THE GEOMETRY OF OPEN MANIFOLDS OF NONNEGATIVE CURVATURE

Ph.D. Thesis, University of Pennsylvania, 1999

Kristopher R. Tapp

Acknowledgments

I wish to thank Wolfgang Ziller for showing me the beauty of Differential Geometry and for beinga wonderfully patient and supportive advisor with a great sense of humor. I am also gratefulfor helpful conversations about this work with Chris Croke, Herman Gluck, Jonathan Block, LuisGuijarro, Gerard Walschap, Vitali Kapovitch, and Jason Cantarella. I owe thanks to Peter Petersenfor his supportive feedback on this work.

I feel very thankful for the huge amount of love, emotional support, playful humor, and un-derstanding which I have received from my fiancee, Charity Hendrickson. She has shared myexcitements and frustrations, gone to my talks, talked to me about souls and manifolds, and helpedme keep my life balanced and filled with love thoughout the preparation of this work.

I wish to thank my parents for all of the care packages, cards, phone conversations, and otherforms of support through college and graduate school. Finally, I am thankful to Peter Selingerfor being a good friend, and for talking me out of joining the Peace Corps after my first year ofGraduate School.

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ABSTRACTTHE GEOMETRY OF OPEN MANIFOLDS OF NONNEGATIVE CURVATURE

Kristopher R. Tapp

In this paper we study the global geometric properties of an open manifold with nonnegative sec-tional curvature. Cheeger and Gromoll’s well-known Soul Theorem states that any such manifold,M , contains a compact totally geodesic submanifold, Σ ⊂ M , called the “soul of M”, whose normalbundle is diffeomorphic to M . In 1994, Perelman proved that the metric projection π : M → Σ is awell defined Riemannian submersion. Perelman’s breakthrough greatly simplifies the study of thisclass of manifolds. The main purpose of this paper is to explore consequences of Perelman’s result.Along the way we develop some general theory for Riemannian submersions which is of interestindependent of its application to nonnegative curvature.

For example, we derive basic properties of Riemannian submersions with compact holonomy. Forthe metric projection onto a soul, we show by example that the holonomy is not necessarily compact,even when the soul is simply connected, and we establish certain rigidity when the holonomy iscompact.

Additionally, we develop some general theory for “bounded Riemannian submersion” (submer-sions whose A and T tensors are both bounded in norm). When applied to the metric projectiononto a soul, π : M → Σ, this theory implies that M is quasi-isometric to any single fiber of π.Perelman’s theorem also enables us to bound the volume growth of M from above and below.

Finally, we study the converse of the Soul Theorem; that is, the question of which vectorbundles over spheres (or more general souls) admit metrics of nonnegative curvature. For example,we prove that only finitely many vector bundles over a given soul admit a nonnegatively curvedmetric satisfying a fixed upper bound for the vertical curvatures at the soul. Also, we translate thequestion of whether a bundle admits nonnegative curvature into the question of whether it admitsa connection and a tensor which together satisfy a certain differential equation.

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Contents

1 Introduction 1

2 Background 32.1 The metric projection onto a soul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 The normal holonomy group of a soul . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 The ideal boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Basic properties of Riemannian submersions . . . . . . . . . . . . . . . . . . . . . . . 7

3 Conditions for nonnegative curvature on vector bundles 83.1 Curvature at the soul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Curvature near the soul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Connection metrics on vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 The warping tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5 Warped connection metrics on vector bundles . . . . . . . . . . . . . . . . . . . . . . 213.6 Summary of conditions for nonnegative curvature . . . . . . . . . . . . . . . . . . . . 26

4 Riemannian submersions with compact holonomy 284.1 Consequences of compact holonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Vertizontal curvature decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 A soul with noncompact normal holonomy . . . . . . . . . . . . . . . . . . . . . . . . 314.4 Measuring size in the holonomy of a vector bundle . . . . . . . . . . . . . . . . . . . 32

5 Volume growth bounds 355.1 Previous results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 An upper bound on volume growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.3 Bounded vertical Jacobi fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.4 A lower bound on volume growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6 Bounded Riemannian submersions 416.1 A bound on the folding of the fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.2 Measuring size in the holonomy of a submersions . . . . . . . . . . . . . . . . . . . . 446.3 The ideal boundary is determined by one fiber . . . . . . . . . . . . . . . . . . . . . 456.4 Two finiteness theorems for Riemannian submersions . . . . . . . . . . . . . . . . . . 47

7 Finiteness theorems for fiber bundles 507.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.2 A finiteness theorem for vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 527.3 A finiteness theorem for principal bundles . . . . . . . . . . . . . . . . . . . . . . . . 55

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Chapter 1

Introduction

In this paper we study the global geometric properties of an open manifold with nonnegative sec-tional curvature. Cheeger and Gromoll’s well-known Soul Theorem states that any such manifold,M , contains a compact totally geodesic submanifold, Σ ⊂ M , called the “soul of M”, whose nor-mal bundle, ν(Σ), is diffeomorphic to M . In 1994, Perelman proved that the metric projectionπ : M → Σ is a well defined Riemannian submersion. Perelman’s breakthrough greatly simplifiesthe study of this class of manifolds. The main purpose of this paper is to explore consequencesof Perelman’s result. Along the way we develop some general theory for Riemannian submersionswhich is of interest independent of its application to nonnegative curvature.

Chapter 2: Background. In this chapter, we review the Soul Theorem, Perelman’s Theorem,and the definition and basic properties of Riemannian submersions. Additionally, we review thedefinition and basic properties of two important geometric objects which, together with the soul,helps one study M ; namely, the ideal boundary, M(∞), of M , and the holonomy group, Φ, of thenormal bundle, ν(Σ), of Σ.

Chapter 3: Conditions for nonnegative curvature on vector bundles. In the firstsection of this chapter, we derive a formula for the curvature of an arbitrary 2-plane at the soul. Inthe second section, we study the curvature of an arbitrary 2-plane “near” the soul. More precisely,imagine a family σ(t) of 2-planes on M beginning with a mixed 2-plane σ(0) at a point of Σ (bya mixed 2-plane, we mean a plane spanned by a tangent vector and a normal vector to Σ; such a2-plane necessarily has zero curvature) and then varying in such a way that the base point movesaway from Σ while simultaneously the plane twists away from its initial “mixed” configuration. Westudy the derivatives of the curvature function t 7→ K(σ(t)).

In the remaining sections of this chapter, we use these curvature formulas to study conditionsunder which a vector bundle admits a metric of nonnegative curvature. More precisely, we translatethe question of whether a vector bundle admits a metric of nonnegative curvature into the questionof whether it admits a connection and a tensor which together satisfy a certain differential equation.This generalizes the work of Walschap and Strake on the question of which vector bundles admitconnection metrics of nonnegative curvature.

Chapter 4: Riemannian submersions with compact holonomy. We say that a Rie-mannian submersion π : M → B has compact holonomy if its holonomy group is a compact finitedimensional Lie group. For the metric projection onto a soul, π : Mn+k → Σn, the holonomy groupof π is just the holonomy group of the normal bundle of Σ, and is therefore a Lie subgroup of theorthogonal group O(k). We show by example that it need not be a compact Lie subgroup of O(k),even when the soul is simply connected. When the metric projection onto a soul does happen tohave compact holonomy, we achieve the following splitting theorem: if the vertizontal curvaturesof M decay towards zero away from Σ, then M splits locally isometrically over Σ. This theorem isbased on the work of Guijarro and Walschap, and generalizes a result of Guijarro and Petersen.

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Chapter 5: Volume growth bounds. In this chapter we use Perelman’s Theorem to studythe volume growth, VG(M), of M . In particular, VG(M) is proven to be bounded above by thedifference between the codimension of the soul and the maximal dimension of an orbit of the actionof the holonomy group Φ on a fiber νp(Σ). Additionally, if M satisfies an upper curvature bound,then we prove that VG(M) is bounded below by one plus the dimension of the ideal boundary,M(∞), of M .

Chapter 6: Bounded Riemannian submersions. In this chapter we generalize away fromnonnegative curvature in order to study Riemannian submersions whose A and T tensors are bothbounded in norm. Suppose that π : Mn+k → Bn is a Riemannian submersion, B is compact andsimply connected, |A| ≤ CA, and |T | ≤ CT . Our main theorem provides a bound on the “folding ofthe fibers” of π. Specifically, if dFp denotes the intrinsic distance function on a fiber Fp = π−1(p),and dM denotes the distance function of M restricted to Fp, then dFp ≤ C · dM , where C dependson B, CA, CT , and k.

Since, by Perelman’s Theorem, the metric projection onto a soul is a bounded Riemanniansubmersion, we have the following application to nonnegative curvature: when the soul is simplyconnected, the ideal boundary of M can be determined completely from a single fiber of π.

We also explore consequences or our “folding of the fibers” bound outside of the field of non-negative curvature; in particular, we prove that there are only finitely many isomorphism typesamong the class of Riemannian submersions whose base space and total spaces both satisfy fixedgeometric bounds. In particular, we generalize a finiteness theorem of J.Y. Wu.

Chapter 7: Finiteness theorems for fiber bundles. In this chapter we prove finitenesstheorems for vector bundles and principal bundles. For example, there are only finitely manyvector bundles of a fixed rank over a fixed compact Riemannian manifold capable of admitting aconnection whose curvature tensor satisfies a fixed bound in norm. An analogous statement holdsfor principal bundles, and the proof of this analog is based on the “folding of the fibers” bound inChapter 6.

We provide two applications of these finiteness theorems to nonnegative curvature. First, thereare only finitely many vector bundles of a fixed rank over a fixed soul (that is, a fixed compact Rie-mannian manifold of nonnegative curvature) which are capable of admitting nonnegatively curvedmetrics for which the vertical curvatures at the soul satisfy a fixed upper bound. Second, we boundthe number of rank k vector bundles over a fixed Bieberbach manifold B which can admit a metricof nonnegative curvature (the bound is in terms of k and the topology of B).

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Chapter 2

Background

In this chapter, we review the Soul Theorem, Perelman’s Theorem, and the definition and basicproperties of Riemannian submersions. Additionally, we review the definition and basic propertiesof two important geometric objects which, together with the soul, helps one study M ; namely, theideal boundary, M(∞), of M , and the holonomy group, Φ, of the normal bundle, ν(Σ), of Σ.

2.1 The metric projection onto a soul

Our primary object of study in this paper is an open Riemannian manifold, M , with nonnegativesectional curvature. The term “open” means complete and noncompact. One might be lead tosuspect rigidity among this class of manifolds by considering the following two facts:

• Any open Riemannian manifold must contain at least one ray.

• A Riemannian manifold with sectional curvature bounded below by δ > 0 is necessarilycompact, and hence cannot contain any rays. Analogously, a Riemannian manifold withnonnegative sectional curvature might be thought of as “wanting to close up”, or “barely ableto contain a ray”.

In 1972, Cheeger and Gromoll proved their celebrated “Soul Theorem”. By harnessing this tensionbetween the necessity that such manifolds contain rays and their tendency to close up, Cheegerand Gromoll proved the following remarkable structure theorem [6]:

Theorem 2.1.1 (Cheeger and Gromoll’s Soul Theorem). Any open manifold, M , withnonnegative sectional curvature contains a compact and totally convex submanifold, Σ ⊂ M(called the “soul of MÔ), whose normal bundle is diffeomorphic to M .

Cheeger and Gromoll conjectured that there should exist a Riemannian submersion fromM ontoits soul (the reader unfamiliar with Riemannian submersions should jump ahead to Section 2.4).The resolution of this conjecture by Perelman in 1994 was a crucial breakthrough in the study ofnonnegatively curved open manifolds. It provides a powerful description of the ways in which thegeometries of the manifold and its soul interrelate. We believe that Perelman’s result opens the fieldto a renewed investigation. The main purpose of our paper is to explore consequences of Perelman’sTheorem. Perelman’s Theorem is based on the following result due to Sharafutdinov [28] (a nicealternative proof due to Croke, Schroeder and Yim is found in [44]):

Theorem 2.1.2 (Sharafutdinov). There exists a distance non-increasing map π : M → Σ.

Since Riemannian submersions are distance non-increasing, π was a good candidate for theconjectured Riemannian submersion. However, Sharafutdinov constructed π as a limit of an infinite

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sequence of rather complicated maps, which made it very difficult to study. Perelman overcamethis difficulty by providing a very simple description of π, while simultaneously proving that π isin fact a Riemannian submersion.

