we will extend the results in [73] proving the existence of

conditional extremals joining two given points of M provided the prior vector field

A is bounded with respect to the Riemannian metric. Study of this function was

initiated in [62], motivated by the problem of interpolating Riemannian manifold


data obtained from an integral curve of an unknown vector field which is thought to

be close to A. From the point of view of mechanics, it is more interesting to study

the function

J2

(x) := 1

2

krtx� A(x)k20

since the resulting minima will be curves which are L2-close to satisfying Newton’s

equation rtx = A. Here we will extend even further to higher order derivatives of

x and non-autonomous vector fields. Consider the restriction of

Jk(x) :=1

2

krk�1

t x� A(t, x)k20

to the submanifold of Hk(I,M) defined as follows. Let F be a map from Hk(I,M)

to the product of Whitney sumsL

k�1

TM ⇥L

k�1

TM defined by

F (x) := (x(0), x(0), . . . ,rk�2

t x(0), x(1), . . . ,rk�2

t x(1))

Again it can be shown that F is a submersion, and therefore given any v, w 2L

k�1

TM we have a submanifold Hk(I,M)v,w := F�1(v, w) consisting of curves

which satisfy the boundary conditions

(x(0), x(0), . . . ,rk�2

t x(0)) = v, (x(1), . . . ,rk�2

t x(1)) = w.

Theorem 6.4.1. If kA(t, p)k is bounded for all p 2M , then Jk satisfies the Palais-

Smale condition on Hk(I,M)v,w.

Proof. The Lagrangian hrk�1

t x�A(x),rk�1

t x�A(x)i is strongly elliptic, and for any

x 2 Hk(I,M) with Jk(x) const we have krk�1

t xk0

Jk(x) + kA(t, x)k0

const.

Furthermore, just as in the proof of Theorem 6.3.6, we can choose h 2 C1(I,M) \Hk(I,M)v,w and then the natural chart centred at h has the submanifold property.

Now by Corollary 6.3.7 Jk|Hk(I,M)v,w satisfies the Palais-Smale condition.

Corollary 6.4.2. Jk attains its infimum on Hk(I,M)v,w, and in any connected com-

ponent there is a critical point which minimises Jk with respect to the component.

Proof. Since Jk is bounded below, this is a standard consequence of the Palais-Smale

condition [67] p. 189.

The category cat(X) of a topological space X is a homotopy type invariant

defined as the minimal number of closed contractible subsets of X which cover X.

Since the restriction of Jk to Hk(I,M)v,w satisfies the Palais-Smale condition it has

at least cat(Hk(I,M)v,w) critical points by the Lusternik-Schnirelmann multiplicity

6.4. EXAMPLES 80

theorem (see eg. [67]). But it is shown in [29] that Hk(I,M)v,w has the same

homotopy type as the based loop space ⌦M . Thus we have the following.

Corollary 6.4.3. There are at least cat(⌦M) critical points of the restriction of Jk

to Hk(I,M)v,w. Moreover if M is not contractible then there are infinitely many

critical points.

Proof. If M is not contractible then cat(⌦M) is infinite by a theorem of Fadell and

Husseini [20] (we have already assumed M is complete and therefore connected).

Remark. If we set A = 0 we reproduce the existence results for Riemannian cubics

and critical points of higher order energy functions proved in [27] and [29].

6.4.2 Riemannian cubics in tension

We now consider the problem of optimal C1 interpolation of n given points on a

Riemannian manifold. If we were to specify velocities at each of the given points then

we can use piecewise Riemannian cubics, existence follows by setting A = 0 in the

previous section (or from [27]). But if we do not then the Palais-Smale condition

fails: suppose we have two points on a sphere and consider the sequence (�i) of

geodesics which pass through the points and wrap around the sphere i times. Each

geodesic is a minimum of J(x) = 1

2

krtxk20

, but there is no convergent subsequence.

The problem is that bounding J does not bound the length of curves unless an initial

velocity is specified, and without such a specification we can not prove J is weakly

proper. If we consider Riemannian cubics in tension instead then the cost function

J (x) := 1

2

(krtxk20

+ ⌧ 2kxk20

) (6.9)

dominates the length, and we will now show that the Palais-Smale condition holds.

More precisely, given points pi 2M, i = 0, . . . , n�1 and corresponding times ti 2 I,

we will prove that the Palais-Smale condition is satisfied when we restrict J to the

submanifold H2(I,M)p := F�1(p), where p = (p0

, . . . , pn�1

) and F : H2(I,M) !Mn, x 7! (x(t

0

), . . . , x(tn�1

)) is a submersion2. Thus we will prove existence of

solutions to what Hussein and Bloch refer to as the ⌧ -elastic variational problem

without motion constraints [34, 35].

Theorem 6.4.4. J satisfies the Palais-Smale condition on H2(I,M)p, therefore

attains its infimum on H2(I,M)p, and in any connected component there is a critical

point which minimises J with respect to the component.

2Moreover, the condition still holds if velocities are specified at the ti too.


Proof. Hk(I,M)p is a submanifold of Hk(I,M)p0 and the natural charts have the

submanifold property. The integrand krtxk2 + ⌧ 2kxk2 is strongly elliptic, and if

J (x) is bounded then so is kxk1

. Thus the result follows from Corollary 6.3.7.

Before applying Ljusternik-Schnirelman category theory we will relate the ho-

motopy type of H2(I,M)p to that of the based loop space ⌦M . For this we require

the following lemma.

Lemma 6.4.5. F : H2(I,M)p !Mn, x 7! (x(t0

), . . . , x(tn�1

)) is a fibration.

The proof will use a result of Earle and Eells [12]:

Proposition 6.4.6. (Earle and Eells [12]) Let X, Y be Finsler manifolds modelled

on Banach spaces, and suppose X is complete. Let f : X ! Y be a surjective map

which foliates X. Furthermore suppose: for every y 2 Y there is a neighbourhood V

and a number ⌘ > 0 such that for every x 2 f�1(V ) there is an sx 2 L(Tf(x)Y, TxX)

such that dfx � sx = 1Tf(x)Y , and |sx| ⌘. Then f is a locally C0-trivial fibration.

Proof. (Lemma 6.4.5). For each ti we choose an open interval B✏i

(ti) such that

\iB✏i

(ti) = ;, and such that x(B✏i

(ti)) is contained in the domain of a chart �i.

Let bi : S ! R be a smooth map such that bi(ti) = 1 and supp bi ⇢ B✏i

(ti). Given

v 2 TF(x)Mn we define

(sxv)(t) =

(bi(ti)d�i(x(t))�1d�i(x(ti))vi t 2 B✏

i

(ti)

0 elsewhere

Then dFx � sxv = v and |sx| is bounded.

Proposition 6.4.7. The inclusion Hk(I,M)p ! C0(I,M)p is a homotopy equiva-

lence.

Proof. It is well known that C0(I,M)! Mn is a fibration when n = 2 (eg. [77] p.

98), and a similar proof shows it is a fibration for any n. The inclusion Hk(I,M)!C0(I,M) is a homotopy equivalence by a theorem of Palais ([65] Theorem 13.14).

If we apply the five lemma to the homotopy sequence for the fibrations

Hk(I,M) ��! C0(I,M)??y

??y

Mn Mn

we see that inclusion of the fibres Hk(I,M)p ! C0(I,M)p induces isomorphisms of

homotopy groups, and is therefore a homotopy equivalence (see eg. [77] p. 405).

6.4. EXAMPLES 82

Corollary 6.4.8. Hk(I,M)p has the same homotopy type as the (n�1)-fold carte-

sian product of the based loop space: (⌦M)n�1.

Proof. C0(I,M)p is homeomorphic to C0(I,M)p0,p1 ⇥ . . . ⇥ C0(I,M)pn�2,pn�1 , and

each C0(I,M)p,q has the same homotopy type as ⌦M .

Corollary 6.4.9. The restriction of J to H2(I,M)p has at least cat((⌦M)n�1)

critical points. If M is not contractible then there are infinitely many critical points.

6.4.3 Closed cubics in tension

Suppose now that we would like to minimize J over H2(S1,M). Since there is no

fixed point that each x 2 H2(S1,M) must pass through, it is not possible to use

the argument from Theorem 6.3.6 to show that J is weakly proper on H2(S1,M).

Just as in the case of geodesics (see [67] p. 251), it is necessary to assume that M

is compact.

Theorem 6.4.10. If M is compact then J : H2(S1,M) ! R satisfies the Palais-

Smale condition.

Proof. From Theorem 6.3.1 we have that J is locally bounding and locally coercive

on H2(S1,M) with respect to Ck�1(S1,M). Suppose U ⇢ H2(S1,M) is such that

J (U) is bounded. Then U is equicontinuous by Lemma 6.3.3 and U(I) ⇢ M

is bounded because M is compact. Thus U is relatively compact in C0(S1,M)

by the Arzela-Ascoli theorem, and by Corollary 6.3.5 U is relatively compact in

C1(S1,M).

Corollary 6.4.11. J attains its infimum on H2(S1,M), and in any connected com-

ponent there is a critical point which minimises J with respect to the component.

Corollary 6.4.12. There are at least cat(C0(S1,M)) critical points of J , and if

M is simply connected and not contractible then there are infinitely many critical

points.

Proof. H2(S1,M) has the same homotopy type as the free loop space C0(S1,M)

([65] Theorem 13.14). If M is simply connected and not contractible then the free

loop space has infinite category [20].

CHAPTER 7

Morse theory for elastica

This chapter has been published as [72]. Minor changes have been made in order to

make it consistent with previous chapters.

Abstract

In Riemannian manifolds the elastica are critical points of the restric-

tion of total squared geodesic curvature to curves with fixed length which

satisfy first order boundary conditions. We verify that the Palais-Smale

condition holds for this variational problem, and also the related prob-

lems where the admissible curves are required to satisfy zeroth order

boundary conditions, or first order periodicity conditions. We also prove

a Morse index theorem for elastica and use the Morse inequalities to give

lower bounds for the number of elastica of each index in terms of the

Betti numbers of the path space.

7.1 INTRODUCTION

The natural generalisation to Riemannian manifolds of the classical elastica problem

studied by Euler and Bernoulli (see [44] or [80] for a historical survey) is the following:

find critical points of the restriction of

F (x) :=

Z `

0

k2(x)ds =

Z1

0

k2kxkdt (7.1)

to the set ⌦`T of immersed curves which have prescribed initial and final points, initial

and final tangent directions, and length `. Here x : I = [0, 1] ! M is a su�ciently

regular curve with length ` on a complete Riemannian manifold M , k(x) = krTTkis the geodesic curvature of x and T = x

kxk the unit tangent vector.

We will use the term elastica to refer to critical points of F |⌦`T . Among the spe-

cial cases of elastica we distinguish pinned elastica and closed elastica as those which

83


are critical subject to zeroth order boundary conditions and first order periodicity

conditions respectively. These are special cases in the sense that they satisfy the

same di↵erential equation with special boundary conditions (see Section 7.2), but of

course not in the sense that they are special cases of the same variational problem

(i.e. the pinned and closed constraints are not special cases of the constraints in

the original problem). In the absence of the length constraint, the resulting critical

curves are known as free elastica. Note that the terms elastica and elastic curve are

often used interchangeably, but the latter is somewhat equivocal so we will avoid it.

In modern times elastica have reappeared in several di↵erent contexts. In ap-

proximation theory they are known as nonlinear splines : a mathematical model for

the drafting tool known as a spline (see eg. [43, 31, 46]). The better known cubic

splines are used for ease of computation, not because they are a good approxima-

tion to drafting splines. Elastica also appear as a model for curve completion in

computer vision [57], and as an important example of an optimal control problem

with nonholonomic constraints [36]. As a consequence of the variety of applications,

the problem has been approached from several di↵erent perspectives. For example

Bryant and Gri�ths used the theory of exterior di↵erential systems to prove a partial

integrability result in homogeneous spaces and study solutions of the Euler-Lagrange

equations for elastica in the Euclidean and hyperbolic planes [8]. At about the same

time, Langer and Singer obtained similar results using Frenet frames and elliptic

functions [40]. Jurdjevic has shown that the Euler-Lagrange equations are com-

pletely integrable in surfaces of constant curvature using techniques from geometric

control theory [36, 37]. This includes in particular elastica on SO(3), by way of the

double cover by S3. Popiel and Noakes also studied elastica in Lie groups, reducing

the problem to the Lie algebra and solving for elastica in SO(3) by quadratures [69].

In [40] Langer and Singer also studied stability properties of the negative gradient

flow of F , which they later termed the curve straightening flow. They proved that

the only stable closed free elastica in S2 are non-trivial closed geodesics. This

motivated the study of the curve straightening flow on Riemannian manifolds as a

method of finding non-trivial closed geodesics. In [41] they showed that this flow is

well behaved on closed curves in R3 with fixed length. They also proved that for

almost all initial curves the flow approaches a circle, i.e. the circles are the only

stable closed elastica in R3. Subsequent work on the curve-straightening flow was

carried out by Linner [47, 48]. Moreover, in [45] Linner investigates free elastica in

the Euclidean plane and gives some conditions for existence and non-existence.

There is also a considerable mathematical physics literature on free elastica and

other functionals depending on geodesic curvature. We mention [58], [5] and ref-

erences therein. However, these authors are mainly interested in finding explicit

85 CHAPTER 7. MORSE THEORY FOR ELASTICA

solutions to initial value problems in semi-Riemannian manifolds, usually without

constraints on the length or speed.

7.1.1 The Palais-Smale condition for total squared curvature

The natural domain for the total squared geodesic curvature F is the set of C1

immersions with square integrable second covariant derivative along the curve. This

set, which we will denote Imm2(I,M), is an open submanifold of the Hilbert manifold

H2(I,M) consisting of C1 curves with square integrable second covariant derivative.

The space ⌦`T of admissible curves can be given the structure of a submanifold of

Imm2(I,M) defined by the boundary conditions and the length constraint.

Let (xi) ⇢ ⌦`T be a sequence which is minimizing for F , i.e. F (xi) ! inf F . It

is possible to prove1, as in [37] p. 17, that such a sequence has a weakly convergent

subsequence and therefore a limiting curve of class H2 exists.

There are good reasons for wanting to prove stronger convergence results, such

as the Palais-Smale (PS) condition. In general, a PS sequence for a smooth real

valued function f on a complete Hilbert manifold X is a sequence (xi) of points on

which f is bounded and |dfxi

| ! 0, and we say f satisfies the PS condition if any

PS sequence has a (strongly) convergent subsequence. In particular, for the elastica

problem this means that a PS sequence for F |⌦`T must have a subsequence which

converges in the H2 metric. If the PS condition holds then the associated negative

gradient flow is a positive semi-group and has at least one critical point as a limit

point (cf. [67] p. 183). We note that this condition provides a semi-constructive

proof of the existence: it ensures that the method of gradient descent will locate

critical points. Moreover, the PS condition makes available the minimax and Morse

theoretic methods of counting critical points.

In this chapter we will verify the PS condition for the elastica and pinned elastica

variational problems on any complete Riemannian manifold M . For closed elastica

we will do so under the additional assumption that M is compact. The relationship

between these and the earlier results of [42] will be discussed at the end of this

section. We will also prove a Morse index theorem for elastica and use the Morse

inequalities to give lower bounds for the number of elastica in terms of the Betti

numbers of the appropriate path space.