In order to state Perelman’s Theorem, we establish the following notation. Let Π : ν(Σ) → Σdenote the normal bundle of Σ in M . Let exp⊥ : ν(Σ) → M denote the normal exponential map.For p ∈ Σ, let νp(Σ) := Π−1(p). If γ is a path in Σ with γ(0) = p and V ∈ νp(Σ), let Pγ(V ) denotethe parallel transport of V along γ. Perelman’s Theorem states [24]:

Theorem 2.1.3 (Perelman).

1. For any p ∈ Σ, X ∈ TpΣ, and V ∈ νp(Σ), the surface (s, t) → exp⊥(s · V (t)) (s, t ∈ R, s ≥0), where V (t) denotes the parallel transport of V along the geodesic with initial tangentvector X, is a flat and totally geodesic half plane (we will refer to these surfaces as“Perelman flatsÔ).

2. π exp⊥ = Π.

3. π is a C1 Riemannian submersion. The horizontal distribution of π is the image underd exp⊥ of the distribution in νp(Σ) which is associated to the natural connection. Saiddifferently, if γ(t) is a path in Σ, p = γ(0), V ∈ νp(Σ), and V (t) := Pγ|[0,t], then the path

t 7→ exp⊥(V (t)) is horizontal with respect to π.

4. The second fundamental forms of the fibers of π are globally bounded in norm.

Some comments about Perelman’s Theorem are in order:

• Perelman’s proof makes no mention of Sharafutdinov’s intricate construction of π; it uses onlythe property that π is distance non-increasing. Thus, π is the unique distance non-increasingmap from M onto Σ.

• Part 2 of Perelman’s Theorem says that π is simply the inverse of the normal exponentialmap of the soul. Said differently, π is the “metric projection”; that is, the map which sendsa point x ∈ M to the point π(x) ∈ Σ to which it is closest. Accordingly, we will refer to πeither as the “Sharafutdinov map” or as the “metric projection onto the soul”.

• Perelman’s Theorem does NOT imply that exp⊥ : ν(Σ) → M is injective; however, if twodifferent vectors of ν(Σ) have the same image under exp⊥, then the two vectors must havethe same base point p ∈ Σ.

• Luis Guijarro has recently proven that π is at least C2 and a.e. smooth [13].

Part 4 of Perelman’s result is of crucial importance for our paper, and we wish to elaborateon it here. Perelman’s precise claim was that the eigenvalues of the second fundamental forms ofthe fibers of π are bounded above, in barrier sense, by the reciprocal of the injectivity radius ofΣ. Since Guijarro has established that π is C2, the second fundamental form of the fibers is welldefined by the usual equation and is continuous, which frees us to ignore the phrase “in barriersense”. Perelman’s proof of this bound actually proves the following more general fact:

Proposition 2.1.4 (Generalization of Perelman’s Bound). Suppose that f : M → B isany C2 Riemannian submersion between Riemannian manifolds. If the curvature of M isbounded below by k and the injectivity radius of B is bounded below by i > 0, then the secondfundamental forms of the fibers of f are bounded in norm by a constant which depends onlyon λ and i.

Since Perelman’s proof of this bound is very abbreviated (it is a single sentence), we choose toinclude an elaboration of his proof here.

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Proof of Proposition 2.1.4. Let p ∈ B and let p ∈ Fp := π−1(p). Let X be any unit-lengthvector which is normal to Fp at p. Let V be any unit-length tangent vector to Fp at p which isan eigenvector of the shape operator SX of Fp in the direction X; that is SXV = λV . We wish tobound λ.

Take any real number r < i. Let α : [0, r] → M denote the horizontal geodesic with α′(0) = X.Let q := α(r). α is distance-minimizing, and in fact it continues to minimize for a short timebeyond p simply because the same is true of its projection α := π α. Therefore the distance sphereS about q of radius r is smooth at p. Also, S intersects Fp only at p; for, if there were another pointx of intersection, then the projection via π of a minimal path from q to x would provide a minimalpath from q := π(q) to p which is different than α, contradicting the injectivity radius bound.

The idea is to compare the second fundamental form of Fp at p to the second fundamentalform of S at p. To assist us in doing so, we reduce to a 2-dimensional picture. Let W denote theexponential image of a small ball in the 2-plane at p spanned by X and V . W clearly meets bothS and Fp transversally. Let γ : (−ε, ε) → W denote the unit-speed parameterization of the curveof intersection of Fp and W , with γ(0) = p. Notice that

λ = 〈SXV, V 〉 = 〈γ(0), X〉,

which is just the curvature at p of γ.Similarly, if S denotes the shape operator of S, then SXV = 〈β(0), X〉, where β is the curve

along which W meets S. By standard comparison theory, S has norm bounded in terms of r andk. Finally, it is straightforward to show that 〈γ(0), X〉 ≤ 〈β(0), X〉 simply because γ and β meetonly at p, and β is the “inner” of the two curves with respect to the orientation of W given by thevector X.

We end this section by mentioning an important consequence of Perelman’s Theorem which wasobserved by Guijarro and Walschap in [17, Proposition 2.4] (the result is also mentioned in [12,Lemma 2.2]):

Proposition 2.1.5 (Guijarro-Walschap). For sufficiently small r > 0, the closed ball BΣ(r) =p ∈ M | d(p,Σ) ≤ r is convex.

Since we will later need to generalize this result, we include Guijarro and Walschap’s proof ofthis proposition.

Proof. It will suffice to prove that the second fundamental form Π of the boundary SΣ(r) of BΣ(r)associated to the inward-pointing unit normal vector field N is positive semidefinite. Let p ∈ SΣ(r),let p := π(p) ∈ Σ, and let γ : [0, r] → M be a minimal geodesic from γ(0) = p to γ(r) = p. ClearlyN(p) = −γ′(r). Let SN denote the shape operator of SΣ(r) associated to N .

By Perelman’s Theorem, N is parallel along any horizontal geodesic. Hence for any X ∈ Hp,Π(X, ∗) = 〈SN (X), ∗〉 = 0.

On the other hand, the set νp(Σ, r) := W ∈ νp(Σ) | |W | = r satisfies:

exp⊥(νp(Σ, r)) = SΣ(r) ∩ ∂Bp(r).

This set is smooth for small r > 0, and it’s tangent space at p is V⊥p := V ∈ Vp | V ⊥ Np. Since

balls of small radius are strictly convex, Π(V, V ) > 0 for any nonzero V ∈ V⊥p . Hence, the second

fundamental form of SΣ(r) is positive semidefinite for sufficiently small r.

2.2 The normal holonomy group of a soul

The holonomy group, Φ, of ν(Σ) plays an important role in the study of M . We remind the readerthat Φ is defined with respect to a fixed p ∈ Σ as the group of all orthogonal endomorphisms of

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νp(Σ) which occur as Pα for some piecewise smooth loop α in Σ at p. Φ is a Lie subgroup of theorthogonal group O(νp(Σ)). The identity component, Φ0, of Φ can be shown to be the subgroupof all endomorphism which occur as Pα for some nulhomotopic loop α in Σ at p. Φ0 is often calledthe “reduced holonomy group”.

Part 3 of Perelman’s Theorem says that the connection in ν(Σ) controls the horizontal distribu-tion of π. Therefore, Perelman’s Theorem implies that, in certain ways, the natural action of Φ onνp(Σ) controls the geometry of M . This will be an important theme of our paper. A well knownresult in this vein is the following splitting theorem:

Theorem 2.2.1 (The Splitting Theorem).

1. If Φ0 = id, then M splits locally isometrically over Σ.

2. If Φ = id, then M splits globally isometrically over Σ.

“M splits locally isometrically over Σ”, means that for any contractible neighborhood U in Σ,π−1(U) is isometric to the Riemannian product U × (Rk, g), where g is some metric of nonnegativecurvature on Rk. “M splits globally isometrically over Σ”, means that M is isometric to Σ×(Rk, g).

Part 2 of this theorem was proven independently by Strake [33], Yim [45] and Marenich [20].Part 1 does not seem to be explicitly stated in the literature, but it is well-known. We will mentionin Remark 3.1.2 an elementary proof of part 1 which is made possible by Perelman’s Theoremtogether with a result of Walschap.

2.3 The ideal boundary

Together with the soul, Σ, and the holonomy group, Φ, a third important space needed to study Mis the “ideal boundary ofM”, denotedM(∞). M(∞) is a compact space which encodes informationabout the geometry of M “at infinity”, or infinitely far away from a fixed soul. In this section wewill review the definition of M(∞).

M(∞) can be defined by choosing any fixed point p ∈ M and declaring M(∞) to be the set ofequivalence classes of unit-speed rays in M from p, endowed with the following distance function:

d∞([γ1], [γ2]) := limt→∞

1

tdt(γ1(t), γ2(t)),

where dt denotes the intrinsic distance function on the sphere of radius t about p. Two rays areconsidered equivalent if they have distance zero.

Since it is difficult to work with dt, we define an alternate metric on M(∞) as follows:

d∞([γ1], [γ2]) =

∞ (if γ1, γ2 lie on different ends of M);

limt→∞1t dM (γ1(t), γ2(t)) (otherwise).

(2.3.1)

Here dM denotes the distance function onM . The relationship between these two distance functionsis derived in [31, Proposition 2.2]. Namely, if rays γ1 and γ2 lie on the same end of M , then:

d∞([γ1], [γ2]) =√

2− 2 cos(d∞([γ1], [γ2])).

In particular, d∞ and d∞ induce the same topology on M(∞).(M(∞), d∞) is an Alexandrov space with curvature bounded below by 1. The proof of this can

be found in [15], and the definition and basic properties of Alexandrov spaces can be found in [4].

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2.4 Basic properties of Riemannian submersions

In this section we review the definition and basic properties of Riemannian submersions, whoserelevance to the study of nonnegative curvature is obvious from Perelman’s Theorem. A goodreference for this material is Chapter 9 of [2].

We begin with the definition. Suppose that M,B are Riemannian manifolds, and that f :M → B is a Ck submersion in the sense of differentiable topology; that is, f is onto and dfx :TxM → Tf(x)B is a surjective linear map for every x ∈ M . This implies that for any p ∈ B,

the fiber Fp := f−1(p) is a Ck submanifold of M . Define the vertical distribution, V, of f tobe the collection of tangent spaces of the fibers. That is, Vx := Tx(Ff(x)). Define the horizontaldistribution H as the distribution orthogonal to V. If for every x ∈ M , dfx|Hx : Hx → Tf(x)B is alinear isometry, then f is called a Riemannian submersion.

If α(t) is a path in B, say from p = α(0) to q = α(1), and x ∈ Fp, then there exists a uniquehorizontal lift, αx(t) of α with αx(0) = x. We define hα : Fp → Fq as the diffeomorphism betweenthe fibers which sends each x ∈ Fp to αx(1) ∈ Fq. hα is a Ck−1 diffeomorphism, and we willsometimes refer to it as the “holonomy diffeomorphism” associated to α.

It is natural to define the “holonomy group”, Φ, of f by fixing a point p ∈ B and defining Φ asthe group of all diffeomorphisms of the fiber Fp which occur as hα for some piecewise smooth loopα in B at p. It is easy to see that Φ is a group and that, up to group isomorphism, Φ does notdepend on the choice of p ∈ B, at least when B is connected.

For the metric projection onto a soul, π : M → Σ, the holonomy group of π is the same as theholonomy group of the normal bundle of the soul, and is therefore a subgroup of the orthogonalgroup O(νp(Σ)). This is immediate from Perelman’s Theorem. For general Riemannian submer-sions, however, the holonomy group need not be a finite dimensional Lie Group. In Chapter 4,we will study Riemannian submersions whose holonomy groups are compact finite dimensional LieGroups. We will show by example that the holonomy group of the metric projection onto a soulneed not be compact, and we will derive rigidity consequences when it is compact.

Associated to any Riemannian submersion are two fundamental tensors, called the A and Ttensors. If E1 and E2 are vector fields on M , then we define:

TE1E2 := H∇VE1VE2 + V∇VE1HE2.

AE1E2 := H∇HE1VE2 + V∇HE1HE2.

It is easy to show that A and T are tensors. The T -tensor is best thought of as the secondfundamental form of the fibers packaged together with its adjoint (the shape operator of the fibers).The best way to think of the A-tensor is provided by the following fact: if X and Y are horizontalvector fields on M then AXY = 1

2V[X,Y ]. Thus, the A-tensor measures the failure of the horizontaldistribution to be integrable.

Curvature information on M is related to curvature information on B by a collection of formulasknown as O’Neill’s formulas [2, Theorem 9.28]. The best known formula from this collection statesthat for x ∈ M , if X,Y ∈ Hx are orthonormal, then

K(X,Y ) = K(dfx(X), dfx(Y ))− 3|AXY |2. (2.4.1)

Equation 2.4.2.1 is often referred to simply as “O’Neill’s formula”. It implies the important factthat Riemannian submersions are curvature non-decreasing. It also implies that if M satisfies alower curvature bound λ, and B satisfies an upper curvature bound µ, then the A-tensor of f is

bounded in norm by√

13 (µ− λ). For the metric projection onto a soul, π : M → Σ, M satisfies the

lower curvature bound λ = 0, and Σ satisfies some upper curvature bound µ simply because Soulsare compact. Therefore, the A-tensor of π is bounded in norm. Part 4 of Perelman’s Theoremsays that the T -tensor of π is also bounded in norm. For this reason, a large portion of our paperis devoted to the study of “bounded Riemannian submersions”; that is, Riemannian submersionsboth of whose fundamental tensors are bounded in norm.

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Chapter 3

Conditions for nonnegativecurvature on vector bundles

In the first two sections of this chapter, we use Perelman’s Theorem to derive simple formulas whichshed light on how the curvature tensor of M behaves at and near the soul of M . In the remainingsections, we use these formulas to translate the question of whether a vector bundle admits a metricof nonnegative curvature into the question of whether it admits a connection and a tensor whichtogether satisfy a certain differential equation. This generalizes the work of Walschap and Strakeon conditions for a vector bundle to admit a connection metric of nonnegative curvature [35].

3.1 Curvature at the soul

In this section, M will denote an open manifold of nonnegative curvature, and π : M → Σ willdenote the metric projection onto its soul. It is obvious that π is smooth in a neighborhoodof Σ. We use the curvature sign convention of [2]. For vector fields Ei on M , we denote by(E1, E2, E3, E4) := 〈R(E1, E2)E3, E4〉, where R is the curvature tensor of M . We denote byK(E1, E2) := (E1, E2, E1, E2). Notice that if E1 and E2 are orthonormal, then K(E1, E2) is thesectional curvature of the 2-plane which they span.

For p ∈ Σ, X,Y ∈ TpΣ, and U, V ∈ νp(Σ), we denote by R∇(X,Y )V the curvature tensor ofν(Σ). We can enlarge the domain of the tensor R∇ somewhat by defining R∇(U, V )X to be thevector in TpΣ for which 〈R∇(U, V )X,Y 〉 = 〈R∇(X,Y )U, V 〉 for any Y ∈ TpΣ. Since Σ is totallygeodesic, the tensor R∇ is simply the restriction of R to a smaller domain; even so, we will oftenuse R∇ in order to point out that a formula depends only on the connection on ν(Σ).

When X is a vector field on Σ, we will use X to denote its basic lift to M .We begin by reviewing a simple description of the A tensor of π. This description is due to

Guijarro [14, Proposition 14], although the main idea first appeared in a paper by Strake andWalschap [34, Proposision 1.7]:

Lemma 3.1.1 (Guijarro). Let p ∈ Σ, X,Y ∈ TpΣ, and W ∈ νp(Σ). Let p := exp⊥(W ). ThenAX(p)Y (p) = J(t), where J(t) is the Jacobi field along γ(t) := exp⊥(tW ) with J(0) = 0 and

J ′(0) = 12R

∇(X,Y )W .

Proof. Write T (ν(Σ)) = H ⊕ V for the splitting of the tangent bundle of ν(Σ) into horizontaland vertical sub-bundles induced by the natural connection on ν(Σ). Remember that Perelman’sTheorem implies that (exp⊥)∗ : H ⊕ V → H ⊕ V preserves the splitting. Extend X,Y to vectorfields on Σ in a neighborhood of p. Let X, Y denote their basic lifts to T (ν(Σ)). Notice that

8

(exp⊥)∗X = X and (exp⊥)∗Y = Y . Then:

AX(p)Y (p) =1

2[X, Y ](p)V = (exp⊥)∗

(

1

2[X, Y ](W )V

)

= (exp⊥)∗(AX(W )Y (W )),

where A denotes the A-tensor associated with the splitting H ⊕ V of T (ν(Σ)).The rightmost term of the above equation can be simplified with the help of a description in [26,

pages 67-70] of the A-tensor of a vector bundle with a connection. Namely:

AX(W )Y (W ) =1

2R∇(X,Y )W. (3.1.1)

In summary:

AX(p)Y (p) =1

2(exp⊥∗ )(R

∇(X,Y )W ),

so the proposition follows from the usual description of the derivative of the exponential map usingJacobi fields.

Remark 3.1.2. It is obvious from Lemma 3.1.1 that A ≡ 0 iff R∇ ≡ 0. But Walschap provedin [41, Theorem 1.3] that, for any Riemannian submersion the total space of which has nonnegativecurvature, if A ≡ 0 then the submersion splits locally isometrically. This proves the local statementof Theorem 2.2.1 (The Splitting Theorem).

Cheeger and Gromoll proved in their original Soul Theorem paper that the curvatures of “mixed”2-planes at the soul vanish [6, Theorem 3.1]. That is:

Lemma 3.1.3 (Cheeger-Gromoll). Let p ∈ Σ, X ∈ TpΣ and V ∈ νp(Σ). Then R(X,V )V =R(V,X)X = 0. In particular K(X,V ) = 0.

Although today it is obvious from Perelman’s Theorem that K(X,V ) = 0, Cheeger and Gro-moll’s original argument was very difficult. The implication “K(X,V ) = 0 ⇒ R(X,V )V =R(V,X)X = 0” is a consequence of the following well-known lemma:

Lemma 3.1.4. Let M be a manifold of nonnegative curvature, p ∈ M and X,V ∈ TpM twovectors such that K(X,V ) = 0. Then R(X,V )V = R(V,X)X = 0.

Proof. Consider the map F : TpM → TpM defined as F (Y ) := −R(Y, V )V . This map is symmetric,which means that 〈F (Y ), Z〉 = 〈F (Z), Y 〉, and nonnegative, which means that 〈F (Y ), Y 〉 ≥ 0. Fromthese two properties one can show that 〈F (Y ), Y 〉 = 0 ⇔ F (Y ) = 0. It follows that R(X,V )V = 0.Swapping X and V in the above argument proves that R(V,X)X = 0 as well.

Next we collect some facts about the A and T tensors and their derivatives at the soul. Thefollowing lemma says that A and T both vanish at points of Σ, as does the first derivative, DT , ofT .

Lemma 3.1.5. Let p ∈ Σ, X,Y ∈ TpΣ, and U, V,W ∈ νp(Σ).

1. AXY = AXU = 0.

2. TUX = TUV = 0.

3. (DV A)XY = 12R

∇(X,Y )V and (DV A)XU = − 12R

∇(V,U)X.

4. (DWT )UV = (DWT )UX = 0

9

Proof. For part 1, AXY = 0 because the soul is totally geodesic. Since 〈AXU, Y 〉 = −〈AXY, U〉 =0, it follows that AXU = 0 as well.

For part 2, TUX = 0 because the “Perelman flat” through X and U is totally geodesic. Since〈TUV,X〉 = −〈TUX,V 〉 = 0, it follows that TUV = 0 as well.

The first statement of part 3 is an immediate consequence of Lemma 3.1.1. The second statementfollows from the first as follows:

〈(DV A)XU, Y 〉 = −〈(DV A)XY, U〉 = −〈12R∇(X,Y )V,U〉 = 〈−1

2R∇(V,U)X,Y 〉.

For part 4, first notice that (DWT )WX = 0 because the Perelman flat through X and W istotally geodesic. Next, since

0 = 〈(DWT )WX,U〉 = −〈(DWT )WU,X〉 = −〈(DWT )UW,X〉,

it follows that (DWT )UW = 0 as well. This is a special case of part 4. To get the general casefrom this special case, we apply O’Neill’s formula and Lemma 3.1.3 as follows:

0 = (X,W,W,U) = (U,W,W,X) = 〈(DWT )UW,X〉 − 〈(DUT )WW,X〉 = −〈(DUT )WW,X〉.

It follows from this that (DUT )WW = 0. Since (DUT )W1W2 is symmetric in W1 and W2, it followsthat DUT = 0.

Corollary 3.1.6. Let p ∈ Σ, X,Y ∈ TpΣ and U ∈ νp(Σ). Then R(X,Y )U = 2R(X,U)Y .

Proof of Corollary 3.1.6. Since the soul is totally geodesic, both R(X,Y )U and R(X,U)Y areelements of νp(Σ). According to one of O’Neill’s formulas ([2, Theorem 9.28]), for any V ∈ νp(Σ),

(X,U, Y, V ) = 〈(DXT )UV, Y 〉 − 〈TUX,TV Y 〉+ 〈(DUA)XY, V 〉+ 〈AXU,AY V 〉

= 0 + 0 +1

2〈R∇(X,Y )U, V 〉+ 0 =

1

2(X,Y, U, V ).

We wish to mention an alternative proof of this Corollary which was described to us by PeterPetersen. Unlike the proof above, it does not rely on Perelman’s Theorem.

Alternative proof of Corollary 3.1.6. From the vanishing of the mixed curvatures and the factthat the soul is totally geodesic, we see that:

0 = R(X + Y, U)(X + Y ) = R(X,U)Y +R(Y, U)X + 0 + 0.

By the Bianchi identity:R(X,Y )U = R(X,U)Y −R(Y, U)X.

From these two observations it follows that R(X,Y )U = 2R(X,U)Y .

We are now ready to study the sectional curvature of an arbitrary 2-plane at the soul. Thefollowing formula appears in [40, page 615]:

Proposition 3.1.7 (Walschap). Let p ∈ Σ, X,Y ∈ TpΣ, and U, V ∈ νp(Σ). Then:

K(X + U, Y + V ) = K(X,Y ) +K(U, V ) + 3(X,Y, U, V )

10

Proof. Expanding the left hand side linearly gives:

K(X + U, Y + V ) = (X,Y,X, Y ) + (X,V,X, V ) + (U, Y, U, Y ) + (U, V, U, V ) +

2(X,Y,X, V ) + (X,Y, U, Y ) + (X,Y, U, V ) +

(X,V, U, Y ) + (X,V, U, V ) + (U, Y, U, V )

Most of the 10 terms in this expression can be simplified. Specifically:

• (X,V,X, V ) = (U, Y, U, Y ) = (X,V, U, V ) = (U, Y, U, V ) = 0 by Lemma 3.1.3

• (X,Y,X, V ) = (X,Y, U, Y ) = 0 because the soul is totally geodesic.

• (X,V, U, Y ) = 12 (X,Y, U, V ) by Corollary 3.1.6

The following corollary also appears in [40, page 615]:

Corollary 3.1.8 (Walschap). Let p ∈ Σ, X,Y ∈ TpΣ and U, V ∈ νp(Σ). Then:

1. 3(X,Y, U, V ) ≤ K(X,Y ) +K(U, V ).

Equivalently,

2. 9(X,Y, U, V )2 ≤ 4K(X,Y ) ·K(U, V ).

Proof. Equation 1 just comes from Proposition 3.1.7, plus the fact that M has nonnegative cur-vature. Equation 2 follows from equation 1 as follows. Choose arbitrary vectors X,Y ∈ TpΣ andU, V ∈ νp(Σ). It is easy to see that if K(X,Y ) = 0 then (X,Y, U, V ) = 0; otherwise rescaling V bya suitably small constant would contradict equation 1, since (X,Y, U, V ) depends linearly on |V |,while K(U, V ) depends quadratically on |V |. Similarly if K(U, V ) = 0 then (X,Y, U, V ) = 0. Itremains to verify equation 2 when neither K(X,Y ) nor K(U, V ) is zero.

Choose λ1, λ2 so that K(X,λ1Y ) = K(U, λ2V ). Then 3(X,λ1Y, U, λ2V ) ≤ K(X,λ1Y ) +K(U, λ2V ) = 2K(X,λ1Y ), so 9(X,λ1Y, U, λ2V )2 ≤ 4K(X,λ1Y )2 = 4K(X,λ1Y )K(U, λ2V ). Inother words, equation 2 is true for the vectors X,λ1Y, U, λ2V . But since equation 2 is scale-invariant, it must also be true of the vectors X,Y, U, V .

In fact equations 1 and 2 are equivalent. To see that equation 2 implies equation 1, just noticethat for any real numbers A,B,C, if A,B ≥ 0 and C2 ≤ 4AB then C ≤ A + B. To see this, setλ := A/B. Since C2 ≤ 4AB = 4λB2,

C ≤ 2√λB ≤ (1 + λ)B = A+B.

The following splitting theorem was observed independently by Marenich [20, Theorem 2] andWalschap [40, Proposition 2.6]:

Corollary 3.1.9. If for every p ∈ Σ and for every U, V ∈ νp(Σ), K(U, V ) = 0, then M splitslocally isometrically over it’s soul.