It is not possible to prove that F satisfies the PS condition on its natural domain

Imm2(I,M) because F is invariant under reparametrization, and the orbits of the

action of reparametrization on immersed curves are not compact. It follows that

any critical point is contained in a non-compact orbit of critical points at the same

1This was pointed out by one of the reviewers.


level of F , which contradicts the PS condition. Several methods of resolving this

kind of problem are discussed in [67] p. 245, with regard to the length functional.

One of these methods is to find a second function which ‘breaks the symmetry’,

meaning it is not invariant under reparametrization, but has the same critical points

as the original function in the following sense: each critical point of the symmetry

breaking function is a critical point of the original function and each orbit of critical

points of the original function contains a critical point of the symmetry breaking

function. This is the preferred method for the length function; the energy function

E =R

1

0

kxk2dt is not parametrization invariant and it is well known that the critical

points of E (geodesics) are arc-length proportionally parametrized critical points of

the length.

We show in Appendix A.3 thatRk4kxk2dt is a symmetry-breaking function for

F in the sense just described. However, it is still only defined for immersions, which

form an open subset of H2(I,M), and therefore has a non-complete domain.

An alternative which is also discussed in [67] is to impose a so-called ‘gauge fixing

condition’ to define a smooth submanifold of the domain which intersects each of

the orbits only once. This method turns out to be the most appropriate for F . The

condition we choose is that the curves should have constant speed2 v. This leads

to a neat simplification of F : on the submanifold ⌃v of constant speed curves F

coincides with the total squared covariant acceleration:

J(x) := 1

2

Z1

0

hrtx,rtxidt = v2

2

F

We show in Lemma 7.2.2 that a curve x which is parametrized proportional to

arc length is a critical point of F |⌦`T if and only if it is a critical point of J |⌃v with

v = `. We therefore carry out all our analysis on J |⌃v.

Langer and Singer have proved related results in [42] but with a di↵erent ob-

jective. They aim to prove that the curve straightening flow on closed curves is

well behaved by showing that F satisfies the PS condition. The parametrization in-

variance of F makes this impossible, so Langer and Singer restrict F to normalised

curves: those parametrized proportional to arc length. They are not interested in

fixing the length, because the curve straightening flow is intended to be used to find

closed non-trivial geodesics whose length may not be known in advance.

The restriction of F to normalized curves still does not satisfy the PS condition

because a subset of curves on which F is bounded does not necessarily have bounded

2We will frequently refer to the speed etc. of a curve, even though strictly speaking this is onlyappropriate when elastica are considered from a dynamical point of view; classically the elasticaproblem is one of shape or equilibrium position, not dynamics.


length (in the terminology of Elıasson, see Section 6.2.1, this means that F is not

weakly proper). Thus we have the counterexamples to the Palais-Smale condition

mentioned in [47] §1.7: the sequence xn of geodesics wrapping around the sphere n

times (we also mentioned this in Section 6.4.2), and circles in the plane with radii

increasing without bound. The sequence of geodesics wrapping around the sphere

is a counterexample to the Palais-Smale condition but it is not an example of a

curve straightening trajectory that has no convergent subsequence. It is still an

open question whether such an example exists on a compact manifold. The success

of Linner [48] in numerically generating periodic geodesics in sphere-like surfaces

seems to suggest that the curve-straightening flow may in fact be convergent in this

case.

To circumvent the di�culties outlined above, Langer and Singer consider the

modified function

F ↵ :=

Z1

0

(k2 + ↵)kxkdt

instead, with ↵ assumed to be positive so that F ↵ bounds the length. They prove

that F ↵, ↵ > 0 satisfies the PS condition on manifolds of closed, normalized curves

on compact Riemannian manifolds. They also remark that their techniques can be

used to prove that F |⌦`T satisfies the PS condition, thus there is some overlap with

our Theorem 7.5.7. Nevertheless it seems worth providing a detailed treatment, par-

ticularly since the techniques used in this paper lend themselves to the development

of a Morse index theorem for elastica.

7.2 LAGRANGE MULTIPLIERS AND ELASTICA

We have already mentioned that the natural domain of F is the set Imm2(I,M),

which is an open submanifold of H2(I,M). The length function L : H2(I,M) !R, L(x) =

R1

0

kxkdt and the speed function ⌫ : H2(I,M) ! H1(I,R), ⌫(x) = kxkare both di↵erentiable on Imm2(I,M).

Suppose p, q 2 M , v 2 TpM and w 2 TqM . We will make use of the following

7.2. LAGRANGE MULTIPLIERS AND ELASTICA 88

subsets of H2(I,M)

H2(I,M)p,q := {x 2 H2(I,M) : x(0) = p, x(1) = q}

H2(I,M)v,w := {x 2 H2(I,M)p,q : x(0) = v, x(1) = w}

H2(I,M)c := {x 2 H2(I,M) : x(0) = x(1), x(0) = x(1)}

Imm2(I,M)⇤ := Imm2(I,M) \H2(I,M)⇤

⌦`⇤ := {x 2 Imm2(I,M)⇤ : L(x) = `}

⌃v⇤ := {x 2 H2(I,M)⇤ : ⌫(x) ⌘ v}

Imm2(I,M)T := {x 2 Imm2(I,M) : T (0) = v, T (1) = w}

⌦`T := {x 2 Imm2(I,M)T : L(x) = `}

where `, v are positive real numbers, v, w are unit vectors, and ⇤ denotes either

(p, q), (v,w), c or void. The space H2(I,M)⇤ is a closed submanifold of H2(I,M)

and Imm2(I,M)⇤ is an open submanifold of H2(I,M)⇤. For now we will assume

that ⌦`⇤ and ⌃v⇤ are submanifolds of Imm2(I,M)⇤ and H2(I,M)⇤ obtained as the

pre-images of regular values ` and v respectively. In section 7.4 we will prove that

this is true for ⌃v⇤ under the assumption that it contains no geodesics.

To begin with we work with the following version of the Lagrange multiplier

theorem which is similar to that in [1] p. 211.

Theorem 7.2.1. (Lagrange multiplier theorem) Let X be a Banach manifold, E a

Banach space and f : X ! R, � : X ! E di↵erentiable maps. Suppose e0

2 E is a

regular value of � so that ⌦ := ��1(e0

) is a submanifold of X, with Tx⌦ = ker d�x

split in TxX, and denote f := f |⌦. Then the following are equivalent for x 2 ⌦:

• x is a critical point of f : ⌦! R

• there is a � 2 E⇤ such that x is a critical point of f � (�,�) : X ! R

Proof. Suppose x is a critical point of f and let � 2 E⇤ be such that �(e) := dfxV

for any V 2 d��1

x e, e 2 E. To see that � is well defined let V 0 2 d��1

x e also, then

V � V 0 2 ker d�x = Tx⌦ and therefore dfx(V 0 � V ) = 0, since x is a critical point of

f . Now (dfx � �d�x)V = 0 for all V 2 TxX. Conversely, if x is a critical point of

f � � � � for some � 2 E⇤ then (dfx � �d�x)V = 0 for all V 2 TxX, which implies

dfxV = 0 for all V 2 Tx⌦.

For elastica we have the following

⌦`⇤ Imm2(I,M)⇤ R

RF |⌦`

⇤F

L


By the Lagrange multiplier theorem x is a critical point of F |⌦`⇤ i↵ there exists

� 2 R such that x is a critical point of

F � :=

Z1

0

k2kxkdt� �

Z1

0

kxkdt =Z `

0

(k2 � �)ds

with domain Imm2(I,M)⇤. From [40] the derivative of F � in arc length propor-

tional parametrization is (taking account of the di↵erent sign given to the Lagrange

multiplier)

dF �x W =

Z `

0

h2r3

TT +rT (3k2 + �)T + 2R(rTT, T )T,W ids

+ [2hrTW,rTT i � hW, 2r2

TT + (3k2 + �)T ]`0

(7.2)

where the higher derivatives of T are understood as weak derivatives. As in [40],

supposing W |t=0,1 = rTW |0,1 = 0, setting dF �

x W = 0 and using the fundamental

lemma of calculus of variations gives the Euler-Lagrange equation

2r3

TT +rT (3k2 + �)T + 2R(rTT, T )T = 0 (7.3)

The fact that weak solutions of this equation are also strong solutions, i.e. the higher

derivatives of T are actually continuous, is a consequence of the regularity theory of

elliptic operators. Alternatively, one can use a so-called bootstrap argument based

on the Du Bois-Reymond lemma to show inductively that weak solutions of (7.3)

are in fact smooth (see Appendix A.4).

The Euler-Lagrange equation (7.3) is not always equivalent to d(F |⌦`⇤)x = 0 or

d(F �| Imm2(I,M)⇤)x = 0. When we consider F |⌦`T the boundary terms in (7.2)

vanish automatically because the tangent space Tx⌦`T consists of fields W along x of

classH2 which satisfyW |t=0,` = 0 andrTW |t=0,` = 0. However, for x to be a critical

point of F |⌦`p,q, we require in addition to (7.3) that x satisfy the natural boundary

conditions rTT |0,` = 0 in order for the boundary terms in (7.2), and therefore dF �x ,

to vanish for all W 2 Tx⌦`p,q. This is the precise sense in which pinned elastica

are a special case of elastica; both satisfy the Euler-Lagrange equation, but pinned

elastica necessarily have vanishing acceleration on the boundary, whereas elastica

do not.

For F |⌦`c, the tangent space Tx⌦`c consists of those fields W along x which satisfy

W (0) = W (`) and rTW (0) = rTW (`). Then from (7.2) the natural boundary

conditions are rTT (0) = rTT (`) and r2

TT (0) = r2

TT (`) (note that T (0) = T (`)

is automatic from the definition of ⌦`c). It then follows from (7.3) and derivatives

thereof that a critical point satisfies rkTT (0) = rk

TT (`) for any k, i.e. the critical

7.2. LAGRANGE MULTIPLIERS AND ELASTICA 90

points are C1-periodic.

Applying the Lagrange multiplier theorem to the restriction of total acceleration

J |⌃v⇤

⌃v⇤ H2(I,M)⇤ H1(I,R)⇤

RJ |⌃v

⇤J

⌫

we have that x 2 ⌃v⇤ is a critical point of J |⌃v

⇤ i↵ there is a � 2 H1(I,R)⇤ such that

x is a critical point of J�(�, ⌫). Equivalently, changing � to its Riesz representative

in H1(I,R), x is a critical point of

J� :=

Z1

0

1

2

hrtx,rtxi � �(t)kxk � � ddtkxkdt

for some � 2 H1(I,R). Writing ⇤ := �� (weakly) we have

dJ�xV =

Z1

0

hr3

t x+R(rtx, x)x+rt(⇤T ), V idt

+ [hrtV,rtxi � hV,r2

t x� ⇤T i]10 (7.4)

with Euler Lagrange equation (cf. [69])

r3

t x+R(rtx, x)x+rt(⇤T ) = 0. (7.5)

Lemma 7.2.2. A curve x 2 ⌦`T is an elastica parametrized proportional to arc

length i↵ x is a critical point of J |⌃vv,w with v = `.

Proof. If kxk = `, then equation (7.3) becomes

r3

t x+R(rtx, x)x+rt(3

2`2(krtxk2 + �)x) = 0

combining (7.5) with kxk = ` and derivatives thereof forces ⇤(t) = 3

2`ddtkrtxk2 (cf.

[69] for ` = 1). Integrating and substituting into (7.5) gives the same equation as

above, with � the constant of integration of ⇤.

Corollary 7.2.3. x 2 ⌦`p,q is a pinned elastica parametrized proportional to arc

length i↵ it is a critical point of J |⌃vp,q with v = `, and x is a closed elastica

parametrized proportional to arc length i↵ it is a critical point of J |⌃vc with v = `.

Proof. Follows from the previous lemma as well as the observation that when x is

parametrized proportional to arc length the natural boundary conditions obtained

from each of (7.2) and (7.4) coincide.


It is tempting now to prove that J� satisfies the PS condition, since then a

sequence (xi) in ⌃vv,w which is a PS-sequence for J�|H2(I,M) has a subsequence

which converges in H2(I,M)v,w to a critical point, and ⌃vv,w is closed in H2(I,M).

However this doesn’t prove that J := J |⌃v1

, or even J�|⌃v1

satisfies the PS condition,

since a PS sequence for J |⌃v1

need not be a PS sequence for J�: in general the

condition |dJxi

| ! 0 does not imply |dJ�xi

| ! 0 because the former has a smaller

domain. For multiplicity results, in particular Morse theory, we need to prove that

J itself satisfies the PS condition.

7.3 LAGRANGE MULTIPLIERS AND THE PS CONDITION

This section will serve as an outline of the method we will use to verify the PS

condition for elastica. We begin with the modifications to Eliasson’s method (section

6.2.1) which are necessary to deal with constraints such as fixed speed. We then

show how these modifications are related to the method of Lagrange multipliers,

and how to use Lagrange multipliers to characterise the nullspace of the Hessian in

the presence of such constraints.

Let ⌦ be a submanifold of X, which is in turn a weak submanifold of X0

. Note

that it is not necessarily true that ⌦ is also a weak submanifold of X0

. Of course at

any point in ⌦ there is a chart for X which restricts to a chart for ⌦, i.e. satisfies

the submanifold property, but in general it is not necessary that this chart is the

restriction of a chart for X0

, viz. a weak chart. For example, consider ⌃1: the

submanifold of H1(I,R2) consisting of unit speed curves in the Euclidean plane.

The natural charts exph �⇠ 7! ⇠, h 2 C1(I,R2) for H1(I,R2) are weak charts with

respect to C0(I,R2). Suppose r1

, r2

2 H1(I,R2) are parametrizations of the upper

and lower halves of a circle respectively, with unit speed and the same initial and

terminal points. Then in the natural chart centred at r1

, the representative of r2

is

r2

� r1

. However, expr11

2

(r2

� r1

) will not have unit speed. This means 1

2

(r2

� r1

) is

not in the local image of ⌃1, and therefore the natural charts for H1(I,R2) do not

satisfy the submanifold property for ⌃1.

For this reason the definitions from section 6.2.1 are not directly applicable and

require the following modifications.

Remark. In this and subsequent sections it will frequently be the case that we are

interested in bounding some quantity by a constant, but the precise value of the

constant is not important. It will therefore be convenient to use the symbol C to

denote a floating constant, i.e. it may change during a calculation but is nevertheless

independent of any variables.

Definition. Let X be a weak submanifold of X0

and ⌦ a submanifold of X (which is

7.3. LAGRANGE MULTIPLIERS AND THE PS CONDITION 92

not necessarily a weak submanifold of X0

), and suppose we have a smooth projection

prT⌦ : TX|⌦! T⌦. We will say prT⌦ is locally bounded with respect to (weak charts

from) X0

if for any !0

in the X0

-closure of ⌦, and any constant L, there is a weak

chart (✓, U) for X at !0

and a constant C such that k prT⌦(⇠)⌘kB Ck⌘kB for all

⇠ 2 ✓(U \ ⌦) with k⇠kB < L, and all ⌘ 2 B (we are adopting a standard abuse of

notation whereby prT⌦ is used to denote both the map and its local representative).

It then follows that | prT⌦(⇠)| C. We call f = f |⌦ locally coercive with respect to

(X,X0

) if for any !0

in the X0

-closure of ⌦, and any constant ↵, there is a weak

chart (U, ✓) at !0

and constants C+

> 0 and C such that

C+

k⇠ � ⌘k2B C|⇠ � ⌘|0

+ (Df(⇠) prT⌦(⇠)�Df(⌘) prT⌦(⌘))(⇠ � ⌘) (7.6)

for all ⇠, ⌘ in ✓(U \⌦) with k⇠kB, k⌘kB < ↵. Note the absence, when compared with

(6.2), of the square on |⇠ � ⌘|0

.