Proof. If the vertical curvatures vanish, then the previous corollary implies that the curvaturetensor R∇ vanishes, and the splitting now follows from Theorem 2.2.1.

11

3.2 Curvature near the soul

In the preceding section we derived a formula for the curvature of an arbitrary 2-plane at a pointp ∈ Σ. In this section we study the derivatives of the function which records the curvatures of afamily of 2-planes initially based at the soul but then drifting away from the soul. The family willbegin with a mixed 2-plane at the soul (which has zero curvature by Lemma 3.1.3), and will thendrift so that the base point moves away from the soul while simultaneously the 2-plane twists awayfrom being a mixed 2-plane.

More precisely, the set up for this section is as follows. Let p ∈ Σ. Let X,Y ∈ TpΣ and letW,V,U ∈ νp(Σ). Let γ(t) := exp⊥(tW ), and let Xt, Yt, Ut, Vt denote the parallel transports ofX,Y, U and V along γ(t). By Perelman’s Theorem, Xt and Yt are horizontal for all t ∈ [0,∞).In other words, parallel translation along the radial geodesic γ preserves the horizontal space. Ittherefore must also preserve the vertical space, and hence Ut and Vt are vertical for all t ∈ [0,∞).Define:

K(t) := KW,X,V,Y,U (t) := K(Xt + tUt, tYt + Vt),

which is the unnormalized sectional curvature of the 2-plane based at γ(t) spanned by Xt+ tUt andtYt + Vt. The special case of this construction when U = Y = 0 was studied by Marenich in [20].Notice that K(0) = K(X,V ) = 0 by Lemma 3.1.3. The goal of this section is to derive formulasfor K ′(0) and K ′′(0). Towards this end, we write:

K(t) = (Xt, Vt, Xt, Vt) + t · 2(Xt, Yt, Xt, Vt) + 2(Xt, Vt, Ut, Vt) (3.2.1)

+t2 · (Xt, Yt, Xt, Yt) + (Ut, Vt, Ut, Vt) + 2(Xt, Yt, Ut, Vt) + 2(Xt, Vt, Ut, Yt)+t3 · 2(Ut, Yt, Ut, Vt) + 2(Xt, Yt, Ut, Yt)+ t4 · (Ut, Yt, Ut, Yt)

Our calculations will require the following Lemma:

Lemma 3.2.1. Let M be a Riemannian manifold and let A denote a tensor of order k onM (that is, for each p ∈ M , Ap : (TpM)k → R). For p ∈ M and X,Y ∈ TpM , if Ap = 0 then(DXDY A)p = (DY DXA)p.

Proof. The proof is straightforward and is left to the reader

Proposition 3.2.2. K ′(0) = 0

Proof. Since M has nonnegative curvature, K ′(0) ≥ 0. But if it were the case that K ′(0) > 0,then replacing W with −W would yield K ′(0) < 0. Hence K ′(0) = 0.

This provides an a priori reason to expect that K ′(0) = 0. In order that our proof generalizesproperly in the next section, we also compute K ′(0) directly. From Equation 3.2.3.1, we see that

K ′(0) = 2(X,Y,X, V ) + 2(X,V, U, V ) +d

dt

∣

∣

∣

∣

t=0

(Xt, Vt, Xt, Vt).

But (X,Y,X, V ) = 0 because the soul is totally geodesic, and (X,V, U, V ) = 0 by Lemma 3.1.3.We use one of O’Neill’s formulas to study the third term:

d

dt

∣

∣

∣

∣

t=0

(Xt, Vt, Xt, Vt) =d

dt

∣

∣

∣

∣

t=0

〈(DXtT )VtVt, Xt〉 − 〈TVtXt, TVtXt〉+ 〈AXtVt, AXtVt〉

=d

dt

∣

∣

∣

∣

t=0

〈(DXtT )VtVt, Xt〉 − 0 + 0

= 〈Ddt

((DXtT )VtVt), X〉 = 〈(DWDXT )V V,X〉 = 〈(DXDWT )V V,X〉 = 0.

In the second equality above, the second and third terms vanish because, by Leibnitz’ rule, thederivative at time t=0 of the inner product of two vector fields both of which vanish at t=0 mustbe zero.

12

Proposition 3.2.3.

K ′′(0) = 2K(X + U, Y + V ) + 2〈(DXR∇)(X,Y )W,V 〉+ 1

2|R∇(W,V )X|2

+〈4(DWDUT )V V − 4(DWDV T )UV + (DXD2WT )V V,X〉

Proof. From Equation 3.2.3.1, we see that:

K ′′(0) = 2(X,Y,X, Y ) + (U, V, U, V ) + 2(X,Y, U, V ) + 2(X,V, U, Y )

+2d

dt

∣

∣

∣

∣

t=0

2(Xt, Yt, Xt, Vt) + 2(Xt, Vt, Ut, Vt)

+d2

dt2

∣

∣

∣

∣

t=0

(Xt, Vt, Xt, Vt).

• The top line of this expression is familiar from the proof of Proposition 3.2.2:

(X,Y,X, Y ) + (U, V, U, V ) + 2(X,Y, U, V ) + 2(X,V, U, Y )

= K(X,Y ) +K(U, V ) + 3(X,Y, U, V )

= K(X + U, Y + V )

• Next, from one of O’Neill’s formulas:

d

dt

∣

∣

∣

∣

t=0

(Xt, Yt, Xt, Vt) =d

dt

∣

∣

∣

∣

t=0

〈(DXtA)XtYt, Vt〉+ 2〈AXtYt, TVtXt〉

=d

dt

∣

∣

∣

∣

t=0

〈(DXtA)XtYt, Vt〉 = 〈Ddt

((DXtA)XtYt), V 〉

= 〈(DWDXA)XY, V 〉 = 〈(DXDWA)XY, V 〉

=1

2〈(DXR∇)(X,Y )W,V 〉

• We apply another of O’Neill’s formulas to simplify the next term:

d

dt

∣

∣

∣

∣

t=0

(Xt, Vt, Ut, Vt) = − d

dt

∣

∣

∣

∣

t=0

(Ut, Vt, Vt, Xt)

= − d

dt

∣

∣

∣

∣

t=0

〈(DVtT )Ut

Vt, Xt〉 − 〈(DUtT )Vt

Vt, Xt〉

= −〈(DWDV T )UV,X〉+ 〈(DWDUT )V V,X〉

• Finally,

d2

dt2

∣

∣

∣

∣

t=0

(Xt, Vt, Xt, Vt)

=d2

dt2

∣

∣

∣

∣

t=0

〈(DXtT )VtVt, Xt〉 − 〈TVtXt, TVtXt〉+ 〈AXtVt, AXtVt〉

= 〈(D2WDXT )V V,X〉 − 2〈(DWT )V X, (DWT )V X〉+ 2〈(DWA)XV, (DWA)XV 〉

= 〈(DXD2WT )V V,X〉 − 0 +

1

2|R∇(W,V )X|2.

13

Corollary 3.2.4. If π has totally geodesic fibers (or more generally if the T -tensor of πvanishes to second order at Σ), then for all X,Y ∈ TpΣ and all V,W ∈ νp(Σ):

1. 2〈(DXR∇)(X,Y )W,V 〉 ≤ 12 |R

∇(W,V )X|2 + 2K(X,Y ).

Equivalently,

2. 〈(DXR∇)(X,Y )W,V 〉2 ≤ |R∇(W,V )X|2 ·K(X,Y ).

Proof. For equation 1, set U = 0 in Proposition 3.2.3, and observe that K ′′(0) ≥ 0 simply becauseM has nonnegative curvature. Equations 1 and 2 are equivalent by the argument of Corollary 3.1.8.

Corollary 3.2.4 should be compared to [35, Corollary 3.2].

3.3 Connection metrics on vector bundles

By a connection metric on a vector bundle Rk → Eπ→ Σ, we mean a metric which arises by

choosing the following data:

1. A metric gΣ on Σ.

2. A Euclidean structure on the bundle (which means a smoothly varying inner product on thefibers).

3. A connection ∇ on the bundle which is compatible with the Euclidean structure.

4. A rotationally symmetric metric g0 on Rk.

Then there is a unique metric gE on E (called a connection metric) for which π : (E, gE) → (Σ, gΣ)is a Riemannian submersion whose horizontal distribution is determined by ∇ and such that eachfiber is totally geodesic and isometric to (Rk, g0).

An arbitrary rotationally symmetric metric g0 on Rk can be expressed in polar coordinates asds2 = dr2 + G2(r)dΘ2, where G(r) is a function for which G(0) = 0, G′(0) = 1, and all evenderivatives of G vanish at 0. The curvature of any 2-plane at the origin of (Rk, g0) equals −G′′′(0).

Given such a connection metric, we will denote by R∇ the curvature tensor associated to theconnection∇, and by R the curvature tensor of (E, gE). By definition, the T -tensor of π : (E, gE) →(Σ, gΣ) vanishes. The A-tensor of this Riemannian submersion can be described as follows. For allp ∈ Σ, X,Y ∈ TpΣ, and W ∈ Ep = π−1(p), if X and Y denote the horizontal lifts of X and Y toTWE, then, as mentioned in Equation 3.1.3.1:

A(X, Y ) =1

2R∇(X,Y )W. (3.3.1)

Also as before, for X,Y ∈ TpΣ and U, V,W ∈ Ep := π−1(p), denote K(t) = KW,X,V,Y,U (t) =K(Xt+tUt, tYt+Vt), where Xt, Yt, Ut, Vt are the parallel translation of X,Y, U, V along the geodesicin the direction of W .

It is still unknown which vector bundles over spheres (or more general souls) admit connectionmetrics of nonnegative curvature. In [35], Strake and Walschap studied this question. They derivedconditions on gB ,∇, g0 under which such a connection metric will have nonnegative curvature.Their main result is as follows:

Theorem 3.3.1 (Strake-Walschap). Suppose (Σ, gΣ) is a compact Riemannian manifold

with strictly positive sectional curvature. Suppose that Rk → Eπ→ Σ is a vector bundle over

Σ with a Euclidean structure and connection ∇. If there exists ε > 0 such that for eachp ∈ Σ, X,Y ∈ TpΣ and V,W ∈ Ep := π−1(p), the following conditions hold:

14

1. 〈(DXR∇)(X,Y )W,V 〉2 ≤ ( 12 − ε) · |R∇(W,V )X|2 ·K(X,Y ).

2. If R∇(W,V )X = 0 then X = 0 or W is parallel to V .

then a rotationally symmetric metric g0 on Rk can be chosen such that the correspondingconnection metric gE on E has nonnegative sectional curvature.

In this section we will use the formulas and observations of the preceding two sections to obtainresults similar to Strake and Walschap’s above theorem.

Perelman’s Theorem demonstrates that an arbitrary open manifold of nonnegative curvature ismuch more similar than previously expected to the total space of a vector bundle with a connectionmetric. For example, the first statement of Perelman’s Theorem (the existence of Perelman flats)is true for any connection metric on any vector bundle:

Lemma 3.3.2. Let π : E → Σ denote a vector bundle over Σ with a connection metric.

1. The zero-section, Σ ⊂ E, is totally geodesic.

2. Let p ∈ Σ, X ∈ TpΣ, and V ∈ Ep. Then the surface (s, t) 7→ exp(s ·V (t)) (s, t ∈ R, s ≥ 0),where V (t) denotes the parallel transport of V along the geodesic with initial tangentvector X, is a flat and totally geodesic half-plane.

3. For sufficiently small r, the closed ball BΣ(r) is convex.

Proof. The first two claims are elementary. The third follows from the second via the argument inthe proof of Proposition 2.1.5.

It should be unsurprising that most of the formulas from the previous two sections are also trueof vector bundles with connection metrics. Specifically:

Proposition 3.3.3. Let Rk → Eπ→ Σ denote a vector bundle with a connection metric. Let

p ∈ Σ, X,Y ∈ TpΣ, and U, V,W ∈ Ep := π−1(p). Then:

1. AXY = AXU = 0.

2. (DV A)XY = 12R

∇(X,Y )V and (DV A)XU = − 12R

∇(V,U)X.

3. R(X,V )V = R(V,X)X = 0 and R(X,Y )U = 2R(X,U)Y .

4. K(X + U, Y + V ) = K(X,Y ) +K(U, V ) + 3(X,Y, U, V ).

5. K ′W,X,V,Y,U (0) = 0.

6. K ′′W,X,V,Y,U (0) = 2K(X + U, Y + V ) + 2〈(DXR∇)(X,Y )W,V 〉+ 1

2 |R∇(W,V )X|2.