Theorem 7.3.1. Let f : X ! R be a smooth function, where X is a weak sub-

manifold of X0

with a locally bounded Finsler structure. Suppose also that ⌦ is a

submanifold of X with a smooth projection prT⌦ : TX|⌦ ! T⌦ which is locally

bounded with respect to X0

. Then if f := f |⌦ is weakly proper with respect to X0

,

f is locally bounding with respect to X0

, and f is locally coercive with respect to

(X,X0

), then f satisfies the Palais-Smale condition.

Proof. Let (xi) ⇢ ⌦ be a sequence for which f(xi) is bounded and |dfxi

|! 0. Since f

is weakly proper we can find a subsequence converging inX0

to some x0

. We choose a

weak chart ✓ : U ! B at x0

, with corresponding trivialisation ⇥ : ⌧�1(U)! B ⇥B

of the tangent bundle, and a subsequence ⇠i := ✓(xi). Then ⇠i is bounded in B

because f is locally bounding. Using the local coercivity of f we have

C+

k⇠i � ⇠jk2B C|⇠i � ⇠j|0

+ (Df(⇠i) prT⌦(⇠i)�Df(⇠j) pr

T⌦(⇠j))(⇠i � ⇠j))

C|⇠i � ⇠j|0

+ (|Df(⇠i)|+ |Df(⇠j)|)Ck⇠i � ⇠jkB(7.7)

where we have used the assumption that the projection is locally bounded. Moreover,

since the Finsler structure for X is locally bounded,

|Df(⇠i)⌘| = |df(xi)⇥�1

xi

⌘| |df(xi)|Ck⌘kB

i.e. |Df(⇠i)| C|df(xi)|. Finally, using the assumption |df(xi)| ! 0, and the

B0

-convergence of (⇠i), we have from (7.7) that (⇠i) is Cauchy in B and then the

corresponding subsequence (xi) converges in ⌦ because it is closed in X.


We consider again the setting of Theorem 7.2.1:

⌦ X E

R˜f

f

�

where e0

2 E is a regular value, meaning Tx� is surjective and has a split kernel for

all x 2 ⌦ := ��1(e0

). Then �⇤TE = X ⇥ E and we have a short exact sequence of

VB morphisms

0 T⌦ TX|⌦ ⌦⇥ E 0�⇤T�

(7.8)

Suppose the above sequence admits a right split, i.e. a VB morphism such that

�⇤T� � r = Id⌦⇥E, or fibrewise: d�x � rx = IdE. Equivalently ([1] p. 183) we have

a splitting TX|⌦ ⇠= T⌦ � im r and a smooth projection prT⌦ : TX|⌦ ! T⌦ given

fibrewise by prT⌦(x) = 1� rxd�x.

Lemma 7.3.2. The sequence (7.8) admits a local right split at any x 2 ⌦.

Proof. Since � is a submersion at x 2 ⌦ = ��1(e0

) there is a chart ✓ : U ! B for X

at x such that:

- ✓(U) ⇠= U1

⇥ U2

⇢ B1

⇥ B2

⇠= B and �✓ : U1

⇥ U2

! E can be factored into

�✓ = � � pr1

: U1

⇥ U2

! U1

! V ⇢ E where � is a di↵eomorphism, and

- ✓|U \ ⌦! {��1(e0

)}⇥ U2

is a chart for ⌦.

Then since �✓(��1(e), w) = e for any e 2 V,w 2 U2

, we also have

D1

�✓(��1(e), w)D��1(e) = IdE .

So defining r✓ : U2

⇥E ! U2

⇥B1

⇥B2

by r✓(w, e) := (w,D��1(e0

)e, 0) gives a local

right split.

If X admits partitions of unity then a split can be constructed from such local

splits.

Lemma 7.3.3. Let f = f � (�,�� e0

) : X ! R, where � : X ! B⇤ is any smooth

map which satisfies �x = dfx � rx for all x 2 ⌦. Then dfxV = dfx prT⌦

x V for all

x 2 ⌦ and V 2 TxX, and therefore x is a critical point of f |⌦ i↵ it is a critical point

of f .

7.3. LAGRANGE MULTIPLIERS AND THE PS CONDITION 94

Proof. For any x 2 ⌦, V 2 TxX we have, with �(x) = e0

,

dfxV = dfxV � (�x, d�xV )� (d�xV,�(x)� e0

)

= dfx(V � rxd�xV ) = dfx prT⌦(x)V (7.9)

Remark. Note that in contrast with Theorem 7.2.1, where the Lagrange multiplier

is treated as an extra variable, � is now a function X ! B⇤ and is defined in

advance on all of ⌦ by a choice of r. This is not necessary in order to write down

Euler-Lagrange equations, but it is needed for Theorem 7.3.5.

Lemma 7.3.4. Let ⌦, X, r be as above with X a weak submanifold of X0

. Suppose

f : X ! R is locally coercive with respect to X0

, and that in a weak chart ✓, U for

X where (6.2) holds we also have

|Df(⇠)r(⇠)D�(⇠)⌘| C|⌘|0

(7.10)

for any ⌘ 2 B, whenever ⇠ 2 ✓(U \ ⌦) with k⇠kB C. Then f |⌦ is locally coercive

with respect to (X,X0

).

Proof. Since prT⌦

x = Id�rxd�x, for any ⌘ 2 B we have

Df(⇠)⌘ = Df(⇠)(prT⌦(⇠) + r(⇠)D�(⇠))⌘

when k⇠kB C. Then since f is locally coercive on X with respect to X0

, using

(7.10),

C+

k⇠ � ⌘k2B C|⇠ � ⌘|20

+ (Df(⇠)�Df(⌘))(⇠ � ⌘)

C|⇠ � ⌘|0

+ (Df(⇠) prT⌦(⇠)�Df(⌘) prT⌦(⌘))(⇠ � ⌘).

Recall that given a critical point x of f and vector fields V,W on X, the Hessian

Hessx f(V,W ) := WxV f is bilinear and symmetric in V and W , and depends only

on the vectors Vx,Wx. The Morse index of a critical point x of f is the dimension

of the maximal subspace on which Hessx f is negative definite. We say f is a

Morse function if Hess f is strongly nondegenerate at every critical point, i.e. if the

associated self-adjoint operator is an isomorphism.

Suppose f is a Morse function on a complete Riemannian manifold X which

satisfies the Palais-Smale condition. Let mi denote the number of critical points of


f with index i, and �i the ith Betti number of X. Then the weak Morse inequalities

state that �i mi (see [67] p. 220).

For f to be a Morse function it is necessary that the nullspace of Hessx f be

trivial. We observe that Vx 2 null Hessx f i↵ Hessx f(V,W ) = WxV f = 0 for all

Wx 2 TxX, i.e. if x is also a critical point of V f . Just as it is necessary to introduce

a Lagrange multiplier in order to write the condition dfx = 0 in strong form (i.e. as

a di↵erential equation), so it is also necessary to introduce a Lagrange multiplier in

order to characterise the nullspace of Hessx f .

Theorem 7.3.5. The following are equivalent statements for x 2 ⌦ a critical point

of f and Vx 2 Tx⌦:

• Vx is in the nullspace of Hessx f , that is, Hessx f(Vx, Y ) = 0 for all Y 2 Tx⌦.

• there exists µ 2 B⇤ such that

WV f � (�x,WV �)� (µ, d�xW ) = 0 (7.11)

for all W 2 TxX.

Proof. Suppose x is a critical point of f and Vx 2 Tx⌦, and let V be an extension

of Vx to X such that V (⌦) ⇢ T⌦. Then from equation (7.9) we see that

V f = (V f)|⌦

= (V f � (V �,�� e0

)� (�, V �))|⌦

= (V f � �(V �))|⌦

Now Vx is in null Hessx f i↵ x is a critical point of V f i↵ x is a critical point of

(V f � (�, V �))|⌦

, which, by the Lagrange multiplier theorem (7.2.1), is equivalent

to

0 = d(V f � (�, V �))xW � (µ, d�xW ) = WV f � (W�, V �)� (�,WV �)� (µ,W�)

for all W 2 TxX, where µ 2 B⇤, and the (W�, V �) term vanishes because V �(x) =

0.

7.4 MANIFOLDS OF CONSTANT SPEED CURVES

We will focus here on conditions which ensure that ⌃v⇤ is the preimage of a regular

value of ⌫ : Imm2(I,M)⇤ ! H1(I,R)⇤, where H1(I,R)⇤ is a suitable submanifold

of H1(I,R). According to Lemma 7.2.2 we will only need to work with ⌃v⇤, and the

proofs for ⌦`⇤ are similar.

7.4. MANIFOLDS OF CONSTANT SPEED CURVES 96

We recall that in order for x 2 Imm2(I,M) to be a regular point of ⌫ the

requirement is that d⌫x should be surjective and have split kernel. For Banach

spaces the latter is not automatic, however the kernel is a closed subspace so in a

Hilbert space it has a closed orthogonal complement, i.e. it splits.

Define the endpoint maps

P0

: H2(I,M)!M ⇥M, P0

(x) := (x(0), x(1))

P1

: H2(I,M)! TM ⇥ TM, P1

(x) := (x(0), x(0), x(1), x(1))

Each of these maps can be shown to be a submersion and then H2(I,M)⇤ is a sub-

manifold of H2(I,M) with TxH2(I,M)⇤ = ker dP⇤(x). For example, TxH2(I,M)v,w

consists of vector fields along x which satisfy V |0,1 = 0 and rtV |

0,1 = 0.

We begin with ⇤ as void. The derivative of ⌫ is d⌫(x)V = 1

kxkhrtV, xi, andevery x 2 Imm2(I,M) is a regular point of ⌫. Indeed suppose w 2 H1(I,R) and

let V 2 H2(x⇤TM) be any solution of rtV = wkxk x, then d⌫(x)V = w, i.e. d⌫(x) is

surjective. In particular, it follows that v is a regular value and thus ⌃v = ⌫�1(v) is

a submanifold of H2(I,M).

Next consider the restriction ⌫p := ⌫| Imm2(I,M)p. Every x 2 Imm2(I,M)p is a

regular point because for any w 2 H1(I,R) there is a solution of rtV = wkxk x with

V (0) = 0 and therefore V 2 TxH2(I,M)p.

For ⌫0 := ⌫| Imm2(I,M)p,q the situation is more complicated because an element

V 2 TxH2(I,M)p,q must satisfy V |t=0,1 = 0. For a given w 2 H1(I,R), in order to

construct such a V which is also in the pre-image d⌫(x)�1w, we look for a solution

of

rtV = wkxk x+

n�1X

i=1

uiEi (7.12)

where {Ei(t)} is a collection of vector fields along x which span the orthogonal

complement of x, and the ui 2 H1(I,R) are functions which we are free to choose.

Equation (7.12) represents a linear time dependent control system (see e.g. [7]). Such

a system is called controllable on [0, 1] if for any initial state V (0) and any V1

2 TqM

there exist controls ui and a corresponding solution V such that V (1) = V1

. If this

system is controllable then x is a regular point of ⌫0. In order to write (7.12) in a

more familiar form we work in an orthonormal parallel frame {ek} along x so that

V = V

kek, x = x

kek, Ei = E

ki ek with repeated indices summed and V, x,Ei 2 Rn.

Then we can write (7.12) as

V = wkxk x+Bu (7.13)

where u 2 H1(I,Rn�1) and B is the n⇥ (n�1) matrix with the coordinates of Ei in

the ith column. We address the question of controllability as follows. First consider


the linear time dependent control system

b = Bu (7.14)

Suppose (7.14) is controllable and a is a solution of a = wkxk x. Then given V

0

,V1

there exists u and a corresponding b such that b(0) = V

0

�a(0), b(1) = V

1

�a(1),

so that V = a + b is a solution to (7.13) with V(0) = V

0

and V(1) = V

1

. Thus

controllability of (7.13) is equivalent to controllability of (7.14). A necessary and

su�cient condition for (7.14) to be controllable on [0, 1] is that the matrix

W :=

Z1

0

B(t)B(t)Tdt,

should be non-singular, in which case a particular control which drives the solution

to b(1) = b

1

is given by u = BTW�1(b1

� b

0

), ([7] p. 76). If W is singular then

there exists a non-zero y 2 Rn such that

y

TWy =

Z1

0

y

TB(yTB)Tdt = 0

Then y

TB(t) = 0 almost everywhere on I. This is only possible if there exists a real

valued function ↵ such that y = ↵(t)x, and then since y is constant ↵x + ↵x = 0,

i.e.

↵x+ ↵rtx = 0 (7.15)

Since we have assumed ↵ 6= 0 it then follows that x is a regular point of ⌫0 if it is

not a reparametrized geodesic.

In particular if ⌫(x) ⌘ v then hrtx, xi = 1

2

ddtkxk2 = 0 and then (7.15) holds i↵

x is a geodesic. Thus if there are no geodesics joining p, q with constant speed v

(and therefore length L(x) =R

1

0

vdt = v) then v is a regular value of ⌫0 and ⌃vp,q is

a submanifold of H2(I,M)p,q.

Next we characterise regular points of the restriction ⌫1 := ⌫| Imm2(I,M)v,w !H1(I,R)kvk,kwk. The codomain is now a submanifold of H1(I,R) with tangent space

H1(I,R)0,0 = {w 2 H1(I,R) : w|t=0,1 = 0}, and an element V of the tangent space

of the domain must satisfy V |t=0,1 = 0 and rtV |t=0,1 = 0. Therefore instead of

(7.12) we look for solutions of

rtV = wkxk x+

n�1X

i=1

ui�(t)Ei (7.16)

where � : I ! R is any smooth function which satisfies �(0) = 0 = �(1) and oth-

7.5. THE PALAIS-SMALE CONDITION FOR ELASTICA 98

erwise �(t) 6= 0. Then since w(0) = 0 = w(1), any solution of (7.16) automatically

satisfies rtV |0,1 = 0. Moreover the system (7.16) is controllable by precisely the

same argument as above, and therefore x is a regular point of ⌫1, provided there is

no solution to (7.15). In particular, it follows that if there are no geodesics in ⌃vv,w

then it is a submanifold.

Finally, let us consider the restriction ⌫c := ⌫| Imm2(I,M)c ! H1(I,R)c =

P�1

0

(diagR2), where the codomain is the submanifold of H1(I,R) consisting of pe-

riodic functions and the tangent space TxH2(I,M)c = {V 2 TxH2(I,M) : V (0) =

V (1),rtV (0) = rtV (1)}. In this case we again look for a solution of (7.16) but

now with x and w periodic, and then rtV is automatically periodic. Moreover, if

the system is controllable then we can set V (0) = V (1) and so x is a regular point.

Again it follows that if there are no geodesics in ⌃vc then it is a submanifold. We

summarize the required results from above in the following Lemma.

Lemma 7.4.1. For ⇤ =void or p, the constant v is a regular value of the restriction

⌫| Imm2(I,M)⇤, and therefore ⌃v⇤ = ⌫�1(v) is a submanifold of Imm2(I,M)⇤. If

⇤ = (p, q), (v,w) or c then the same is true provided there are no geodesics in ⌃v⇤.

Proof. See preceding discussion.

7.5 THE PALAIS-SMALE CONDITION FOR ELASTICA

Our goal in this section is to prove that J |⌃v⇤ satisfies the PS condition using Theorem

7.3.1. We will assume henceforth that v, (p, q), (v,w) where relevant, are such that

⌃v⇤ contains no geodesics and is therefore a submanifold by Lemma 7.4.1.

For ⌃vv,w, the short exact sequence corresponding to (7.8) is

0 T⌃vv,w T Imm2(I,M)v,w|⌃v

v,w ⌃vv,w ⇥H1(I,R)

0,0 0⌫⇤T⌫

(7.17)

and we will begin by constructing a right split r for the above sequence. For x 2 ⌃vv,w

and w 2 H1(I,R) we will define rxw as follows. We have already observed that a

solution V of (7.16) will satisfy d⌫xV = w. Setting rxw := V where V is a solution of

(7.16) with V |t=0,1 = 0 will then satisfy the desired property: d⌫xrx = Id. However

we haven’t specified the frame {Ei} for x? or the controls ui and so rx is not yet

well-defined.

First we will show how to construct a particular frame for x? for any x 2 ⌃vv,w.

Fix an orthonormal basis { 1

vv, ei} for TpM and solve

rtEi = � 1

vhEi,rtxix, Ei(0) = ei. (7.18)


Then 0 = hrtEi, xi + hEi,rtxi = ddthEi, xi and therefore hEi, xi = 0 because

hei,vi = 0. It also follows that hrtEi, Eji = 0, then ddthEi, Eji = 0 and hEi, Eji = �ij

where � is the Kronecker delta function. Hence (7.18) defines an orthonormal frame

{ 1

vx, Ei} along x.

We will use this adapted frame in (7.16) and as in (7.13) we (temporarily) work

in an orthonormal parallel frame along x and write (7.16) as

V = wvx+

n�1X

i=1

ui�Ei (7.19)

with V, x,E 2 Rn. We will also assume that � is normalised toR

1

0

�dt = 1. To

construct a solution to the above with V(0) = 0 = V(1) we first let a be the solution

to a = wvx with a(0) = 0. Then we look for a solution of ˙

b =P

i ui�Ei, such that

b(0) = 0 and b(1) = �a(1) and let V = a+ b. In matrix form b = Bu where

B := �[E1

. . .En�1

]. According to [7] p. 76, a control which drives b(0) = 0 to

b(1) = �a(1) is given by u = BT⌘ where ⌘ is any solution ofR

1

0

BBTdt⌘ = a(1).

Since the Ei are orthonormal and � is normalized we haveR

1

0

BBTdt = n � 1. So

we let ⌘ = 1

n�1

a(1) and then u = 1

n�1

BTa(1), i.e. ui =

�n�1

(Ei · a(1))Ei.

In covariant terms this means we define

rxw := a+ b (7.20)

where a and b are the solutions of

rta = wvx, a(0) = 0 (7.21)

rtb =n�1X

i=1

�n�1

hEi,P�1

1�ta1iEi, b(0) = 0, a1

= a(1) (7.22)

and where by Pt we mean parallel translation along x for time t beginning at p =

x(0).

In order to apply Theorem 7.3.1 we will need to prove that the projection induced

by r is locally bounded. This will require some estimates for krxwk2

.

First we estimate kak2

. From (7.21) we have krtak2 = w2. Then since a(0) = 0,

using the fundamental theorem of calculus and the Cauchy-Schwarz and Holder

inequalities gives

kak2 =Z t

0

dd⌧kak2dt = 2

Z t

0

hrta, aidt 2

Z1

0

|w|kakdt 2kak0

kwk0

(7.23)

from which we observe kak0

2kwk0

and also |a|0

2kwk0

. Di↵erentiating (7.21)


gives

r2

ta = wvrtx+ w

vx (7.24)

and therefore kr2

tak2 = (w

v)2krtxk2 + w2. Now overall we have

kak2

kak0

+ krtak0

+ kr2

tak0

Ckwk1

+ Ckwk0

krtxk0

(7.25)

From (7.22), the Cauchy-Schwarz inequality, and recalling that parallel transla-

tion gives isometries

krtbk2 = ( �n�1

)2X

i

hEi,P�1

1�ta1i2 Cka(1)k2 Ckwk20

where the last step uses the inequality |a|0

2kwk0

proved above. Then since

b(0) = 0 the same argument used for a gives kbk0

Ckwk0

. Di↵erentiating (7.22)

gives

r2

t b =�

n�1

X

i

(hrtEi,P�1

1�ta1iEi + hEi,P�1

1�ta1irtEi) (7.26)

From (7.18) krtEik2 = hEi,rtxi2 krtxk2, and also hrtEi, Eji = 0, so

kr2

t bk2

= ( �n�1

)2X

i

(hrtEi,P�1

1�ta1i2 + hEi,P�1

1�ta1i2krtEik2)

CkP�1

1�ta1k2

0

krtEik2 Ckwk20

krtxk2

Combining the preceding estimates for b shows that kbk2

Ckwk0

+ Ckwk0

krtxk0

,

which together with (7.25) yields

krxwk2

Ckwk1

+ Ckwk0

krtxk0

(7.27)

The next task is to prove that the projection prT⌃

v

v,w = Id�rxd⌫x, henceforthabbreviated to pr, is locally bounded with respect to (H2, C1). For this we will need

to infer bounds on the local expression for pr in a trivialisation induced by a weak

chart, from bounds obtained in tangent spaces, such as (7.27) above. In order to do

so we require the following auxiliary lemmas.

Lemma 7.5.1. Let ✓h, Uh be the natural chart for H2(I,M) centred at h. Then

for any constant c1

there is a constant c2

such that x = ✓�1

h (⇠) satisfies kxk1

c2

whenever ⇠ 2 ✓h(Uh) with k⇠k2

c1

.

Proof. From [15] Theorem 11 (or equation (6.5)) the local expressions for x and rtx

with respect to the induced trivialisation ⇥h are @h⇠ = rt⇠ +Q1

(⇠) and (rt@)h⇠ =

r2

t ⇠ + Q2

(⇠) where Q1

, Q2

are polynomial di↵erential operators of order 0 and 1


respectively. Using these local expressions, and the fact that the Finsler structure

on H0(H2(I,M)⇤TM) is locally bounded (cf. Lemma 6.2.2) :

kxk20

k⇥h(⇠, @h⇠)k20

Ck@h⇠k20

Ck⇠k21

and similarly krtxk20

Ck(rt@)h⇠k20

Ck⇠k22

.

Lemma 7.5.2. Let ⇥h be the local trivialisation for TH2(I,M) induced by the

natural chart ✓h, Uh. Then for any constant c1

there is a positive constant c+

such

that for any ⌘ 2 H2(h⇤TM), we have that v = ⇥�1

h (⇠, ⌘) satisfies kvk2

� c+

k⌘k2

whenever k⇠k2

c1

.

Proof. Similar to the proof of Lemma 6.2.2, we have:

kvk22

=2X

i=0

Z

I

g(x)(ritv,ri

tv) =2X

i=0

Z

I

g(h)(G(⇠)(rit)h⌘, (ri

t)h⌘) � C+

k⌘k22

using the fact that G(⇠) is positive definite, the assumption that |⇠|0

is bounded,

and the local formula (6.5) for (rit)h.

Proposition 7.5.3. The projection pr : TH2(I,M)v,w ! T⌃vv,w, obtained from the

right split r (7.20) as pr = 1� rxd⌫x, is locally bounded with respect to C1(I,M).

Proof. From d⌫xV = 1

vhrtV, xi we have d

dt(d⌫xV ) = 1

v(hr2

tV, xi + hrtV,rtxi) andthen

kd⌫xV k 1

vkrtV kkxk = krtV k

k ddt(d⌫xV )k kr2

tV k + 1

vkrtV kkrtxk

Thus kd⌫xV k1

kr2

tV k0

+ CkrtV k0

(1 + krtxk0

) and using (7.27):

krxd⌫xV k2

Ckd⌫xV k1

+ Ckd⌫xV k0

krtxk0

Ckr2

tV k0

+ CkrtV k0

(1 + 2krtxk0

)(7.28)

Let ✓h, Uh be the natural chart for H2(I,M) centred at h. Then for any ⇠ 2✓h(Uh) with k⇠k

2

C, writing x = ✓�1

h (⇠), we have by Lemma 7.5.1 that kxk1

c

and so from (7.28) krxd⌫xV k2

CkV k2

. Thus

kprx V k2

= k(1� rxd⌫x)V k2

kV k2

+ krxd⌫xV k2

CkV k2

(7.29)

Now we will write pr(⇠)⌘ = ⇥h(x, prx V ), where ⇥h is the local trivialisation corre-


sponding to ✓h, and ⌘ = ⇥h(x, V ), then by Lemma 7.5.2 and (7.29) we have

kpr(⇠)⌘k2

Ckprx V k2

CkV k2

Ck⌘k2

where we have also used the local boundedness of the Finsler structure on TH2(I,M).

Since the above inequality has been shown to hold for any ⇠ 2 ✓h(Uh) such that

k⇠k2

C, we have shown that pr is locally bounded.

Proposition 7.5.4. J |⌃vv,w is locally coercive with respect to (H2, C1).

Proof. The derivative of J at x 2 H2(I,M) is

dJxV = hr2

tV,rtxi0

+ hR(rtx, x)x, V i0

so for any V 2 TxH2(I,M)v,w and x 2 ⌃vv,w,

dJxrxd⌫xV = hr2

t (rxd⌫xV ),rtxi0

+ hR(rtx, x)x, rxd⌫xV i0

(7.30)

From (7.24) and (7.26)

r2

t (rxw) =wvrtx+ w

vx+ �

n�1

X

i

(hrtEi,P�1

1�ta1iEi + hEi,P�1

1�ta1irtEi)

and therefore, recalling from (7.18) that hrtEi,rtxi = 0 and hEi,rtxi = �hrtEi, xi

hr2

t (rxd⌫xV ),rtxi = 1

v2hrtV, xikrtxk2 � �

n�1

X

i

hrtEi,P�1

1�ta1ihrtEi, xi

Now using the estimate for |a|0

from (7.23), kP�1

1�ta1k = ka1

k |a|0

Ckd⌫xV k0

CkrtV k

0

. Moreover from (7.18) we have krtEik krtxk, hence

hr2

t (rxd⌫xV ),rtxi0

Z

1

0

1

v2hrtV, xikrtxk2 + CkrtV kkrtxk2dt

C|rtV |0

krtxk20

Using the bounds obtained for kak0

, kbk0

we have krxd⌫xV k0

Ckd⌫xV k0

CkrtV k0

,

and then from (7.30)

|dJxrxd⌫xV | C|rtV |0

krtxk20

+ kR(rtx, x)xk0

krtV k0

(7.31)

Now suppose we work in a natural chart (✓h, Uh) centred at h. Then for any ⇠ 2�hUh with k⇠k

2

C we have from Lemma 7.5.1 that x := ��1

h ⇠ satisfies kxk1

C.

Moreover, x(I) is contained in a compact subset of M because the length and x(0) =


p are fixed. Thus, from (7.31) we have |dJxrxd⌫xV | C|V |1

, and then locally, for

any ⌘ 2 H2(h⇤TM),

|DJh(⇠)rh(⇠)D⌫h(⇠)⌘| = |dJxrxd⌫x⇥�1

h (⇠, ⌘)| C|⇥�1

h (⇠, ⌘)|1

C|⌘|1

because the Finsler structure |.|1

is locally bounded by a very similar argument to

the proof of Lemma 6.2.2. Since J is locally coercive with respect to C1 by Theorem

6.3.1, the result now follows from Lemma 7.3.4.

Lemma 7.5.5. ⌃v⇤ is a compact subset of C0(I,M) for ⇤ = (p, q), (v,w), c, provided

M is compact in the case ⇤ = c.

Proof. ⌃vp,q is equicontinuous by Lemma 6.3.3, and since each x 2 ⌃v

p,q has length

v and a fixed initial point there exists a closed and bounded K ⇢ M such that

x(I) ⇢ K for all x 2 ⌃vp,q. K is compact by the Hopf-Rinow theorem and therefore

⌃vp,q is pointwise relatively compact (i.e. given a sequence (xi) ⇢ ⌃v

p,q and fixed t1

,

(xi(t1)) has a convergent subsequence). Hence by the Arzela-Ascoli theorem ⌃vp,q is

a compact subset of C0(I,M), which contains ⌃vv,w as a closed subset. For ⌃v

c the

initial point is not fixed and so we assume that M is compact in this case.

Lemma 7.5.6. The restriction J |⌃v⇤ is weakly proper with respect to C1(I,M),

provided M is compact in the case ⇤ = c.

Proof. If U ⇢ ⌃v⇤ then U is relatively compact in C0(I,M) by Lemma 7.5.5. Fur-

thermore if J(x) is bounded for all x 2 U we have kxk21

= v2 + 2J(x) also bounded

and U is relatively compact in C1(I,M) by Corollary 6.3.5.

Theorem 7.5.7. J |⌃vv,w satisfies the Palais-Smale condition.

Proof. Recalling that ⌃vv,w is the inverse image of a regular value of the speed

⌫1 : Imm2(I,M)v,w ! H1(I,R) (Lemma 7.4.1) and Imm2(I,M)v,w is open in

H2(I,M)v,w which is in turn a weak submanifold of C1(I,M) (Lemma 6.3.2), we

check the conditions of Theorem 7.3.1:

- By Proposition 7.5.3, pr is locally bounded with respect to C1

- J |⌃vv,w is weakly proper with respect to C1 by Lemma 7.5.6.

- J is locally bounding with respect to C1 by Theorem 6.3.1

- J |⌃vv,w is locally coercive with respect to (H2, C1) by Proposition 7.5.4.


We now consider the pinned elastica, i.e. J |⌃vp,q. In this case it is not possible

to use exactly the same right split r because (7.18) required a fixed initial adapted

basis for TpM but we are now allowing the direction of x(0) to vary. Moreover, by

the hairy ball theorem a global smoothly x(0)-dependent choice of adapted basis for

TpM may be impossible. Fortunately, as we will see below, a global definition will

not be needed. We define r0 in a C1 neighbourhood of x0

2 ⌃vp,q as follows. Suppose

x0

(0) = v0

and let U be a neighbourhood of v0

in the sphere of radius v in TpM such

that the orthonormal frame bundle is trivial over U . Fix a smooth section f of the

orthonormal frame bundle over U . Then for any x 2 ⌃vp,q with x(0) 2 U solve (cf.

(7.18))

rtFi = � 1

vhFi,rtxix, Fi(0) = fi(x(0))

to obtain an adapted orthonormal frame { xv, Fi} along x adapted to x. Then as

before we define r0x by (7.20)-(7.22) (although � is actually no longer needed), but

now using Fi instead of Ei.

Theorem 7.5.8. J |⌃vp,q satisfies the PS condition.

Proof. Let (xi) ⇢ ⌃vp,q be a PS sequence for J |⌃v

p,q. Then since J |⌃vp,q is weakly

proper with respect to C1(I,M) by Lemma 7.5.6, there is a subsequence, still de-

noted (xi), such that (xi) converges in C1 to x0

2 C1(I,M). We may therefore

choose a natural chart ✓h, U centred at h 2 C1(I,M) and containing x0

, and a

subsequence (xi) ⇢ U with ⇠i := ✓(xi). If necessary we may then further restrict

attention (and take a further subsequence) to a subet U 0 ⇢ U such that for any

x 2 U 0, x(0) is contained in a neighbourhood of v0

:= x0

(0) in the sphere of radius

v in TpM which has trivial orthonormal frame bundle. We then define r0 on U 0 as

described above. The estimates (7.23)� (7.27) are also valid with this definition of

r0 (on U 0). Moreover, the proofs of Propositions 7.5.3 and 7.5.4 also carry through

to prove that on U 0 the corresponding projection prT⌃

v

p,q = 1� r0 is locally bounded

with respect to C1, and J |U 0 is almost locally coercive. As in the proof of Theo-

rem 7.3.1 it follows that ⇠i is Cauchy and converges in H2(h⇤TM), and therefore xi

converges in ⌃vp,q.