Proof. Item 1 is obvious. Item 2 is clear from Equation 3.3.3.1. For the first part of item 3, useO’Neill’s formulas plus the vanishing of the T -tensor to show that (X,V, V, Y ) = (X,V, V, U) =(V,X,X, Y ) = (V,X,X,U) = 0. Then the second part of item 3 follows from the first by theargument of Corollary 3.1.6. Items 4,5, and 6 are proven exactly like their analogs in the previoustwo sections.

We will ask a slightly easier question than did Strake and Walschap. Instead of seeking aconnection metric on a bundle which has nonnegative curvature, we will only seek a connectionmetric which, up to second order calculations at the soul, looks like it has nonnegative curvature.This notion is captured by the following definition:

Definition 3.3.4. A metric gE on the total space E of a vector bundle Rk → E → Σ is said to be“nonnegatively curved to second order” if:

15

1. The zero section Σ ⊂ E is totally geodesic.

2. For every p ∈ Σ, X,Y ∈ TpΣ and U, V ∈ Ep, K(X + U, Y + V ) ≥ 0 and = 0 iff X is parallelto Y and U is parallel to V .

3. If p ∈ Σ, X,Y ∈ TpΣ and U, V,W ∈ Ep, then K ′W,X,V,Y,U (t) = 0 and K ′′

W,X,V,Y,U (0) ≥ 0.Further, when W,X, V 6= 0, K ′′

W,X,V,Y,U (0) = 0 iff V and U are both parallel to W , and Y isparallel to X, in which case KW,X,V,Y,U (t) = 0 for all t.

Item 2 of this definition says that all 2-planes at Σ have nonnegative sectional curvature, andonly the mixed 2-planes are flat. Therefore, if our goal is to insure that a neighborhood of Σ inE is nonnegatively curved, then the only trouble spot is a neighborhood (in the Grassmannian of2-planes on M , which we denote by G) of the set, Z, of mixed 2-planes at Σ. Item 3 says thatalong any “straight line” l(t) in G beginning in Z, the curvature function K(t) = K(l(t)) satisfiesK ′(0) = 0 andK ′′(0) ≥ 0 and = 0 iff all of l(t) lies on a single “Perelman flat”. Thus, a metric whichis nonnegatively curved to second order is a very good candidate for having nonnegative curvaturein a neighborhood of its zero-section. We do not know whether a connection metric which isnonnegatively curved to second order is necessarily nonnegatively curved in a neighborhood of itszero-section. This issue is of interest because of the following proposition:

Proposition 3.3.5. Suppose that Eπ→ Σ is a vector bundle with a connection metric gE on

E which is nonnegatively curved in a neighborhood of the zero-section Σ ⊂ E. Then thereexists a complete metric g′E on E which is everywhere nonnegatively curved.

Proof. According to part 3 of Lemma 3.3.2, BΣ(r) is convex for sufficiently small r > 0. Thestatement then follows from [12, Theorem A] (or from [22, Theorem 1.2], on which the previousreference is based). In fact, it is easy to see from Guijarro’s construction that the new metric g′Ewill also be a connection metric.

Thus, a connection metric which is nonnegatively curved to second order can very likely bedeformed to a connection metric of nonnegative curvature. This motivates our interest in Defini-tion 3.3.4. We prove the following:

Theorem 3.3.6. Suppose (Σ, gΣ) is a compact Riemannian manifold with strictly positive

curvature, and Rk → Eπ→ Σ is a vector bundle over Σ. Then there exists a connection

metric gE on E with soul (Σ, gΣ) which has nonnegative curvature to second order iff thebundle admits a Euclidean structure and a connection ∇ such that the following conditionholds for each p ∈ Σ, X,Y ∈ TpΣ and V,W ∈ Ep:

(*) 〈(DXR∇)(X,Y )W,V 〉2 ≤ |R∇(W,V )X|2 ·K(X,Y ) with equality iff Y is parallel to X orW is parallel to V .

Lemma 3.3.7. Condition (*) of Theorem 3.3.6 is equivalent to:

(**) 2〈(DXR∇(X,Y )W,V 〉 ≤ 12 |R

∇(W,V )X|2 + 2K(X,Y ). Further, when X,W, V 6= 0, thereis equality iff Y is parallel to X and W is parallel to V .

Proof. The inequalities in conditions (*) and (**) are equivalent by Corollary 3.2.4, so it remainsonly to discuss the equality cases. First we prove that (**) implies (*). Setting Y = 0 in (**)implies the following:

If R∇(W,V )X = 0 then X = 0 or W is parallel to V . (3.3.2)

Suppose that 〈(DXR∇)(X,Y )W,V 〉2 = |R∇(W,V )X|2 · K(X,Y ). Suppose also that Y is notparallel to X, and W is not parallel to V . Since Σ has strictly positive curvature, K(X,Y ) 6= 0.

16

By 3.3.3.2, |R∇(W,V )X|2 6= 0. Choose λ1, λ2 such that |R∇(W,λ2V )X|2 = 4 ·K(X,λ1Y ). Sincecondition (*) is invariant under rescalling of the vectors,

〈(DXR∇)(X,λ1Y )W,λ2V 〉2 = |R∇(W,λ2V )X|2 ·K(X,λ1Y ) = 4K(X,λ1Y )2,

and hense,

〈(DXR∇(X,λ1Y )W,λ2V 〉 = 2K(X,λ1Y ) =1

4|R∇(W,λ2V )X|2 +K(X,λ1Y ),

which contradicts (**).Next we prove that (*) implies (**). First notice that (*) implies 3.3.3.2. Assume that

2〈(DXR∇(X,Y )W,V 〉 = 12 |R

∇(W,V )X|2 + 2K(X,Y ), and that X,V,W 6= 0. Since Σ has strictlypositive curvature, if W is parallel to V then X must be parallel to Y . By 3.3.3.2, if X is parallel toY , then W must be parallel to V . So it remains to contradict the hypothesis that X is not parallelto Y AND W is not parallel to V . Assume that this is the case. Define:

f(t) := −2〈(DXR∇(X, tY )W,V 〉+ 1

2|R∇(W,V )X|2 + 2K(X, tY ).

f(t) is a nonnegative valued quadradic function with f(1) = 0, and hense its discriminant equals0. This means that 〈(DXR∇)(X,Y )W,V 〉2 = |R∇(W,V )X|2 ·K(X,Y ), which contradicts (*).

Lemma 3.3.8. Conditions (*) and (**) are also equivalent to the following equivalent con-ditions:

(*ε) For some ε > 0, 〈(DXR∇)(X,Y )W,V 〉2 ≤ (1− ε)|R∇(W,V )X|2 ·K(X,Y ) with equality iffY is parallel to X or W is parallel to V .

(**ε) For some ε > 0, 2〈(DXR∇(X,Y )W,V 〉 ≤ 12 |R

∇(W,V )X|2 + 2(1 − ε)K(X,Y ). Further,when X,W, V 6= 0, there is equality iff Y is parallel to X and W is parallel to V .

Proof. Clearly (*ε)⇒(*) and (**ε)⇒(**). Also, (*ε)⇔(**ε) by the argument of the previous lemma.So it remains to prove that (*)⇒(*ε). Assuming (*), a compactness argument provides an ε > 0for which

〈(DXR∇)(X,Y )W,V 〉2 ≤ (1− ε)|R∇(W,V )X|2 ·K(X,Y )

whenever |X| = |Y | = |W | = |V | = 1, X ⊥ Y and W ⊥ V . But this equation is not sensitiverescallings of the vectors, nor is it sensitive to replacing Y (respectively V ) with its componentorthogonal to X (respectively W ). Hense, this equation holds for all X,Y,W, V with no restrictions.

Proof of Theorem 3.3.6. If the bundle admits a connection metric which is nonnegatively curvedto second order, then condition (**) is satisfied by part 6 of Proposition 3.3.3. Conversely, supposethat condition (**ε) is satisfied. We can choose a radially symmetric metric g0 on Rk such that thecurvature of every 2-plane at the origin of (Rk, g0) is arbitrarily large; in particular, we can makethese curvatures larger than:

sup

9(X,Y, U, V )2

4εK(X,Y )

∣

∣

∣

∣

X,Y ∈ TpΣ and U, V ∈ Ep orthonormal

Then 9(X,Y, U, V )2 ≤ 4εK(X,Y ) · K(U, V ) for all orthonormal vectors X,Y, U, V , and hencefor all (not necessarily orthonormal) vectors as well. This is equivalent to the following inequality:

3(X,Y, U, V ) ≤ εK(X,Y ) +K(U, V ) (3.3.3)

17

(see the discussion of equivalence in Corollary 3.1.8). So by part 4 of Proposition 3.3.3, all 2-planesat Σ are nonnegatively curved, and it is easy to see that only the mixed 2-planes are flat.

Next, combining (**ε) with Equation 3.3.3.3 gives:

2〈(DXR∇)(X,Y )W,V 〉 ≤ 1

2|R∇(W,V )X|2 + 2K(X,Y )− 2εK(X,Y )

≤ 1

2|R∇(W,V )X|2 + 2K(X,Y )− 23(X,Y, U, V )−K(U, V )

=1

2|R∇(W,V )X|2 + 2K(X − U, Y + V ).

It follows from part 6 of Proposition 3.3.3 that K ′′W,X,V,Y,U (0) ≥ 0. Assuming that W,X, V 6= 0,

the second inequality above becomes an equality iff Y is parallel to X and U is parallel to V . Inthis case, the first inequality above becomes an equality iff V and W are parallel.

3.4 The warping tensor

As before, let M be an open manifold of nonnegative curvature and let π : M → Σ denote themetric projection onto its soul. We define the “warping function” and the “warping tensor” of M ,both of which reflect how the metric on M near Σ differs from the connection metric with flat fiberson ν(Σ). The second derivatives of the T -tensor of π (which appear in Proposition 3.2.3) can beexpressed nicely in terms of the warping tensor.

For p ∈ Σ and W,U, V ∈ νp(Σ), define:

F (W,U, V ) := 〈(d exp⊥)WU, (d exp⊥)WV 〉.

We will call F the warping function of M . Since the metric on M is smooth, F is smooth in thedomain where |W | < the cut-distance of Σ. The following properties of F are obvious from thedefinition:

Lemma 3.4.1. For any W,U, V ∈ νp(Σ),

1. F (0, U, V ) = 〈U, V 〉

2. If |W | <the cut-distance of Σ, then F (W, ·, ·) is a symmetric positive-definite bilinearform.

3. F (W,W,U) = 〈W,U〉.

4. ddt |t=0F (tW,U, V ) = 0

5. d2

dsdt |s=t=0F (tW1 + sW2, U, V ) = d2

dsdt |s=t=0F (tU + sV,W1,W2).

The last two properties follow from the smoothness of the metric (or equivalently from thesmoothness of F ).

Next, for every quadruple (W1,W2, U, V ) of vectors in νp(Σ), define:

Υ(W1,W2, U, V ) :=d2

dsdt

∣

∣

∣

∣

s=t=0

F (tW1 + sW2, U, V ).

We will call Υ the warping tensor of M . It records second derivative information about how themetric onM differs near the soul from the connection metric with flat fibers on ν(Σ). For fixed U, V ,think of Υ as the second derivative of the function from νp(Σ) to R which sends W 7→ F (W,U, V ).The following properties of Υ are clear from definition:

18

Lemma 3.4.2.

1. Υ is multi-linear, symmetric in (W1,W2), and symmetric in (U, V ).

2. Υ(W,W,W,U) = 0.

3. Υ(W1,W2, U1, U2) = Υ(U1, U2,W1,W2)

Our next goal is to express the second derivatives of the T -tensor (which appear in Proposi-tion 3.2.3) in terms of Υ. In order to state the result, we need to establish some notation. Wedescribe a method for extending a vector U ∈ νp(Σ) to a vertical vector field U on a neighborhood

of p in M . Fist, for q ∈ Σ near p, define Uq as the parallel transport of U to q along the minimal

geodesic from p to q in Σ. This defines an extension U of U to a neighborhood of p in Σ. Thenfor each q ∈ Σ near p, extend the vector Uq to a vertical vector field U on the fiber Fq by defining

Uexp⊥(W ) := (d exp⊥)W Uq for all vectors W ∈ νq(Σ) with small norm.Additionally, when X ∈ TpΣ, we will denote by X an extension of X to a basic vector field on

M in a neighborhood of p.For fixed vectors W,U, V ∈ νp(Σ), we can think of F (W,U, V ) as a real-valued function on

a neighborhood of p in Σ, by parallel transporting W,U, V along radial geodesics from p. Inother words, the function is q 7→ F (Wq, Uq, Vq). With this interpretation, it makes sense towrite gradF (W,U, V ) ∈ TpΣ for the gradient of this function and XF (W,U, V ) for the deriva-tive of this function in the direction X ∈ TpΣ. Similarly, Υ(W1,W2, U, V ) can be thought of asa real-valued function on a neighborhood of p in Σ, and we can write gradΥ(W1,W2, U, V ) orXΥ(W1,W2, U, V ) = DXΥ(W1,W2, U, V ). By hessΥ(W1,W2,U,V )(X) we mean the hessian of thisfunction in the direction X ∈ TpΣ.