Theorem 7.5.9. J |⌃vc satisfies the PS condition, provided M is compact.

Proof. For the same reasons as those given above for ⌃vp,q, we can only define rc

locally. We mimic the construction of the adapted orthonormal frame Fi above.

Then again we define rcx by (7.20)-(7.22) using Fi, and the periodicity of w and x

ensures that rcxw is C1-periodic as required. We may then follow the same argument

as in the proof of Theorem 7.5.8, because J |⌃vc is weakly proper when M is compact

(Lemma 7.5.6).


Corollary 7.5.10. Provided M is compact in the case ⇤ = c, J |⌃v⇤ obtains its

infimum on ⌃v⇤ and in any connected component there is a critical point which

minimises J with respect to the component. Furthermore there are at least cat(⌃v⇤)

critical points altogether, where cat denotes the Ljusternik-Schnirelman category.

Proof. Both statements are standard consequences of the Palais-Smale condition

(see [67] p. 188-190).

7.6 MORSE THEORY

In this section we prove a Morse index theorem for elastica and use the Morse

inequalities to give lower bounds for the number of elastica with each index. There

is no hope of proving that either of J |⌃vp,q or J |⌃v

c are Morse functions; in the first

case considerM = E3 where any pinned elastica can be varied through critical curves

by rotating about the line through the endpoints. As for J |⌃vc there is a degeneracy

in the parametrization because the curve will be critical regardless of which point

corresponds to t = 0. We therefore focus on J |⌃vv,w which we will denote by J to

correspond with the notation in Theorem 7.3.5.

We will now use Theorem 7.3.5 to derive the Jacobi equation for elastica. We

proceed by calculating each term in equation (7.11) separately. For the derivatives

of J we calculate

V J = hr2

tV +R(V, x)x,rtxi0

WV J = hr2

trWV +rt(R(W, x)V ) +R(W, x)rtV +rW (R(V, x)x),rtxi0

+ hr2

tV +R(V, x)x,r2

tW +R(W, x)xi0

(7.32)

For ⌫ we have V ⌫ = 1

kxkhrtV, xi, and

WV ⌫ = 1

kxk

⇣1

kxk2 hrtW, xihrtV, xi+ hrtrWV +R(W, x)V, xi+ hrtV,rtW i⌘

(7.33)

where the first term vanishes if Vx 2 Tx⌃vv,w. Now supposing x is a critical point of

J |⌃vv,w, notice that the only terms in (7.32) and (7.33) which depend on the values of

V away from x are those involving rWV . When we calculate WV J � �WV ⌫ these

terms group together to form dJxrWV , which is zero since x is also a critical point

of J . It will be convenient to represent the Lagrange multipliers �, � as elements of

H1(I,R) and write ⇤ := �� ,� := � � � (weakly). After repeated integration by

7.6. MORSE THEORY 106

parts and several applications of Bianchi identities we find that

WV J � (�,WV ⌫)� (�,W⌫)

= hr4

tV + F (V, x) + ⇤

vR(V, x)x+rt(

⇤

vrtV ) +rt(

�

vx),W i

0

(7.34)

where F (V, x) is the same large collection of curvature terms that appears in [10] eq.

(9). Thus by Theorem 7.3.5 and the fundamental lemma of calculus of variations,

we have that Vx 2 Tx⌃vv,w is in the nullspace of Hessx J i↵

r4

tV + F (V, x) + ⇤

vR(V, x)x+rt(

⇤

vrtV ) +rt(

�

vx) = 0 (7.35)

which we call the Jacobi equation for elastica. From Lemma 7.2.2 the value of ⇤ is

known. Similiarly, if we take the inner product of equation (7.35) with x and use

the constraints kxk = 1, hrtV, xi = 0, derivatives thereof, and the Euler-Lagrange

equation (7.5) to simplify we find (after several manipulations)

� = 3

vddthr2

tV +R(V, x)x,rtxi (7.36)

It then follows that the nullspace of Hessx J at a critical point x of J , being the

intersection of Tx⌃vv,w with the space of solutions of the system (7.35),(7.36), is

finite dimensional.

Lemma 7.6.1. If x is a critical point of J then Hessx J is strongly nondegenerate

i↵ the associated self-adjoint operator hessx J : Tx⌃vv,w ! Tx⌃v

v,w has trivial kernel.

Proof. We have seen above that ker hessx J is finite dimensional. Since it is self-

adjoint we have ker hessx J = coker hessx J , and therefore if ker hessx J is trivial

then hessx J is an isomorphism.

Corollary 7.6.2. hessx J is Fredholm with Fredholm index zero.

Theorem 7.6.3. (Uhlenbeck [81]) Let B be a bilinear form on a Hilbert space H,

and H0

⇢ Ht ⇢ H1

= H, 0 t 1 an increasing family of closed subspaces. Denote

B|Ht

⇥Ht

by Bt and let Nt be the nullspace of Bt. If

(i) the dimension of the maximal subspace on which B is non-positive is finite

(ii) Nt \Nk = 0 for t 6= k

(iii) B is Fredholm of finite index

(iv) [t<kHt = Hk = \t>kHt


then there are only finitely many conjugate points, i.e. t 2 [0, 1] such that n(t) :=

dimNt is non zero. Furthermore indexB � indexB0

=P

0t<1

n(t) where indexBt

is the dimension of the maximal subspace on which Bt is negative definite.

Note that (i) does not appear in the statement of this theorem in [81] but it is

assumed earlier in the paper.

We let H1

= Tx⌃vv,w, Ht = {V 2 Tx⌃v

v,w : suppV ⇢ (0, t)}. Then (iv) is satisfied

and (iii) has just been proved. For (ii), suppose there exists V 2 Nt \ Nk, k > t,

then V (⌧) = 0 for all ⌧ 2 (t, k) and V satisfies the Jacobi equation for elastica. But

then by local uniqueness of solutions of the Jacobi equation and the compactness of

I, V = 0 on the entire unit interval. As for (i) we proceed as follows. At a critical

point, Hessx J is equal to the restriction of Hessx J to Tx⌃vv,w ⇥ Tx⌃v

v,w. Now since

J is the integral of a strongly elliptic PDO, from (6.8) we have the inequality:

Hessx J(V, V ) = D2J(V, V ) � ↵kV k22

� ckV k21

where ↵ > 0, for all V 2 Tx⌃vv,w. If suppV ⇢ (0, ") then by the Holder inequal-

ity kV k20

=R "0

1kV k2dt (R "0

1dt)1

2 (R

1

0

kV k4)1

2 p"kV k2

0,4 Similarly krtV k20

p"krtV k2

0,4 and therefore kV k21

p"kV k2

1,4. But the Sobolev imbedding H2 ⇢W 4,1 gives kV k

1,4 const kV k22

, hence

Hessx J(V, V ) � ↵kV k22

� Cp"kV k2

2

Then with " su�ciently small we have that Hessx J is positive definite on (0, ").

Now choose ti 2 I, i = 0, 1 . . . N such that Hessx J is positive definite on the sub-

space Y := {w 2 Tx⌃vv,w : w(ti) = 0 = rtw(ti)}. The map P : H2(x⇤TM) !

⇥i(Tx(ti

)

M)2 defined by Pi(V ) := (V (ti),rtV (ti)) is surjective so kerP has finite

codimension. Then Y = kerP \ Tx⌃vv,w has finite codimension as a subspace of

Tx⌃vv,w, and (i) holds. We have now proved the following theorem.

Theorem 7.6.4. Each elastica has a finite number of conjugate points ti and finite

Morse index equal toP

i n(ti).

Now J is a Morse function i↵ 0 and 1 are not conjugate along any elastica

satisfying the given first order boundary conditions, in which case we say that the

boundary conditions themselves are non-conjugate.

Theorem 7.6.5. Suppose the boundary conditions defining ⌃vv,w are non-conjugate,

and let �i denote the ith Betti number of ⌃vv,w over a field. Then there are at least

�i elastica in ⌃vv,w with Morse index i.

Proof. This is an application of the Morse inequalities (see eg. [67] p. 220).

7.6. MORSE THEORY 108

Concluding remarks In section 7.5 we have cat(⌃v⇤) as a lower bound for the

total number of critical points. Typically we would compare the homotopy type (and

therefore category) of this path space with that of the based loop space. However

in the case of elastica it is not clear that any such general statements can be made,

since the based loop space may contain homotopy classes of curves which all have

length greater than v. It might be interesting to study the topology of ⌃v⇤. At

the beginning of section 7.6 it was explained that J |⌃vp,q and J |⌃v

c are not Morse

functions. However, we have not excluded the possibility that they are Morse-

Bott functions; it may be that the critical sets are nondegenerate critical manifolds.

Finally, it is possible that Theorem 7.3.1, or some variant thereof, will be useful for

other constrained variational problems.

CHAPTER 8

Towards variational problems in linearvelocity constraints

8.1 INTRODUCTION

We recall the general one dimensional variational problem: minimise an action/cost

function f : X ! R, where X is an infinite dimensional manifold of curves on a

manifold M , over a subset ⌦ ⇢ X consisting of curves which satisfy some given

constraints. Chapter 6 was concerned with problems where ⌦ is a submanifold of

X determined only by boundary conditions. Chapter 7 treated a specific problem

where ⌦ was determined by a non-linear constraint applied to the velocities at all

points on the path. We now consider the case where ⌦ is determined by linear

velocity constraints which must be satisfied at almost all points on the path. Our

main example is the sub-Riemannian geodesic problem.

We remark that in contrast to the three previous chapters this work is not yet

developed to the stage of publication. The reader should expect only preliminary

explorative results and some speculation.

8.1.1 Sub-Riemannian geometry

A sub-Riemannian (SR) manifold (a.k.a Carnot-Caratheodory space) consists of a

manifold M , a distribution (sub-bundle) D ⇢ TM and a smooth fibrewise inner

product onD. Interesting examples of SR geometries are abundant and the literature

is correspondingly vast. We have used the excellent book by Montgomery [56] as a

reference.

A vector field or curve on M is called horizontal if it is tangent to D almost

everywhere. This terminology originates in the important example where D is the

horizontal sub-bundle of a connection. The SR distance between two points p, q 2M

is defined as d(p, q) := inf l(x) = infRkxkdt where the infimum is taken overH1Dp,q:

the set of all horizontal paths of class H1 with endpoints p, q. A path in H1Dp,q

109


which realises this distance is called a minimising SR geodesic. Thus the SR geodesic

problem:

Minimise E(x) = kxk20

: H1(I,M) ! R subject to the boundary condi-

tions x(0) = p, x(1) = q 2 M and also the constraint x(t) 2 D ⇢ TM

almost everywhere, where D is a distribution on M .

If H1Dp,q is empty then the distance is defined to be infinite, and if it is non-

empty we say that p and q are horizontally path connected. For example, if D is

involutive then by Frobenius’ theorem it is integrable and p, q are horizontally path

connected i↵ they are in the same leaf of the distribution. Supposing that p, q are

in the same leaf, we can restrict to this leaf and the SR geodesic problem reduces to

the standard Riemannian geodesic problem. At the opposite extreme is the bracket-generating (a.k.a Hormander’s) condition. This condition requires that at any point

p 2 M there exists a local frame {Xi} spanning D such that the Xi together with

their iterated Lie brackets [Xi, Xj], [Xi, [Xj, Xk]] . . . span TpM when evaluated at

p. For bracket-generating distributions every pair p, q 2 M is horizontally path

connected (Chow’s theorem).

The set H1Dp of horizontal curves with initial point p is a smooth Hilbert man-

ifold ([56] Theorem D.1), but this is not always the case for H1Dp,q. In fact it is

possible for the endpoint map H1Dp ! M,x 7! x(1) to have critical points. Thus

if q is a critical value of the endpoint map the extrema of E|H1Dp,q need not be

stationary points, i.e they do not necessarily satisfy the condition d(E|H1Dp,q) = 0

nor the corresponding Euler-Lagrange equation. In fact, several authors [56], [49]

have provided examples of minima which are not stationary. These curves are called

abnormal or singular geodesics.

Quite a lot has been written about existence of SR geodesics already, primarily

under the assumption that D is bracket-generating (see [56] and references therein).

A notable exception is the work of Giambo, Giannoni, Piccione and Tausk [26, 28,

68], where: the distribution is assumed to be the orthogonal complement of a non-

vanishing vector field; and, instead of a single point, the endpoint is constrained

to a one-dimensional submanifold which is transversal to the distribution. Another

exception is the work of Vakhrameev [82, 83] on control systems. These systems

require admissible curves to be tangent to the span of a collection of vector fields

{X1

, . . . Xm} which may or may not be linearly independent. Vakhrameev assumes

that the control system has constant rank, meaning that for any p 2M the endpoint

mapping from admissible curves starting at p has a derivative of constant rank.

In this chapter we investigate the smoothness of the horizontal path spaceH1Dp,q.

We find conditions on D, p, q which ensure thatH1Dp,q is a submanifold ofH1(I,M),

111 CHAPTER 8. TOWARDS VARIATIONAL PROBLEMS IN LINEAR VELOCITY CONSTRAINTS

and at the opposite extreme, conditions which ensure there are no H1-rigid (i.e. iso-

lated) horizontal curves.

8.2 THE PATH SPACE IN SUB-RIEMANNIAN GEOMETRY

Let (M,D) be a finite dimensional SR manifold, and h i a Riemannian metric whose

restriction to D gives the SR inner product (such metrics, known as taming metrics,

always exist). We denote by ↵ : TM ! D? the projection onto the orthogonal

complement of D induced by the Riemannian metric so that D = ker↵. Projection

onto the distribution will be denoted � : TM ! D. Setting X := Hk(I,M)

composition with ↵ induces a smooth map

↵� : Hk(I, TM) ⇠= Hk(X ⇤TM)! Hk(I,D?) ⇠= Hk(X ⇤D?)

which is in fact a VB morphism. The isomorphisms are explained in Section 4.5.

This map extends to a VB morphism Hj(X ⇤TM) ! Hj(X ⇤D?) for any j < k.

We may therefore define a smooth constraint function � : X ! Hk�1(X ⇤D?) by

� := ↵ � @, where @ : X ! Hk�1(X ⇤TM) is the tangent lift from section 4.6. The

pre-image by � of the zero section 0X ⇢ Hk�1(X ⇤D?) is the set of curves tangent to

D, viz. the horizontal curves, of class Hk which we will write as HkD := ��1(0X ).

This set will be a submanifold of X if � is transverse to 0X , or equivalently, given a

connection on Hk�1(X ⇤D?), if r�(x) is surjective for any x 2 ��1(0X ).

A connector for D? is given by KD? := ↵ � KTM |TD? ⇢ T 2M ! D? where

KTM is the connector for M . Moreover KTM� is a connector for X (see [14]) and

therefore ↵ �KTM� is a connector for Hk(X ⇤D?) which extends to a connector for

Hk�1(X ⇤D?)

So we have that � is transverse to the zero section i↵ r�(x) = KD? � T�(x) issurjective for any x 2 ��1(0X ). To calculate T� we consider a smooth a : I ! X(i.e. a variation) with a(0) = x and @

@s|s=0

a(s) = v 2 TxX . Then

(Tx�)v = @@s|0

↵ � (@a(s)) = T↵ � @@s|0

@@ta(s) = T↵ � Tv( d

dt)

and therefore, using the definition of covariant di↵erentiation from section 2.6

r�(x)v = KD? � T↵ � Tv( ddt) = ↵ �KTM � T↵ � Tv( d

dt) = ↵ � rt(↵(v))

where rt is covariant di↵erentiation in TM .

Thus � is transverse to the zero section if for any x 2 ��1(0X ) and w 2

8.2. THE PATH SPACE IN SUB-RIEMANNIAN GEOMETRY 112

Hk�1(x⇤D?) there is a solution v 2 Hk(x⇤TM) to

↵ � rt(↵(v)) = w (8.1)

If we also impose first order boundary conditions x(0) = p, x(1) = q, then we

need to know whether the restriction �|Xp,q ! Hk�1(X ⇤p,qD?)

0

is transverse to the

zero section. Here Hk�1(X ⇤p,qD?)

0

denotes the restricted subbundle of Hk�1(X ⇤D?)

where the fibre over x 2 Xp,q is the subspace Hk�1(x⇤TM)0

consisting of liftings of x

which vanish at 0 and 1. Given that v 2 TxXp,q i↵ v(0) = 0 = v(1), the condition for

transversality of �|Xp,q is that for any w 2 Hk�1(x⇤TM)0

there be a solution of (8.1)

which vanishes at the endpoints. In order to obtain a more concrete characterisation

of this transversality condition, and relate it to the bracket generating condition, we

introduce a special kind of local frame.

8.2.1 Allotted frames

A basis {Xi} for TmM is called adapted (to D) if the first r elements are a basis

for Dm. It will be convenient to write {Xi, Yj} in order to distinguish the bases for

D and D?. Given x 2 Hk(I,M) an adapted, orthonormal time dependent frame

{Xi(t,m), Yj(t,m)} defined for m in a local neighbourhood of of x(t) will be called

an allotted frame at x.

Lemma 8.2.1. For any x 2 Hk(I,M) there exists an allotted frame at x.

Proof. First we show how to construct an adapted time dependent orthonormal

frame {Xi(t), Yj(t)} along x 2 Hk(I,M). Fix an orthonormal adapted basis {X0

i , Y0

j }for Dx(0) � D?

x(0). Let Xi(t) be the parallel transport of X0

i along x with re-

spect to the connection KD := � � KTM |TD, and Yj(t) parallel transport of Y 0

i

with respect to KD? . Then, identifying Xi with its inclusion in TM we have

0 = rDXi = KD � TXi = � �KTM � TXi = � � rXi. Therefore

ddthXi, Xki = hrtXi, Xki+ hXi,rtXki = h� � rtXi, Xki+ hXi, � � rtXki = 0

and similarly ↵ �rYj = 0 so ddthYj, Y`i = 0 also. Then since {Xi, Yj} is orthonormal

at t = 0 it is orthonormal for all t.

Next suppose �Xi

(t) : [0, ") ! M is the unique geodesic with initial conditions

�Xi

(t)(0) = x(t) and �Xi

(t)(0) = Xi(t) and use the same procedure along �. Since exp

is a local di↵eomorphism, the result is an orthonormal D-adapted time dependent

frame {Xi(t, p), Yj(t, p)} defined for p in a neighbourhood of x(t).


8.2.2 Regular points of the endpoint map

Let x 2 Hk(I,M)p,q and {Xi, Yj} an allotted frame at x. Then �(x) =P

jhYj, xiYj

and

r�(x)v =X

j

(hYj,rtvi+ hrvYj, xi)Yj. (8.2)

Suppose �(x) = 0, we want to know whether r�(x) is surjective, i.e. given

w =P

i wiYi 2 Hk�1(x⇤D?)

0

can we find rxw =P

i �iwXi +

Pj ⇢

jwYj such that

r�(x)rxw = w. For convenience we will henceforth omit the subscripts on � and

⇢ and use the Einstein summation convention: repeated indices are summed. First

we calculate rtrxw = �iXi + �irtXi + ⇢jYj + ⇢jrtYj and then from (8.2) we have

(r�(x)rxw)k = �ihrtXi, Yki+ ⇢k + ⇢jhrtYj, Yki+ �ihrXi

Yk, xi+ ⇢jhrYj

Yk, xi

Since x is horizontal we write x = x`X`, and then using the identities

hrX`

Xi, Yji = �hrX`

Yj, Xii, hrYk

Xi, Yji = �hrYk

Yj, Xii

obtained as derivatives of hXi, Yji = 0, we have

�ihrtXi, Yki = �ix`hrX`

Xi, Yki

�ihrXi

Yk, xi = �ix`hrXi

Yk, Xì = ��ix`hYk,rXi

Xì

⇢jhrtYj, Yki = ⇢jx`hrX`

Yj, Yki

⇢jhrYj

Yk, xi = ⇢jx`hrYj

Yk, Xì = �⇢jx`hYk,rYj

Xì

Therefore

(r�(x)rxw)k = ⇢k + �ix`h[X`, Xi], Yki+ ⇢jx`h[X`, Yj], Yki (8.3)

and r�(x)rxw = w can be written as ⇢k = wk+⇢jx`h[Yj, X`], Yki+�ix`h[Xi, X`], Ykior in matrix form as

⇢ = w + A⇢+B�

where Akj = x`h[Yj, X`], Yki and Bki = x`h[Xi, X`], Yki. Let � = bu where b : I ! Ris some smooth function with b(0) = 0 = b(1) and otherwise b(t) > 0. Now if there

is a solution to

⇢ = w + A⇢+Bbu (8.4)

with ⇢(0) = 0 = ⇢(1) then r�(x) is surjective. The existence of such a solution

is a question of controllability of the system (8.4) (from 0 to 0). This is equivalent

to controllability of the system z = Az + Bbu, because if the latter is controllable


then we can solve � = w + A�,�(0) = 0, control z = Az + Bbu from z(0) = 0 to

z(1) = ��(1), and then let ⇢ = �+ z. Since w is arbitrary, we really do need total

controllability from zero here. The controllability Gramian (cf. [7] p. 77) for this

system is

W (0, 1) =

Z1

0

�(1, t)b2B(t)B(t)T�(1, t)Tdt

where �(t, t0

) is the state transition matrix for y = Ay. The system is controllable

if W is non-singular. Since � is invertible, if W is singular then there exists y(t)

such thatR

1

0

y(t)TB(t)B(t)Ty(t)dt = 0 and therefore BTy(t) = 0 for almost all

t 2 [0, 1]. Then we have

(BTy(t))i = Bkiyk = ykx`h[Xi, X`], Yki.

Thus BTy(t) = 0 i↵ x`[Xi, X`] is orthogonal to y for every Xi and for (almost) all

t 2 I. It follows that a su�cient1 condition for the controllability of 8.4 is that there

be a measurable set U ⇢ [0, 1] such that for all t 2 U the span of {x`[Xi, X`]x(t)}contains D?

x(t).

Definition. A vector v 2 Dm will be called strongly bracket generating if Dm +

[D, V ]m = TmM , where V is any horizontal extension of v.

To see that this condition is independent of the extension V , let {Xi} be a local

frame for D so that v = viXi(m) and let V = V iXi be an extension of v, i.e.

V i(m) = vi. Then [Xj, V ]m = V i(m)[Xj, Xi]m + (XjV i)Xi(m) = vi[Xj, Xi]m +

(XjV i)Xi(m). Similarly, if V = V iXi is a another extension then [Xj, V ]m =

vi[Xj, Xi]m+(XjV i)Xi(m). Since it is only the second term in each expression that

depends on the extension, it follows that Dm + [D, V ]m = TmM i↵ Dm + [D, V ]m =

TmM .

Definition. The distribution D is called strongly bracket generating (a.k.a one-step

bracket generating or fat) if for any m 2 M , every non-zero v 2 TmM is strongly

bracket generating.

According to the discussion above, we have the following theorem.

Theorem 8.2.2. A horizontal curve x 2 H1Dp,q is a regular point of the endpoint

map H1Dp ! M if there is a measurable set U ⇢ I such that x(t) is strongly

bracket generating for all t 2 U . Moreover, if D is strongly bracket generating then

every such curve is a regular point, q is a regular value, and H1Dp,q is a smooth

submanifold of H1Dp.

1this is not a necessary condition because we took arbitrary y instead of �(t, 0)y0


It is proved in [56] (p. 243) that the endpoint map on H1Dp is an open mapping

if the distribution is bracket-generating, but it is not proved to be a submersion.

However, it is proved (p. 70) that a strongly bracket generating distribution admits

no singular curves. The proof is based on work by Hsu [33] on exterior di↵erential

systems and is quite di↵erent to the proof given above.

8.2.3 Rigid curves?

Following Montgomery [56] p. 39, a horizontal curve x0

is called C1-rigid if there is

a C1 neighbourhood of x0

such that every horizontal curve in this neighbourhood is

a reparametrization of x0

. Results of Bryant-Hsu [9] and Liu-Sussman [49] showed

that the well-known examples of singular geodesics are C1-rigid. Agrachev and

Sarychev [3, 4] proved that they are L11-rigid. However, as Montgomery argues, the

natural domain of the energy functional has the H1 topology. Montgomery proves

that bracket-generating distributions admit no H1-rigid curves and suggests that

the same is true for any distribution. Here we will investigate this suggestion.

To be precise: x 2 H1Dp,q is called H1-rigid if there is an H1 neighbourhood of

x0

such that every horizontal curve in this neighbourhood is a reparametrization of

x. Equivalently, x is H1-rigid if kerr�(x)\TxH1(I,M)p,q is equal to the span of x.

From (8.3), this means that to prove x is not rigid it is su�cient to find a non-zero

V = �iXi + ⇢jYj satisfying

(r�(x)V )k = ⇢k + �ix`h[X`, Xi], Yki+ ⇢jx`h[X`, Yj], Yki = 0

and V |t=0,1 = 0, which is not parallel to x. We observe two conditions which ensure

that x is not rigid:

• Suppose there is an Xi (say, i = 1) which is not parallel to x and satisfies

[X1

, x]x(t) 2 Dx(t) for any horizontal extension x of x and any t in some open

neighbourhood of I. Let V = b(t)X1

where b is a smooth real valued function

with non-empty support contained in said neighbourhood. Then (r�(x)V )k =

bh[x, X1

], Yki = 0.

• Suppose that for t in some open neighbourhood U , [D, x]x(t) has a nonzero

component in D?, say in the direction Y1

. Let V = �iXi+ ⇢1Y1

and look for a

solution of the control system ⇢1 + buix`h[X`, Xi], Y1

i+ ⇢1x`h[X`, Y1

], Y1

i = 0.

It follows from the assumption and the arguments in the previous section that

this system is controllable, so �i = bui exist for which the solution satisfies

⇢(0) = 0 = ⇢(1) (to avoid the solution ⇢ ⌘ 0 we could control from ⇢(0) = 0

to ⇢(t0

) = 1, where t0

2 U , and then to ⇢(1) = 0).


The corresponding global (i.e. x independent) versions of these conditions are

as follows:

• D contains an involutive sub-distribution of dimension at least two.

• In any local horizontal frame there is an X1

such that dim[X1

,D] > dimD.

These conditions are not obviously exhaustive, so the question of existence of H1-

rigid curves remains open, but we have narrowed the search for examples.

Concluding remarks We have laid the foundation here for future work verifying

the Palais-Smale condition for sub-Riemannian geodesics, and perhaps other cost

functions in sub-Riemannian constraints. At least when the path space is smooth,

one might use Theorem 7.3.1, with a suitable modification of Lemma 7.3.4 and a

right split obtained as solutions of the control problem from Section 8.2.2.

CHAPTER 9

Conclusion

9.1 REVIEW

Suppose N is either a closed interval or a circle, M a complete Riemannian man-

ifold, and consider a variational problem min f |⌦, where f : Hk(N,M) ! R and

⌦ ⇢ Hk(N,M). It was explained in the proof of Lemma 6.2.2 that the natural charts

(Theorem 4.4.3) for Ck�1(N,M) are weak charts for Hk(N,M) which is therefore a

weak submanifold of Ck�1(N,M). According to Theorem 6.2.1 if ⌦ is also a weak

submanifold of Ck�1(N,M) and f |⌦ is weakly proper, locally bounding and locally

coercive with respect to Ck�1(N,M) then f |⌦ satisfies the Palais-Smale condition.

Now if f is strongly elliptic of order k (i.e. it is the integral of a strongly

elliptic polynomial di↵erential operator), then it is locally bounding and locally

coercive on Hk(N,M) with respect to Ck�1(N,M) (Theorem 6.3.1). If ⌦ is such

that the natural charts for Hk(N,M) satisfy the submanifold property for ⌦, then

it is also a weak submanifold of Ck�1(N,M) and the restriction f |⌦ is also locally

bounding and locally coercive with respect to Ck�1(N,M) (Lemma 6.3.2). This is

not true in general, but seems to hold whenever ⌦ is defined by boundary conditions.

Furthermore, if for all x 2 ⌦ the initial values ritx(0), 0 j ↵ are fixed (or

perhaps contained in a compact manifold), and f(x) < C implies that kritxk

0

< C

for all ↵ < i k, then f |⌦ is weakly proper. In summary, if f is strongly elliptic,

the natural charts for Hk(N,M) have the submanifold property for ⌦, and the

initial conditions for ⌦ are balanced by the energy bounding of f in the sense just

described, then f |⌦ satisfies the Palais-Smale condition (Corollary 6.3.7).

For example, consider the Riemannian cubics, a.k.a biharmonic curves, which

are the critical points of the strongly elliptic functional J : H2(I,M) ! R, J(x) =1

2

krtxk20

. The restriction J |Hk(I,M)v,w, where v, w 2 TM , to paths satisfy-

ing first order boundary conditions satisfies the Palais-Smale condition, however

J |H2(I,M)p,q with p, q 2 M does not because kxk0

is not bounded by J . On

the other hand, the function J (x) = 1

2

(krxk0

+ ⌧kxk0

), which has the Rieman-

117

9.2. FUTURE WORK 118

nian cubics in tension as critical points, does satisfy the Palais-Smale condition on

H2(I,M)p,q.

When ⌦ is not a weak submanifold of Ck�1(N,M) more work is required. The

natural charts for Hk(N,M) do not have the submanifold property for ⌦, and there-

fore local boundedness and coercivity of f do not persist for the restriction f |⌦.Nevertheless it may be the case that in a natural chart for Hk(N,M) the image

of ⌦ is, loosely speaking, not too badly behaved. More precisely: we need a pro-

jection prT⌦ : THk(N,M)|⌦ ! T⌦ which is locally bounded in natural charts for

Hk(N,M) and which can be used to prove that f |⌦ is locally coercive with respect

to Hk(N,M), Ck�1(N,M) (see Section 7.3). If such a projection exists, f is strongly

elliptic, and f |⌦ is weakly proper then the Palais-Smale condition holds.

This procedure was carried out for elastica in Section 7.5. Elastica are critical

points of the restriction of J to fixed speed curves ⌃v := ⌫�1(v) where ⌫ is the

speed function. A projection prT⌃

v

was constructed from a right inverse for the

derivative d⌫, and then both J |⌃vp,q and J |⌃v

v,w satisfy the Palais-Smale condition.