Lemma 3.4.3. Let W,W1,W2, U, V ∈ νp(Σ). Assume |W | < the cut-distance of Σ. Letp := exp⊥(W ).

1. 〈TV X, U〉p = 12XF (W,U, V ) for any X ∈ TpΣ.

2. (TV U)p = − 12gradF (W,U, V ).

3. (DW1DW2T )UV = − 12gradΥ(W1,W2, U, V ).

Proof. First notice that X and U commute simply because their preimages under d exp⊥ in T (ν(Σ))commute. So, using the standard coordinate-free expression for the connection,

2〈TV X, U〉p = 2〈∇V X, U〉p= X〈V , U〉p + V 〈U , X〉p − U〈X, V 〉p

−〈[X, U ], V 〉p − 〈[U , V ], X〉p − 〈[X, V ], U〉p= X〈V , U〉p = XF (W,U, V ).

This proves part 1. Part 2 is proven as follows:

〈TV U , X〉p = −〈TV X, U〉p = −1

2XF (W,U, V )

= 〈−1

2gradF (W,U, V ), X〉 = 〈−1

2gradF (W,U, V ), X〉p.

For part 3, it will suffice to prove that (D2WT )UV = − 1

2gradΥ(W,W,U, V ), which is done as follows.Let γ(t) = exp⊥(tW ). Then:

19

DW (DWTUV ) =D

dt

∣

∣

∣

∣

t=0

(DWTU V )γ(t) − 0− 0

=D

dt

∣

∣

∣

∣

t=0

(

D

dr

∣

∣

∣

∣

r=0

(TU V )γ(t+r) − (T(∇W U)V )γ(t) − (TU (∇W V ))γ(t)

)

=D2

dt2

∣

∣

∣

∣

t=0

(TU V )γ(t) − 0− 0

= −1

2

D2

dt2

∣

∣

∣

∣

t=0

(gradF (tW,U, V ))γ(t) = −1

2

D2

dt2

∣

∣

∣

∣

t=0

gradF (tW,U, V )

= −1

2grad

(

d2

dt2

∣

∣

∣

∣

t=0

F (tW,U, V )

)

= −1

2gradΥ(W,W,U, V ).

To justify the third equality above, notice that:

D

dt

∣

∣

∣

∣

t=0

(T(∇W U)V )γ(t) = (DWT )(∇W U)V + T(∇W∇W U)V + T(∇W U)(∇W V )

= 0 + 0 + 0 = 0

Next, we describe how to calculate the curvature of a 2-plane at the soul in terms of the warpingtensor.

Lemma 3.4.4. Let W,V ∈ νp(Σ). Then K(W,V ) = − 32Υ(W,W, V, V ).

Proof. It will suffice to prove this when W,V are orthonormal. Let S denote the surface in Mobtained as the exponential image of the plane spanned by W and V . K(W,V ) is the Gausscurvature of S at p. Write the metric on S in polar coordinates:

ds2 = dr2 + f2(r, θ)dθ2,

where θ = 0 corresponds to the direction of W . Let γ(r) = exp⊥(rW ) (in polar coordinates,γ(r) = (r, 0)). Along γ, f can be expressed as:

f(r, 0) = r√

F (rW,U,U).

The Gauss curvature of S at (r, 0) equals −frr(r,0)f(r,0) , where frr denotes the second partial with respect

to r. The result now follows by performing the differentiation and taking the limit as r → 0.

It is now possible to express K ′′W,X,V,Y,U (0) in terms only of the connection, the warping tensor,

and the metric of the soul:

Proposition 3.4.5.

K ′′W,X,V,Y,U (0) = 2K(X,Y ) + 6〈R∇(X,Y )U, V 〉 − 3Υ(U,U, V, V )

+2〈(DXR∇)(X,Y )W,V 〉+ 1

2|R∇(W,V )X|2

−2DXΥ(W,U, V, V ) + 2DXΥ(W,V,U, V )− 1

2hessΥ(W,W,V,V )(X).

Proof. The formula is obtained from the formula of Proposition 3.2.3 by using Lemma 3.4.4 tore-writing the term K(U, V ) and using Lemma 3.4.3 to re-write the three T -tensor terms.

20

Corollary 3.4.6. For all X,Y ∈ TpM and all V,W ∈ νp(Σ):

1. 2〈(DXR∇)(X,Y )W,V 〉 ≤ 12 |R

∇(W,V )X|2 + 2K(X,Y )− 12hessΥ(W,W,V,V )(X).

Equivalently,

2. 〈(DXR∇)(X,Y )W,V 〉2 ≤ (|R∇(W,V )X|2 − hessΥ(W,W,V,V )(X)) ·K(X,Y ).

Proof. Part 1 is obtained by setting U = 0 in Proposition 3.4.5, and noting that K ′′(0) ≥ 0. Theequivalence of part 1 and part 2 is proven as in Corollary 3.2.4.

3.5 Warped connection metrics on vector bundles

In Section 3.3, we studied conditions for the existence of a connection metric of nonnegative cur-vature on a vector bundle. But connection metrics are a very special class of metrics. It mightbe possible for a bundle to admit a metric of nonnegative curvature, but fail to admit a connec-tion metric of nonnegative curvature (deciding whether this phenomenon occurs remains an openproblem).

In Section 3.4, we began to generalize away from connection metrics; specifically, Corollary 3.4.6provides a necessary condition for the existence of an arbitrary metric of nonnegative curvature ona bundle (namely, the bundle must admit a connection ∇ and a tensor Υ which together satisfythe differential equation of the corollary). The goal of this section is to prove that this conditionis almost sufficient, at least for the existence of a metric which is nonnegatively curved to secondorder. We begin by constructing warped connection metrics on vector bundles.

By a “warped connection metric” on a vector bundle Rk → Eπ→ Σ, we mean a metric which

arises by choosing the following data:

1. A metric gΣ on Σ.

2. A Euclidean structure on the bundle.

3. A connection ∇ on the bundle which is compatible with the Euclidean structure.

4. A smooth “warping function” F (for each p ∈ Σ, Fp : Ep × Ep × Ep → R) which satisfies:

(a) F (0, U, V ) = 〈U, V 〉.(b) For fixed W , F (W, ·, ·) is a symmetric positive-definite bilinear form.

(c) F (W,W,U) = 〈W,U〉.

Then there is a unique metric gE on E (called a warped connection metric) for which π : (E, gE) →(Σ, gΣ) is a Riemannian submersion whose horizontal distribution is determined by ∇ and such that〈(d exp⊥)WU, (d exp⊥)WV 〉 = F (W,U, V ) for every W,U, V ∈ Ep. To construct gE explicitly, onecan be begin with the connection metric with flat fibers on the bundle, and then alter the metricon the vertical space according to the warping function F .

In other words, a warped connection metric is an arbitrary metric for which the bundle projectionis a Riemannian submersion and exp⊥ : ν(Σ) → E is a diffeomorphism.

The following properties of F are consequences of the definition of a warping function (partic-ularly of the smoothness of F ):

• ddt |t=0F (tW,U, V ) = 0

• d2

dsdt |s=t=0F (tW1 + sW2, U, V ) = d2

dsdt |s=t=0F (tU + sV,W1,W2).

21

The warping tensor Υ corresponding to F is then defined as in the previous section, and has theproperties of Lemma 3.4.2.

Warped connection metrics are much more general than connection metrics. In fact, it is clearfrom Section 3.4 that any metric of nonnegative curvature on a vector bundle agrees inside of thecut-locus of Σ with a warped connection metric:

Proposition 3.5.1. Let M be an open manifold of nonnegative curvature with soul Σ. ThenM agrees in a neighborhood Σ with a warped connection metric on its normal bundle.

The following examples exhibit three simple classes of warped connection metrics.

Example 3.5.2 (Connection metrics). Let G : R → R be a function for which G(0) = 0,G′(0) = 1, and all even derivatives of G vanish at 0. Consider the following warping function:

F (W,U, V ) =G2(|W |)|W |2

〈U, V 〉 for all U, V ⊥ W

(because of properties (b) and (c) in our definition of a warping function, F is completely determinedby the values F (W,U, V ) when U, V ⊥ W ). The corresponding warped connection metric on thebundle is just the connection metric whose radially-symmetric fiber metric is ds2 = dr2+G2(r)dΘ2.The corresponding warping tensor is completely determined by the following case:

Υ(W,W,U,U) =2

3G′′′(0) · |W |2 · |U |2 for all W ⊥ U.

Example 3.5.3 (Simply-warped connection metrics). Let Gp (p ∈ Σ) be a smoothly varyingfamily of functions such that for all p ∈ Σ, Gp(0) = 0, G′

p(0) = 1 and all even derivatives of Gp

vanish at 0. For example, we could take:

G2p(r) :=

ε2(p)r2

ε2(p) + r2, (3.5.1)

where ε : Σ → R is a smooth positively-valued function. Consider the following warping function:

Fp(W,U, V ) =G2

p(|W |)|W |2

〈U, V 〉 for all U, V ⊥ W.

We call the corresponding metric a “simply-warped connection metric”. The fiber over p ∈ Σ willhave the radially symmetric metric ds2 = dr2 + G2

p(r)dΘ2. The curvature of any vertical 2-plane

at p is −G′′′p (0). The corresponding warping tensor is determined by the following case:

Υp(W,W,U,U) =2

3G′′′

p (0) · |W |2 · |U |2 for all W ⊥ U.

For example, equation 3.5.3.1 gives:

Υp(W,W,U,U) = − 2

ε2(p)· |W |2 · |U |2 for all W ⊥ U. (3.5.2)

Example 3.5.4 (Quadratically-warped connection metrics). Let Υ be a warping tensor, bywhich we mean a tensor of order 4 on the vector bundle which satisfies the following conditions:

1. Υ(W1,W2, U, V ) is symmetric in (W1,W2), and symmetric in (U, V ).

2. Υ(W,W,W,U) = 0.

3. Υ(W1,W2, U1, U2) = Υ(U1, U2,W1,W2)

22

Consider the warping function:

F (W,U, V ) := 〈U, V 〉+ 1

2Υ(W,W,U, V ).

The corresponding metric will be called a “quadratically-warped connection metric”. The corre-sponding warping tensor is Υ. Notice that the metric in this example might not be positive-definiteeverywhere, but it is at least positive-definite in a neighborhood of the zero-section. For exam-ple, in the quadratically warped connection metric associated to the warping tensor defined byequation 3.5.3.2, the fiber Ep has the radially-symmetric metric ds2 = dr2 + G2

p(r)dΘ2 where

G2p(r) = r2 − 1

ε2(p)r4, which is a well-defined metric for r < ε(p). Notice that G2

p(r) is just the

fourth degree Taylor polynomial of the function G2p(r) defined by equation 3.5.3.1.

We list below some rigidity properties which warped connection metrics have in common withnonnegatively curved metrics on vector bundles:

Lemma 3.5.5. Let π : E → Σ denote a vector bundle over Σ with a warped connectionmetric. Let F denote the warping function and let Υ denote the warping tensor. Let A, Tdenote the fundamental tensors of π. Let p ∈ Σ, X,Y ∈ TpΣ and let U, V,W,W1,W2 ∈ Ep.Then:

1. The zero-section, Σ ⊂ E, is totally geodesic.

2. The surface (s, t) 7→ exp(s · V (t)) (s, t ∈ R, s ≥ 0), where V (t) denotes the paralleltransport of V along the geodesic with initial tangent vector X, is a flat and totallygeodesic half-plane.

3. For sufficiently small r, the closed ball BΣ(r) is convex.

4. Ap = 0 and Tp = 0.

5. (DV A)XY = 12R

∇(X,Y )V and (DV A)XU = − 12R

∇(V,U)X.