Notice that in this case it is the constraint (not the functional) which ensures that

kxk0

is bounded.

9.2 FUTURE WORK

There are many avenues adjoining this work that remain relatively unexplored. I

would like to highlight some that I find particularly interesting.

• In Chapter 7 the Palais-Smale condition was verified for closed elastica, but

no attempt was made at Morse-theoretic results because the critical points

are degenerate. However it may be the case that they are contained in non-

degenerate critical manifolds and Morse-Bott theory is applicable.

• Chapter 8 is directed at verifying the Palais-Smale condition for sub-Riemannian

geodesics, at least when the path space is regular. The strategy would be to

prove this using Theorem 7.3.1, a suitable modification of Lemma 7.3.4, and

a right split obtained as solutions of the control problem from Section 8.2.2.

• Finally, on a much more speculative note, I wonder whether the technique of

imposing a “gauge-fixing condition” in concert with Theorem 7.3.1 (as was

used for elastica) might also be useful for higher dimensional variational prob-

lems with parametrization invariance.

APPENDIX A

A.1 PROOF OF LEMMA 5.3.1

(i) Let ⇠ be a smooth vector field on H1(I,M), then r⇠(A�)(x) = K�T (A�)(⇠(x))andr⇠(x)A(x) = K�TA(⇠(x)). Therefore it is su�cient to show that T (A�)(⇠(x)) =TA � ⇠(x). Suppose expX = x in a natural chart centred at h, then

T (A�)(⇠(x)) = dds(A�)(exp(X + s⇠h(X)))|s=0

= ddsA(exp(X + s⇠h(X)))|s=0

= TA(D2

exp(X, ⇠h(X)))

= TA(⇠(x))

(ii) Given vector fields ⌫, ⇠ on H1(I,M) we have from (3.5)

rY (x)(⌫(x), ⇠(x)) = K � TY � (⌧, T ⌧, K)�1(⇠(x), ⌫(x), 0)

The bundle map for H1(H1(I,M)⇤TM) is ⌧�, the connection map is K� and using

the chain rule TH1(⌧) = H1(T ⌧). Hence r(Y �)(x)(⌫, ⇠) = K�T (Y �)�(⌧, T ⌧, K)�1�(⇠, ⌫, 0). By similar reasoning to that used for (i) we have T (Y �)(⇠(x)) = TY � ⇠(x),and this completes the proof.

(iii) Follows immediately from (i) and (ii).

(iv) Since R(⇠, ⌘, ⌫)(x) depends only on ⇠(x), ⌘(x), ⌫(x), we may extend ⌫(x) to a

vector field V on M and

R(⇠, ⌘, ⌫)(x) = R(⇠, ⌘, (V �))(x) = r2(V �)(⇠, ⌘)(x)� r2(V �)(⌘, ⇠)(x)

= r2V (⇠(x), ⌘(x))�r2V (⌘(x), ⇠(x)) = R(⇠(x), ⌘(x), ⌫(x))

(v) It is shown in [17] that r⇠@(x) = rt⇠(x), we will show that the same is true

of r@(x)⇠. We work in a natural chart centred at h with expX = x, and denote the

119

A.2. SOME COMMENTS ON THE TERMINOLOGY OF ELIASSON 120

local expressions for @ and ⇠ as @h and ⇠h respectively. We will calculate r@(x)⇠ in

terms of coordinates on M using the fact that ⇤ is the local connector for r, the

definition of ⇤ (3.13), and the local formula (3.11).

r@(x)⇠ = (x, @2

exp(X, d⇠h(X) · @h(X) + ⇤(X)(@h(X), ⇠h(X))))

= (x, @2

exp(X, @@s⇠h(X + s@h(X))|s=0

)) +r2

r2

exp(X, @h(X), ⇠h(X))

= (x, @@s@2

exp(X + s@h(X), ⇠h(X + s@h(X)))|s=0

+ �(expX)(@2

exp(X, @h(X)), @2

exp(X, ⇠h(X))))

= (x, @@s⇠(exp(X + s@h(X)))|s=0

+ �(x)(x, ⇠(x)))

= (x, d⇠(x) · @2

exp(X, @h(X)) + �(x)(x, ⇠(x)))

= (x, ddt⇠(x) + �(x)(x, ⇠(x)))

which is of course the local expression for rt⇠(x).

A.2 SOME COMMENTS ON THE TERMINOLOGY OF ELIASSON

We recall that a map between two topological spaces is called proper if the in-

verse images of (relatively) compact sets are (relatively) compact. Thus f : X ! Ris proper if the inverse image of any bounded (i.e. relatively compact) set is rela-

tively compact in X. In section 6.2.1 we have the additional assumption that X is

contained in another space X0

, and that the relative topology is weaker than the

topology on X. Thus weakly proper : the pre-image of any bounded set is relatively

compact in X0

. We emphasize that this is not necessarily the weak topology.

The notion of a coercive bilinear form is an important part of the Lax-Milgram

lemma, which is a generalisation of the Riesz-Representation theorem to bilinear

forms that are not necessarily symmetric.

Lemma A.2.1. (Lax-Milgram, cf. [70] pg. 290) Let H be a Hilbert space and let

B : H⇥H ! R be a bilinear mapping. Suppose there exist positive constants c1

, c2

such that

(i) |B(x, y)| c1

kxkHkykH for all x and y in H, and

(ii) B(x, x) � c2

kxk2H for all x 2 H.

Then for every f 2 H⇤ there exists a unique y 2 H such that B(x, y) = f(x) for all

x 2 H.

121 APPENDIX A.

A bilinear form which satisfies (ii) is called coercive. Comparing (6.3) we see

that f : X ! R is locally coercive with respect to X0

if D2f is locally a coercive

bilinear form modulo the local X0

-norm.

A.3 A SYMMETRY-BREAKING FUNCTION FOR THE TOTAL SQUARED CURVATURE

Using the notation of Chapter 7 we define F : Imm2(I,M)! R by

F(x) :=

Z1

0

k4(x)v2dt

Consider a variation x(w, t) : (�", ") ⇥ I ! M with variation vector field W =

@wx|w=0

. From Lemma 1.1 in [40] we have

@wv = hrTW,T iv

@wk2 = 2hr2

TW,rTT i � 4k2hrTW,T i+ 2hR(W,T )T,rTT i

thus

dF(x)W =

Z1

0

2k2v2@wk2 + 2k4v@wvdt

= 2

Z1

0

k2v(hr2

TW,rTT i � 4k2hrTW,T i+ 2hR(W,T )T,rTT i)

+ k4vhrTW,T ids

Notice the change in variable for the integration. Integrating by parts, and using a

Bianchi identity, gives

dF(x)W = 2

Z1

0

2hW,r2

T (k2vrTT )i+3hW,rT (k

4vT i+2k2vhR(rTT, T )T,W ids

+ 2⇥2k2vhrTW,rTT i � 3hW,rT (k

4vrTT )i � hW, 4k4vT i⇤1

0

We may assume that W |t=0,1 = 0 = rTW |t=0,1 so that the boundary terms disap-

pear, and then by the fundamental lemma of calculus of variations we obtain the

Euler-Lagrange equation:

2r2

T (k2vrTT ) + 3rT (k

4vT ) + 2k2vR(rTT, T )T = 0 (A.1)

The Euler-Lagrange equation associated with F =R

1

0

k2ds is equation 7.3 with

� = 0, i.e.

2r3

TT + 3rT (k2T ) + 2R(rTT, T )T = 0 (A.2)

A.4. REGULARITY OF WEAK SOLUTIONS TO ODE 122

Observe that a solution of (A.2) with k2v constant will be a solution of (A.1). In fact,

any solution of (A.1) has constant k2v and is therefore also a solution of (A.2). To

see this we will require the following identities which are obtained by di↵erentiating

hT, T i = 1:

hrTT, T i = 0

hr2

TT, T i = �k2

hr3

TT, T i = �3

2

@sk2

Now taking the inner product of (A.1) with T and using the above identities:

0 = 2hr2

T (k2vrTT ), T i+ 3hrT (k

4vT ), T i

= 2h@2s (k2v)rTT + 2@s(k2v)r2

TT + k2vr3

TT, T i+ 3h@s(k4v)T + k4vrTT, T i

= �4k2@s(k2v)� 3k2v@s(k

2) + 3@s(k4v)

= �k2@s(k2v)

Multiplying by 2v gives 0 = 2k2v@s(k2v) = @s(k2v)2, so k2v must be constant.

A.4 REGULARITY OF WEAK SOLUTIONS TO ODE

Let Q be a polynomial di↵erential operator of order at most 2k � 1 and suppose

x 2 Hk(I,M) satisfies the following ODE r2k�1

t x+Q(x) = 0 weakly : i.e.

hr2k�1

t x+Q(x), V i0

= 0

for all V 2 Hk(x⇤TM) : V |t=0,1 = 0. Let Q be the solution of rkt Q = Q,rj

tQ(0) = 0

for all j < k, so that repeated integration by parts in the above gives:

hrk�1

t x+ Q(x),rkt V i

0

= 0 (A.3)

Now let ⌘ be the solution of

rkt ⌘ = rk�1

t x+ Q(x), rjt⌘(0) = 0, for all j < k

Our intention is to use ⌘ (or something like it) in place of V in (A.3) and therefore

conclude that rk�1

t x + Q(x) = 0, from which it would follow that rk�1

t x has the

same Hk regularity as Q. However, ⌘ cannot be substituted for V because it doesn’t

satisfy the required boundary conditions. We begin the process of correcting for

123 APPENDIX A.

this: for each i k let ⇠i be the i-parallel (rit⇠i = 0) field along x with

⇠1

(1) = ⌘(1)

rjt⇠i(1) = 0 for j 6= i� 1

ri�1

t ⇠i(1) = ri�1

t ⌘(1)�i�1X

j=1

k!

(k � i+ j)!rj�1

t ⇠j(1)

Now define � := ⌘ � tkPk

i=1

⇠i. The conditions above have been chosen so that

rit�|t=0,1 = 0 for all 0 j < k, and so V = � is allowed. Moreoverrk

t � = rkt ⌘�P (⇠)

for P satisfying rktP (⇠) = 0, and therefore applying integration by parts k times

gives hrkt �, P i

0

= �|10

= 0. Then from (A.3)

0 = hrk�1

t x+ Q,rkt �i

0

= hrk�1

t x+ Q� P,rkt �i

0

= krk�1

t x+ Q� Pk0

and it follows that rk�1

t x = P � Q (almost everywhere). Since P, Q are of class Hk,

x must be of class H2k. So the original ODE holds in strong form and x is C1 by

induction.

A.4. REGULARITY OF WEAK SOLUTIONS TO ODE 124

Bibliography

[1] R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, tensor analysis, and appli-

cations, vol. 75 of Applied Mathematical Sciences, 2nd edition, Springer-Verlag,

New York, 1988, URL http://dx.doi.org/10.1007/978-1-4612-1029-0.

[2] R. A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt

Brace Jovanovich, Publishers], New York-London, 1975, Pure and Applied

Mathematics, Vol. 65.

[3] A. A. Agrachev and A. V. Sarychev, Strong minimality of abnormal geodesics

for 2-distributions, J. Dynam. Control Systems, 1 (1995), 139–176, URL http:

//dx.doi.org/10.1007/BF02254637.

[4] A. A. Agrachev and A. V. Sarychev, Abnormal sub-Riemannian geodesics:

Morse index and rigidity, Ann. Inst. H. Poincare Anal. Non Lineaire, 13

(1996), 635–690.

[5] J. Arroyo, O. J. Garay and J. Mencıa, Elastic circles in 2-spheres, J. Phys.

A, 39 (2006), 2307–2324, URL http://dx.doi.org/10.1088/0305-4470/39/

10/005.

[6] T. Aubin, Nonlinear analysis on manifolds. Monge-Ampere equations, vol. 252

of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles

of Mathematical Sciences], Springer-Verlag, New York, 1982, URL http:

//dx.doi.org/10.1007/978-1-4612-5734-9.

[7] R. Brockett, Finite dimensional linear systems, Series in decision and control,

Wiley, 1970.

[8] R. Bryant and P. Gri�ths, Reduction for constrained variational problems andR

1

2

k2 ds, Amer. J. Math., 108 (1986), 525–570, URL http://dx.doi.org/

10.2307/2374654.

[9] R. L. Bryant and L. Hsu, Rigidity of integral curves of rank 2 distribu-

tions, Invent. Math., 114 (1993), 435–461, URL http://dx.doi.org/10.

1007/BF01232676.

125

http://dx.doi.org/10.1007/978-1-4612-1029-0

http://dx.doi.org/10.1007/BF02254637

http://dx.doi.org/10.1007/BF02254637

http://dx.doi.org/10.1088/0305-4470/39/10/005

http://dx.doi.org/10.1088/0305-4470/39/10/005

http://dx.doi.org/10.1007/978-1-4612-5734-9

http://dx.doi.org/10.1007/978-1-4612-5734-9

http://dx.doi.org/10.2307/2374654

http://dx.doi.org/10.2307/2374654

http://dx.doi.org/10.1007/BF01232676

http://dx.doi.org/10.1007/BF01232676

BIBLIOGRAPHY 126

[10] M. Camarinha, F. Silva Leite and P. Crouch, On the geometry of Riemannian

cubic polynomials, Di↵erential Geom. Appl., 15 (2001), 107–135.

[11] P. Crouch and F. Silva Leite, The dynamic interpolation problem: on Rieman-

nian manifolds, Lie groups, and symmetric spaces, J. Dynam. Control Systems,

1 (1995), 177–202, URL http://dx.doi.org/10.1007/BF02254638.

[12] C. J. Earle and J. Eells Jr., Foliations and fibrations, J. Di↵erential Geometry,

1 (1967), 33–41.

[13] J. Eells Jr., On the geometry of function spaces, in Symposium internacional de

topologıa algebraica International symposium on algebraic topology, Universidad

Nacional Autonoma de Mexico and UNESCO, Mexico City, 1958, 303–308.

[14] H. I. Elıasson, Geometry of manifolds of maps, J. Di↵erential Geometry, 1

(1967), 169–194.

[15] H. I. Elıasson, Variation integrals in fiber bundles, in Global Analysis (Proc.

Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968), Amer. Math. Soc.,

Providence, R.I., 1970, 67–89.

[16] H. I. Elıasson, Condition (C) and geodesics on Sobolev manifolds, Bull. Amer.

Math. Soc., 77 (1971), 1002–1005.

[17] H. I. Elıasson, Morse theory for closed curves, in Symposium on Infinite-

Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967),

Princeton Univ. Press, Princeton, N. J., 1972, 63–77. Ann. of Math. Stud-

ies, No. 69.

[18] H. I. Elıasson, Introduction to global calculus of variations, in Global analysis

and its applications (Lectures, Internat. Sem. Course, Internat. Centre Theoret.

Phys., Trieste, 1972), Vol. II, Internat. Atomic Energy Agency, Vienna, 1974,

113–135.

[19] E. Fadell and S. Husseini, A note on the category of the free loop space, Proc.

Amer. Math. Soc., 107 (1989), 527–536, URL http://dx.doi.org/10.2307/

2047844.

[20] E. Fadell and S. Husseini, Category of loop spaces of open subsets in Euclidean

space, Nonlinear Anal., 17 (1991), 1153–1161.

[21] L. E. Fraenkel, On regularity of the boundary in the theory of Sobolev spaces,

Proc. London Math. Soc. (3), 39 (1979), 385–427, URL http://dx.doi.org/

10.1112/plms/s3-39.3.385.

http://dx.doi.org/10.1007/BF02254638

http://dx.doi.org/10.2307/2047844

http://dx.doi.org/10.2307/2047844

http://dx.doi.org/10.1112/plms/s3-39.3.385

http://dx.doi.org/10.1112/plms/s3-39.3.385

127 BIBLIOGRAPHY

[22] S. Gabriel and J. Kajiya, Spline interpolation in curved space, in State of the

Art in Image Synthesis, vol. 11 of SIGGRAPH ’85 Course Notes, ACM Press,

New York, 1985, 1–14.

[23] P. Garrett, Non-existence of tensor products of hilbert spaces, 2010, URL

http://www.math.umn.edu/

~

garrett/m/v/nonexistence_tensors.pdf.

[24] F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X. Vialard,

Invariant higher-order variational problems, Comm. Math. Phys., 309 (2012),

413–458, URL http://dx.doi.org/10.1007/s00220-011-1313-y.

[25] F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X. Vialard,

Invariant higher-order variational problems II, J. Nonlinear Sci., 22 (2012),

553–597, URL http://dx.doi.org/10.1007/s00332-012-9137-2.

[26] R. Giambo, F. Giannoni, P. Piccione and D. V. Tausk, Morse theory for nor-

mal geodesics in sub-Riemannian manifolds with codimension one distributions,

Topol. Methods Nonlinear Anal., 21 (2003), 273–291.

[27] R. Giambo, F. Giannoni and P. Piccione, An analytical theory for Riemannian

cubic polynomials, IMA J. Math. Control Inform., 19 (2002), 445–460.

[28] R. Giambo, F. Giannoni and P. Piccione, Existence, multiplicity, and regularity

for sub-Riemannian geodesics by variational methods, SIAM J. Control Op-

tim., 40 (2002), 1840–1857 (electronic), URL http://dx.doi.org/10.1137/

S0363012900367242.

[29] R. Giambo, F. Giannoni and P. Piccione, Optimal control on Riemannian

manifolds by interpolation, Math. Control Signals Systems, 16 (2004), 278–

296.

[30] Y. E. Gliklikh, Global and stochastic analysis with applications to mathematical

physics, Theoretical and Mathematical Physics, Springer-Verlag London, Ltd.,

London, 2011, URL http://dx.doi.org/10.1007/978-0-85729-163-9.

[31] M. Golomb and J. Jerome, Equilibria of the curvature functional and manifolds

of nonlinear interpolating spline curves, SIAM J. Math. Anal., 13 (1982), 421–

458, URL http://dx.doi.org/10.1137/0513031.

[32] K. Grove, Condition (C) for the energy integral on certain path spaces and

applications to the theory of geodesics, J. Di↵erential Geometry, 8 (1973),

207–223.

http://www.math.umn.edu/~garrett/m/v/nonexistence_tensors.pdf

http://dx.doi.org/10.1007/s00220-011-1313-y

http://dx.doi.org/10.1007/s00332-012-9137-2

http://dx.doi.org/10.1137/S0363012900367242

http://dx.doi.org/10.1137/S0363012900367242

http://dx.doi.org/10.1007/978-0-85729-163-9

http://dx.doi.org/10.1137/0513031

BIBLIOGRAPHY 128

[33] L. Hsu, Calculus of variations via the Gri�ths formalism, J. Di↵erential

Geom., 36 (1992), 551–589, URL http://projecteuclid.org/euclid.jdg/

1214453181.

[34] I. I. Hussein and A. M. Bloch, Dynamic interpolation on riemannian manifolds:

an application to interferometric imaging, in American Control Conference,

2004. Proceedings of the 2004, vol. 1, IEEE, 2004, 685–690.

[35] I. Hussein and A. Bloch, Dynamic coverage optimal control for multiple space-

craft interferometric imaging, Journal of Dynamical and Control Systems, 13

(2007), 69–93, URL http://dx.doi.org/10.1007/s10883-006-9004-2.

[36] V. Jurdjevic, Non-Euclidean elastica, Amer. J. Math., 117 (1995), 93–124,

URL http://dx.doi.org/10.2307/2375037.

[37] V. Jurdjevic, Integrable Hamiltonian systems on complex Lie groups, Mem.

Amer. Math. Soc., 178 (2005), viii+133, URL http://dx.doi.org/10.1090/

memo/0838.

[38] I. Kolar, P. W. Michor and J. Slovak, Natural operations in di↵erential ge-

ometry, Springer-Verlag, Berlin, 1993, URL http://dx.doi.org/10.1007/

978-3-662-02950-3.

[39] S. Lang, Fundamentals of di↵erential geometry, vol. 191 of Graduate Texts in

Mathematics, Springer-Verlag, New York, 1999, URL http://dx.doi.org/

10.1007/978-1-4612-0541-8.

[40] J. Langer and D. A. Singer, The total squared curvature of closed curves, J.

Di↵erential Geom., 20 (1984), 1–22.

[41] J. Langer and D. A. Singer, Curve straightening and a minimax argument for

closed elastic curves, Topology, 24 (1985), 75–88, URL http://dx.doi.org/

10.1016/0040-9383(85)90046-1.

[42] J. Langer and D. A. Singer, Curve-straightening in Riemannian manifolds,

Ann. Global Anal. Geom., 5 (1987), 133–150, URL http://dx.doi.org/10.

1007/BF00127856.

[43] E. H. Lee and G. E. Forsythe, Variational study of nonlinear spline curves,

SIAM Rev., 15 (1973), 120–133.

http://projecteuclid.org/euclid.jdg/1214453181

http://projecteuclid.org/euclid.jdg/1214453181

http://dx.doi.org/10.1007/s10883-006-9004-2

http://dx.doi.org/10.2307/2375037

http://dx.doi.org/10.1090/memo/0838

http://dx.doi.org/10.1090/memo/0838

http://dx.doi.org/10.1007/978-3-662-02950-3

http://dx.doi.org/10.1007/978-3-662-02950-3

http://dx.doi.org/10.1007/978-1-4612-0541-8

http://dx.doi.org/10.1007/978-1-4612-0541-8

http://dx.doi.org/10.1016/0040-9383(85)90046-1

http://dx.doi.org/10.1016/0040-9383(85)90046-1

http://dx.doi.org/10.1007/BF00127856

http://dx.doi.org/10.1007/BF00127856

129 BIBLIOGRAPHY

[44] R. Levien, The elastica: a mathematical history, Technical Report

UCB/EECS-2008-103, EECS Department, University of California, Berke-

ley, 2008, URL http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/

EECS-2008-103.html.

[45] A. Linner, Existence of free nonclosed Euler-Bernoulli elastica, Nonlinear Anal.,

21 (1993), 575–593, URL http://dx.doi.org/10.1016/0362-546X(93)

90002-A.

[46] A. Linner, Unified representations of nonlinear splines, J. Approx. Theory, 84

(1996), 315–350, URL http://dx.doi.org/10.1006/jath.1996.0022.

[47] A. Linner, Curve-straightening and the Palais-Smale condition, Trans.

Amer. Math. Soc., 350 (1998), 3743–3765, URL http://dx.doi.org/10.

1090/S0002-9947-98-01977-1.

[48] A. Linner, Periodic geodesics generator, Experiment. Math., 13

(2004), 199–206, URL http://projecteuclid.org/getRecord?id=euclid.

em/1090350934.

[49] W. Liu and H. J. Sussman, Shortest paths for sub-Riemannian metrics on

rank-two distributions, Mem. Amer. Math. Soc., 118 (1995), x+104.

[50] L. Machado, F. Silva Leite and K. Krakowski, Higher-order smoothing splines

versus least squares problems on Riemannian manifolds, J. Dyn. Control Syst.,

16 (2010), 121–148, URL http://dx.doi.org/10.1007/s10883-010-9080-1.

[51] J. E. Marsden and T. S. Ratiu, Introduction to mechanics and symmetry,

vol. 17 of Texts in Applied Mathematics, 2nd edition, Springer-Verlag, New

York, 1999, URL http://dx.doi.org/10.1007/978-0-387-21792-5, A basic

exposition of classical mechanical systems.

[52] J. Mawhin and M. Willem, Critical point theory and Hamiltonian systems,

vol. 74 of Applied Mathematical Sciences, Springer-Verlag, New York, 1989.

[53] J. Milnor, Morse Theory, Princeton University Press, Princeton, N.J., 1963.

[54] S. Montaldo and A. Ratto, Biharmonic curves into quadrics, Glasg. Math. J.,

57 (2015), 131–141.

[55] S. Montaldo and I. I. Onnis, Biharmonic curves on an invariant surface,

J. Geom. Phys., 59 (2009), 391–399, URL http://dx.doi.org/10.1016/j.

geomphys.2008.11.011.

http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-103.html

http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-103.html

http://dx.doi.org/10.1016/0362-546X(93)90002-A

http://dx.doi.org/10.1016/0362-546X(93)90002-A

http://dx.doi.org/10.1006/jath.1996.0022

http://dx.doi.org/10.1090/S0002-9947-98-01977-1

http://dx.doi.org/10.1090/S0002-9947-98-01977-1

http://projecteuclid.org/getRecord?id=euclid.em/1090350934

http://projecteuclid.org/getRecord?id=euclid.em/1090350934

http://dx.doi.org/10.1007/s10883-010-9080-1

http://dx.doi.org/10.1007/978-0-387-21792-5

http://dx.doi.org/10.1016/j.geomphys.2008.11.011

http://dx.doi.org/10.1016/j.geomphys.2008.11.011

BIBLIOGRAPHY 130

[56] R. Montgomery, A tour of subriemannian geometries, their geodesics and ap-

plications, vol. 91 of Mathematical Surveys and Monographs, American Math-

ematical Society, Providence, RI, 2002.

[57] D. Mumford, Elastica and computer vision, in Algebraic geometry and its

applications (West Lafayette, IN, 1990), Springer, New York, 1994, 491–506.

[58] V. V. Nesterenko, A. Feoli and G. Scarpetta, Dynamics of relativistic particles

with Lagrangians dependent on acceleration, J. Math. Phys., 36 (1995), 5552–

5564, URL http://dx.doi.org/10.1063/1.531332.

[59] L. Noakes, Conditional extremals, 2011, Manuscript submitted for publication.

arXiv:1104.2647v1.

[60] L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces, IMA

J. Math. Control Inform., 6 (1989), 465–473.

[61] L. Noakes and T. Popiel, Null riemannian cubics in tension in SO(3), IMA J.

Math. Control Inform., 22 (2005), 477–488.

[62] L. Noakes, Conditional extremals, Mathematics of Control, Signals, and Sys-

tems (MCSS), 24 (2012), 295–320.

[63] R. S. Palais and S. Smale, A generalized Morse theory, Bull. Amer. Math.

Soc., 70 (1964), 165–172.

[64] R. S. Palais, Lusternik-Schnirelman theory on Banach manifolds, Topology, 5

(1966), 115–132.

[65] R. S. Palais, Foundations of global non-linear analysis, W. A. Benjamin, Inc.,

New York-Amsterdam, 1968.

[66] R. S. Palais, Critical point theory and the minimax principle, in Global

Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif, 1968), Amer.

Math. Soc., Providence, R.I., 1970, 185–212.

[67] R. S. Palais and C.-L. Terng, Critical point theory and submanifold geometry,

vol. 1353 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1988.

[68] P. Piccione and D. V. Tausk, Variational aspects of the geodesics problem in

sub-Riemannian geometry, J. Geom. Phys., 39 (2001), 183–206, URL http:

//dx.doi.org/10.1016/S0393-0440(01)00011-0.

http://dx.doi.org/10.1063/1.531332

http://dx.doi.org/10.1016/S0393-0440(01)00011-0

http://dx.doi.org/10.1016/S0393-0440(01)00011-0

131 BIBLIOGRAPHY

[69] T. Popiel and L. Noakes, Elastica in SO(3), J. Aust. Math. Soc., 83 (2007),

105–124.

[70] M. Renardy and R. C. Rogers, An introduction to partial di↵erential equations,

vol. 13 of Texts in Applied Mathematics, 2nd edition, Springer-Verlag, New

York, 2004.

[71] P. Schrader, Existence of variationally defined curves with higher order elliptic

Lagrangians, Nonlinear Anal., 115 (2015), 1–11, URL http://dx.doi.org/

10.1016/j.na.2014.11.016.

[72] P. Schrader, Morse theory for elastica, J. Geom. Mech., 8 (2016), 235–256,

URL http://dx.doi.org/10.3934/jgm.2016006.

[73] P. Schrader and L. Noakes, Conditional extremals in complete Riemannian

manifolds, Nonlinear Anal., 75 (2012), 2177–2188, URL http://dx.doi.org/

10.1016/j.na.2011.10.018.

[74] J. T. Schwartz, Nonlinear functional analysis, Gordon and Breach Science

Publishers, New York-London-Paris, 1969, Notes by H. Fattorini, R. Niren-

berg and H. Porta, with an additional chapter by Hermann Karcher, Notes on

Mathematics and its Applications.

[75] F. Silva Leite, M. Camarinha and P. Crouch, Two higher order variational

problems on Riemannian manifolds and the interpolation problem, in Proceed-

ings of the third European control conference, Rome, 1995, 3269–3274.

[76] F. Silva Leite, M. Camarinha and P. Crouch, Elastic curves as solutions

of Riemannian and sub-Riemannian control problems, Math. Control Signals

Systems, 13 (2000), 140–155.

[77] E. H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York, 1966.

[78] M. E. Taylor, Partial di↵erential equations. I, vol. 115 of Applied Mathematical

Sciences, Springer-Verlag, New York, 1996, URL http://dx.doi.org/10.

1007/978-1-4684-9320-7, Basic theory.

[79] A. Trouve and F.-X. Vialard, Shape splines and stochastic shape evolutions:

a second order point of view, Quart. Appl. Math., 70 (2012), 219–251, URL

http://dx.doi.org/10.1090/S0033-569X-2012-01250-4.

[80] C. Truesdell, The influence of elasticity on analysis: the classic heritage, Bull.

Amer. Math. Soc. (N.S.), 9 (1983), 293–310, URL http://dx.doi.org/10.

1090/S0273-0979-1983-15187-X.

http://dx.doi.org/10.1016/j.na.2014.11.016


http://dx.doi.org/10.3934/jgm.2016006



http://dx.doi.org/10.1007/978-1-4684-9320-7

http://dx.doi.org/10.1007/978-1-4684-9320-7

http://dx.doi.org/10.1090/S0033-569X-2012-01250-4

http://dx.doi.org/10.1090/S0273-0979-1983-15187-X

http://dx.doi.org/10.1090/S0273-0979-1983-15187-X

BIBLIOGRAPHY 132

[81] K. Uhlenbeck, The Morse index theorem in Hilbert space, J. Di↵erential

Geometry, 8 (1973), 555–564.

[82] S. A. Vakhrameev, Morse theory and Lyusternik-Shnirelman theory in geomet-

ric control theory, in Current problems in mathematics. Newest results, Vol. 39

(Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn.

i Tekhn. Inform., Moscow, 1991, 41–117, 180, Translated in J. Math. Sci. 71

(1994), no. 3, 2434–2485.

[83] S. A. Vakhrameev, Geometrical and topological methods in optimal control

theory, J. Math. Sci., 76 (1995), 2555–2719, Analysis, 5.

[84] J. Vilms, Connections on tangent bundles, J. Di↵erential Geometry, 1 (1967),

235–243.

global analysis of one-dimensional variational … · introduction the basic variational principle...

Documents