6. DWT = 0 and (DW1DW2T )UV = − 12gradΥ(W1,W2, U, V ).

7. R(X,V )V = R(V,X)X = 0 and R(X,Y )U = 2R(X,U)Y .

8. K(X + U, Y + V ) = K(X,Y ) +K(U, V ) + 3(X,Y, U, V ).

9. K ′W,X,V,Y,U (0) = 0.

10. K ′′W,X,V,Y,U (0) is given by the equation of Proposition 3.4.5.

Proof. The only claim which is not immediate from previous arguments is that DWT = 0, whichis justified as follows. Let γ(t) = exp⊥(tW ). Then:

DWTUV =D

dt

∣

∣

∣

∣

t=0

(TU V )γ(t) = −1

2

D

dt

∣

∣

∣

∣

t=0

(gradF (tW,U, V ))γ(t)

= −1

2

D

dt

∣

∣

∣

∣

t=0

gradF (tW,U, V ) = −1

2grad

(

d

dt

∣

∣

∣

∣

t=0

F (tW,U, V )

)

= 0

As a consequence of part 3 of Lemma 3.5.5, we get the following generalization of Proposi-tion 3.3.5:

23

Proposition 3.5.6. Suppose that Eπ→ Σ is a vector bundle with a warped connection metric

gE on E which is nonnegatively curved in a neighborhood of the zero-section Σ ⊂ E. Thenthere exists a complete metric g′E on E which is everywhere nonnegatively curved.

Theorem 3.5.7. Suppose (Σ, gΣ) is a compact Riemannian manifold with strictly positive

sectional curvature. Suppose that Rk → Eπ→ Σ is a vector bundle over Σ. Then there exists

a warped connection metric gE on E with soul (Σ, gΣ) which has nonnegative curvature tosecond order iff the bundle admits a Euclidean structure, a connection ∇, and a warpingtensor Υ such that the following condition holds for all p ∈ Σ, X,Y ∈ TpΣ and V,W ∈ Ep:

(*) 〈(DXR∇)(X,Y )W,V 〉2 ≤(

|R∇(W,V )X|2 − hessΥ(W,W,V,V )(X))

·K(X,Y ) with equality iffY is parallel to X or W is parallel to V .

Lemma 3.5.8. Condition (*) of Theorem 3.5.7 is equivalent to any one of the following:

(*ε) 〈(DXR∇)(X,Y )W,V 〉2 ≤ (1 − ε)(

|R∇(W,V )X|2 − hessΥ(W,W,V,V )(X))

·K(X,Y ) for someε > 0, with equality iff Y is parallel to X or W is parallel to V .

(**) 2〈(DXR∇(X,Y )W,V 〉 ≤ 12 (|R

∇(W,V )X|2−hessΥ(W,W,V,V )(X))+2K(X,Y ). Further, whenX,W, V 6= 0, there is equality iff Y is parallel to X and W is parallel to V .

(**ε) 2〈(DXR∇(X,Y )W,V 〉 ≤ 12 (|R

∇(W,V )X|2−hessΥ(W,W,V,V )(X))+2(1−ε)K(X,Y ) for someε > 0. Further, when X,W, V 6= 0, there is equality iff Y is parallel to X and W isparallel to V .

Proof. This is proven exactly like Lemmas 3.3.7 and 3.3.8.

Lemma 3.5.9. A warped connection metric gE on the total space E of a vector bundleRk → E

π→ Σ is nonnegatively curved to second order iff:

1. For every p ∈ Σ, X,Y ∈ TpΣ and U, V ∈ Ep, K(X + U, Y + V ) ≥ 0 and = 0 iff X isparallel to Y and U is parallel to V .

2. If p ∈ Σ, X,Y ∈ TpΣ, and U, V,W ∈ Ep satisfy |X| = |V | = |W | = 1, W ⊥ V , X ⊥ Y ,and U ⊥ V , then K ′′

W,X,V,Y,U (0) > 0.

Proof. If gE is nonnegatively curved to second order, then it satisfies hypothesis 1 and 2 of thelemma by definition. Conversely, assume that gE satisfies hypotheses 1 and 2 of the lemma. Letp ∈ Σ, X,Y ∈ TpΣ and U, V,W ∈ Ep be arbitrary vectors with W,V,X 6= 0. If V is a multipleof W , then by Proposition 3.4.5, K ′′

W,X,V,Y,U (0) = 2K(X + U, Y + V ), which by property 1 of thelemma is ≥ 0, and equals zero only when Y is parallel to X and U is parallel to V . So to provethat gE has nonnegative curvature to second order, it remains to prove that if V is not a multipleof W , then K ′′

W,X,V,Y,U (0) > 0.So assume that V is not a multiple of W . First, by Proposition 3.4.5, it is easy to see that

K ′′W,X,V,Y,U (0) = K ′′

W⊥,X,V,Y ⊥,U⊥(0),

where W⊥ denotes the component of W which is orthogonal to V , Y ⊥ denotes the component ofY which is orthogonal to X, and U⊥ denotes the component of U which is orthogonal to V .

Finally, we use the following identities, each of which can be easily checked either from thedefinition of K ′′

W,X,V,Y,U (0) or from it’s formula in Proposition 3.4.5:

• K ′′aW,X,V,aY,aU (0) = a2 ·K ′′

W,X,V,Y,U (0).

• K ′′W,aX,V,Y,aU (0) = a2 ·K ′′

W,X,V,Y,U (0).

• K ′′W,X,aV,aY,U (0) = a2 ·K ′′

W,X,V,Y,U (0).

24

Using these properties, it is easy to choose λ1, λ2 such that:

K ′′W,X,V,Y,U (0) = K ′′

W⊥,X,V,Y ⊥,U⊥(0) = K ′′W⊥|W⊥|

, X|X| ,

V|V | ,λ1Y ⊥,λ2U⊥(0).

The last term of this equation is > 0 by hypothesis 2 of the lemma.

proof of Theorem 3.5.7. If the bundle admits a warped connection metric which is nonnegativelycurved to second order, then condition (**) is clearly satisfied. Conversely, suppose that condition(**ε) is satisfied. For fixed C, let ΥC denote the warping tensor on the vector bundle which isdetermined by the following equation:

ΥC(W,W,U,U) = −2

3· C · |W |2 · |U |2 for all W ⊥ U.

In other words, ΥC is the warping tensor associated to the connection metric for which the curvatureof any vertical 2-plane at the zero-section is C (see Example 3.5.2). We will prove that for sufficientlylarge C, the quadradically-warped connection metric associated to the warping tensor Υ = Υ+ΥC

(see Example 3.5.4) is nonnegatively curved to second order.By choosing C large, one can make the sectional curvature of all vertical 2-planes at Σ arbitrarily

large. It is therefore easy to choose C sufficiently large that condition 1 of Lemma 3.5.9 is satisfied,exactly as done in the proof of Theorem 3.3.6. It remains to satisfy condition 2 of Lemma 3.5.9.

Let p ∈ Σ, X,Y ∈ TpΣ and W,U, V ∈ Ep. Assume that W ⊥ V , X ⊥ Y , U ⊥ V , and|W | = |X| = |V | = 1. We must show that for sufficiently large C, K ′′

W,X,V,Y,U (0) > 0. Since

ΥC is parallel, DXΥ = DXΥ and hessΥ = hessΥ. Hence, Proposition 3.4.5 applied to the metricassociated to Υ, gives the following formula:

K ′′W,X,V,Y,U (0) = 2εK(X,Y ) + 6〈R∇(X,Y )U, V 〉 − 3Υ(U,U, V, V ) (3.5.3)

−2DXΥ(W,U, V, V ) + 2DXΥ(W,V,U, V )

+2(1− ε)K(X,Y )− 1

2hessΥ(W,W,V,V )(X)

+2〈(DXR∇)(X,Y )W,V 〉+ 1

2|R∇(W,V )X|2.

Condition (**ε) says that the sum of the terms on the last two lines of equation 3.5.3.3 is ≥ 0.Further, setting Y = 0 in (**) and then using a compactness argument gives,

1

2|R∇(W,V )X|2 − 1

2hessΥ(W,W,V,V )(X) > δ > 0,

where δ depends on ∇, Υ, and the metric gΣ on Σ. It follows that there exist constants δ1, δ2 > 0(depending on ∇, Υ, gΣ, and the Euclidean structure on the bundle) such that if |Y | ≤ δ1 then thesum of the terms on the last two lines of equation 3.5.3.3 is > δ2.

Next let H denote the sum of the terms on the first two lines of equation 3.5.3.3. Since−3Υ(U,U, V, V ) = −3Υ(U,U, V, V ) + 2C · |U |2, we have:

H = 2εK(X,Y ) + 6〈R∇(X,Y )U, V 〉 − 3Υ(U,U, V, V ) + 2C · |U |2

−2DXΥ(W,U, V, V ) + 2DXΥ(W,V,U, V ).

To complete the proof, it will suffice show that for sufficiently large C:

1. if |Y | ≤ δ1 then H + δ2 ≥ 0, and

2. if |Y | > δ1 then H > 0.

It is straightforward to choose C (depending on δ1, δ2, the minimal sectional curvature of Σ, andthe norms of the tensors R∇, Υ, and DXΥ) such that these two conditions are met.

25

Example 3.5.10. If the warping tensor Υ in Theorem 3.5.7 can be taken to be taken to be of thesimple form:

Υ(W,W,U,U) = −C · |W |2 · |U |2 for all W ⊥ U,

then the resulting metric will be a connection metric (see Example 3.5.2). In this case, Theo-rem 3.5.7 reduces to Theorem 3.3.6.

Corollary 3.5.11. Suppose (Σ, gΣ) is a compact Riemannian manifold with strictly positive

sectional curvature. Suppose that Rk → Eπ→ Σ is a vector bundle over Σ. Then there exists

a warped connection metric gE on E with soul (Σ, gΣ) which has nonnegative curvature tosecond order and radially-symmetric fibers iff the bundle admits a Euclidean structure, aconnection ∇, and positive real valued function f : Σ → R such that the following conditionholds for all p ∈ Σ, X,Y ∈ TpΣ and V,W ∈ Ep:

(*) 〈(DXR∇)(X,Y )W,V 〉2 ≤(

|R∇(W,V )X|2 + |W ∧ V |2 · hessf (X))

· K(X,Y ) with equalityiff Y is parallel to X or W is parallel to V .

Proof. If a warped connection metric has radially symmetric fibers, then the warping tensor hasthe form:

Υ(W,W,U,U) = −f(p) · |W |2 · |U |2 for all W ⊥ U,

where f is a positive real-valued function on Σ. Conversely, the quadradically-warped connectionmetric associated to a warping tensor of this form has radially-symmetric fibers.

3.6 Summary of conditions for nonnegative curvature

The following is a review of Theorem 3.3.6, Theorem 3.5.7 and Corollary 3.5.11:

Theorem 3.6.1 (Review of Results in this Chapter). Suppose (Σ, gΣ) is a compact Rie-

mannian manifold with strictly positive sectional curvature. Suppose that Rk → Eπ→ Σ is a

vector bundle over Σ.

1. There exists a connection metric gE on E with soul (Σ, gΣ) which has nonnegativecurvature to second order iff the bundle admits a Euclidean structure and a connection∇ such that the following condition holds for each p ∈ Σ, X,Y ∈ TpΣ and V,W ∈ Ep:

〈(DXR∇)(X,Y )W,V 〉2 ≤ |R∇(W,V )X|2 · K(X,Y ) with equality iff Y is parallel toX or W is parallel to V .

2. There exists a warped connection metric gE on E with soul (Σ, gΣ) which has nonneg-ative curvature to second order and radially-symmetric fibers iff the bundle admits aEuclidean structure, a connection ∇, and positive real valued function f : Σ → R suchthat the following condition holds for all p ∈ Σ, X,Y ∈ TpΣ and V,W ∈ Ep:

〈(DXR∇)(X,Y )W,V 〉2 ≤(

|R∇(W,V )X|2 + |W ∧ V |2 · hessf (X))

·K(X,Y ) with equal-ity iff Y is parallel to X or W is parallel to V .

3. There exists a warped connection metric gE on E with soul (Σ, gΣ) which has nonneg-ative curvature to second order iff the bundle admits a Euclidean structure, a connec-tion ∇, and a warping tensor Υ such that the following condition holds for all p ∈ Σ,X,Y ∈ TpΣ and V,W ∈ Ep:

〈(DXR∇)(X,Y )W,V 〉2 ≤(

|R∇(W,V )X|2 − hessΥ(W,W,V,V )(X))

·K(X,Y ) with equal-ity iff Y is parallel to X or W is parallel to V .

26

In order to translate the above theorem into a necessary and sufficient condition for a bundleto admit a metric of nonnegative curvature (instead of a metric which is nonnegatively curved tosecond order), one must answer the following question:

Question 3.6.2. Is it true that a vector bundle admits a metric of nonnegative curvature iff itadmits a metric which is nonnegatively curved to second order?

Neither direction is obvious. If a vector bundle admits a metric of nonnegative curvature, itis not obvious that it admits one for which K ′′

W,X,V,Y,U (0) = 0 only when X is parallel to Y andboth U and V are parallel to W . The requirement that V is parallel to W seems especially strong;perhaps this hypothesis must be removed from the definition of “nonnegatively curved to secondorder” in order that Question 3.6.2 is true.

On the other hand, if a vector bundle admits a metric which is nonnegatively curved to secondorder, it seems computationally possible to check whether the quadradically-warped connectionmetric on the bundle associated to its warping tensor has nonnegative curvature in a neighborhoodof the zero-section. The author is currently working on this question.

We end with an example, due to Walschap, which helps justify the need for our generalizationfrom connection metrics to warped connection metrics.

Example 3.6.3. Consider the manifold M = (S2 × R2) ×R R, where R acts on S2 by rotationabout the north and south poles, on R2 by rotation, and on R by translations. M is diffeomorphicto S2 ×R2, but the induced submersion metric on M is a metric of nonnegative curvature which isdifferent from the product metric. In particular, the fibers of the Sharafutdinov map are not totallygeodesic, and the connection on the normal bundle of the soul is not a flat connection (see[39, page529]). This connection is not capable of inducing a connection metric of nonnegative curvature onthe bundle; this follows from [39, Theorem 1.5]. Thus, when a connection on a bundle is fixed,it is possible that this connection is capable if inducing a warped connection metric, but not aconnection metric on the bundle.

27

Chapter 4

Riemannian submersions withcompact holonomy

We say that a Riemannian submersion π : M → B has compact holonomy if its holonomy group,Φ, is a compact finite dimensional Lie group. We show by example that the metric projection ontoa soul does not necessarily have compact holonomy. When the metric projection onto a soul doeshappen to have compact holonomy, we achieve the following splitting theorem: if the vertizontalcurvatures of M decay to zero away from the soul, the M splits locally isometrically over its soul.

4.1 Consequences of compact holonomy

Our first consequence of compact holonomy is essentially due to Schroeder and Strake [27, Propo-sition 1]:

Lemma 4.1.1. Let π : M → B be a Riemannian submersion. If π has compact holonomy,then there exists a constant b1 = b1(π) such that the holonomy group is:

Φ = hα | α is a piecewise smooth loop in B at p of length ≤ b1.

In other words, the entire holonomy group can be represented by loops in B of bounded length.Since Schroeder and Strake stated this only for the case of the holonomy group of the normal bundleof the soul, we include here a translation of their proof into the above generality. As Schroeder andStrake point out, the proof is essentially a modification of [21, Appendix 4].

Proof. Fix p ∈ B. Consider a loop α0 at p in B for which hα0 = id ∈ Φ. For example, this occurswhen α0 is the trivial loop, but may occur for nontrivial loops as well. If αt, t ∈ (−ε, ε), is avariation of α0, then V := ∂

∂t |t=0hαt ∈ G, where G denotes the Lie algebra of Φ.

Our first goal is to prove that every vector of G occurs in this way. To establish this, leth ⊂ G denote the set of all vectors in G which do occur in this way. We first argue that h is a Liesub-algebra of G. Let αi

t (i = 1, 2) denote two families of loops of the type described above. Let

V i := ∂∂t |t=0h

αit be the corresponding elements of h. By re-parameterizing α1

t one can demonstratethat λV 1 ∈ h for any λ ∈ R. By considering the concatenation α1

t α2t , one can demonstrate that

V 1+V 2 ∈ h. Finally, if we let βt denote that path for which βt(s2) = (α1

t α2t (α1

t )−1 (α2

t )−1)(s),

then ∂∂t |t=0h

βt = [V 1, V 2] ∈ h.Since h is a Lie sub-algebra of G, the distribution g 7→ (Lg)∗(h) is involutive, where Lg denotes

left translation by g ∈ Φ. The maximal integral manifold through id ∈ Φ, denoted H, is the Liesubgroup of Φ corresponding to h.

28

We wish to prove that H = Φ0. Take any g ∈ Φ0. Let α denote a loop in B at p for whichhα = g, and let αt (t ∈ [0, 1]) denote a nulhomotopy of α0 = α. Let γ(t) := hα1−t , which is a pathin G from id to g. We show that for each t0 ∈ [0, 1], γ′(t0) ∈ (Lγ(t0))∗(h), and hence that g ∈ H.Consider the following family of loops:

βt := (αt0)−1 αt0+t.

Since hβ0 = id, V := ∂∂t |t=0h

βt ∈ h. Further, (Lγ(t0))∗(V ) = γ′(t0) by construction. This completesour proof that H = Φ0.

Thus, we can choose a finite collection, α1t , ..., α

Nt , of families of closed loops of the type described

above, such that the corresponding vectors, V 1, ..., V N , form a basis of G.Consider the map f : (−ε, ε)N → G defined by:

f(t1, ..., tN ) := hα1t1

···αNtN .

By construction f is nonsingular at (0, ..., 0), and therefore f([− ε2 ,

ε2 ]

N ) contains a closed neigh-borhood U of the identity element of Φ. By compactness, the lengths of the loops α1

t1 · · · αNtN

are bounded when |ti| ≤ ε2 . Finally, since Φ is compact, it can be covered by finitely many left-

translates of U . In other words, there is a finite collection of loops β1, ..., βl at p in B such that anyelement g ∈ Φ can be written as:

g = hβi · hα1t1

···αNtN = hβiα1

t1···αN

tN

for some i = 1, .., l and for some ti ∈ [− ε2 ,

ε2 ]. The statement of the proposition follows.

Our second consequence of compact holonomy is a global Lipschitz bound on all holonomydiffeomorphisms, at least when π has a bounded T -tensor. It is known that the holonomy diffeo-morphism hα satisfies a Lipschitz bound depending on the length of α. More precisely, Guijarroand Walschap proved the following in [17, Lemma 4.2]:

Lemma 4.1.2 (Guijarro-Walschap). Let π : M → B denote a Riemannian submersionwith bounded T -tensor (|T | ≤ CT ). Let α be a path in B from p to q. Then the holonomydiffeomorphism hα : Fp → Fq satisfies the Lipschitz constant eCT ·length(α).

Proof. Assume that α is parameterized proportional to arclength such that α(0) = p and α(1) = q.Let x ∈ Fp and let V ∈ Vx (=the tangent space to the fiber at x) be a unit-length vector. Let b(s)(s ∈ (−ε, ε)) denote a path in Fp with b′(0) = V . For each s let t 7→ αs(t) denote the horizontal liftof α with αs(0) = b(s). Let V (t) denote the vector field along α0 corresponding to this variation.V (t) is an everywhere vertical vector field with V (0) = V and V (1) = (hα)∗(V ). But it’s easy tocompute that V ′(t) = Aα′

0(t)V (t) + TV (t)α

′0(t), so that if we let f(t) = |V (t)|2, then

f ′(t) ≤ 2〈V (t), TV (t)α′0(t)〉 ≤ 2 · CT · length(α) · f(t).

Thus |V (1)|2 = f(1) ≤ e2CT length(α). Taking the square root of both sides completes the proof.

We prove that in the case of compact holonomy, the holonomy diffeomorphisms satisfy a Lips-chitz bound which does not depend on the length of the path. In the case of non-compact holonomywe at least find a Lipschitz bound which depends linearly, rather than exponentially, on the lengthof α:

Proposition 4.1.3.

1. Let π : M → B be a Riemannian submersion with a bounded T -tensor (|T | ≤ CT ). If πhas compact holonomy, then there exists a constant L = L(π) such that all holonomydiffeomorphisms of π satisfy the Lipschitz constant L.

29

2. Even if Φ is non-compact, there exists a constant L = L(CT , diam(B)) such that theholonomy diffeomorphism associated with any loop of length l satisfies the Lipschitzconstant 1 + l · L.

Proof. For part 1, if c is a loop at p in B of arbitrary length, one can find a different loop cof length ≤ b1 which generates the same holonomy diffeomorphism: hc = hc, where b1 is theconstant from Lemma 4.1.1. It follows that if α is a path of arbitrary length in B between pointsp and q, then hα = hα for a properly chosen path α of length ≤ diam(B) + b1. For example,take α equal to a minimal path β from p to q followed by a loop at q which generates the sameholonomy diffeomorphism as β−1 ∗ α. Therefore L := eCT (diam(B)+b1) is as required for part 1 ofthe proposition.

For part 2, first notice that for any loop α of length l ≤ D := 2 · diam(B) + 1, hα satisfies theLipschitz constant eCT ·l ≤ 1+l·P , where P := CT ·eCT ·D (that is, P equals the maximum derivativeof the function eCT ·l between l = 0 and l = D). But for a unit-speed loop α at p in B of arbitrarylength l, it is possible to find l loops, α1, ..., αl, each of length ≤ D, such that hα = hα1 · · · hαl ,where l denotes the smallest integer which is ≥ l. To do this, define αi to be the composition of aminimal path in B from p to α(i− 1), followed by α|[i−1,i], followed by a minimal path from α(i)to p. Since for each i, hαi satisfies the Lipschitz constant 1 + P ·D, it follows that hα satisfies thefollowing Lipschitz constant:

l(1 + P ·D) ≤ (l + 1)(1 + P ·D) = 1 + (1 + PD)l + PD ≤ 1 + 2(1 + PD)l.

The final inequality holds whenever l ≥ 1, but if l < 1 then hα satisfies the Lipschitz constant1+ (CT · eCT ·1)l (as above). So the choice L := max2(1+PD), CT · eCT is as required for part 2of the proposition.

We do not know whether the holonomy diffeomorphisms of the metric projection onto a soulalways obey a global Lipschitz bound (this question was first asked by Guijarro and Walschap).However, for general Riemannian submersions, we have the following:

Example 4.1.4. For a general Riemannian submersion with bounded T -tensor, the holonomydiffeomorphisms need not satisfy a global Lipschitz bound. For example, consider the projectionπ : S2 × R2 → S2, where S2 has the round metric, R2 has the flat metric, and S2 × R2 has theproduct metric. Let X denote the vector field on S2 whose flow is rotation about the axis throughthe north and south poles N,S. Let Y be a vector field on S2 which is orthogonal to X and vanishesat N,S. Let ∂

∂θ (v) (v ∈ R2) denote the radial vector field on R2, and let W (v) = µ(|v|) · ∂∂θ (v),

where µ is a smooth bump function with support [1, 2]. We can consider X,Y to be horizontalvector fields and W to be a vertical vector field on S2 × R2 in the obvious way. Define a 2-planedistribution H on S2 × R2 as follows:

H(p, v) = span

Y (p),X(p)

|X(p)|+ |X(p)| ·W (v)

.

This distribution clearly extends continuously to the fibers over N,S. There is a unique metricon S2 × R2 for which π becomes a Riemannian submersion with horizontal distribution H suchthat the fibers are isometric to flat R2. The holonomy group Φ is isomorphic to R, and its actionon the fiber π−1(N) = R2 is simply the flow along the vector field W . It is easy to see thatarbitrarily long loops in S2 are necessary to achieve arbitrarily large time parameters for this flow,and hence arbitrarily bad Lipschitz bounds for the associated holonomy diffeomorphism. But sincethe T -tensors vanishes outside of a compact set in S2 × R2, it clearly has bounded norm.

4.2 Vertizontal curvature decay

In this section, we explore a consequence of compact holonomy for the metric projection onto asoul. The application is related to a splitting theorem of Guijarro and Petersen which states that

30

if the curvatures of all 2-planes on M decay towards zero away from the soul then the soul must beflat [16]. By O’Neill’s formula, one can then conclude that the A-tensor of π vanishes, and hencethat M splits locally isometrically over the soul (see Remark 3.1.2). Guijarro and Walschap havedemonstrated that if M has the property that all holonomy diffeomorphism of π obey a globalLipschitz bound, then one only needs to know that the curvatures of all vertizontal 2-planes (thatis, 2-planes spanned by a horizontal and a vertical vector) decay towards zero away from the soulin order to conclude that M splits locally isometrically over its soul [17, Theorem 4.3.2]. Thus, wehave as a corollary to Proposition 4.1.3:

Corollary 4.2.1. If Φ is compact and the curvatures of vertizontal 2-planes on M decaytowards zero away from Σ, then M splits locally isometrically over Σ.

We do not know whether this corollary is true even when Φ is noncompact. A more difficultquestion is whether the holonomy diffeomorphism of the metric projection onto a soul must satisfya global Lipschitz bound, even when the holonomy group is non-compact.

4.3 A soul with noncompact normal holonomy

In this section we provide an example of a simply connected open manifold, M , of nonnegativecurvature for which the metric projection onto the soul has noncompact holonomy. Consider thefollowing action of the Lie group R on S2 × C2 × R:

((ϕ, θ), z1, z2, t0)t7→ ((ϕ, θ + t), eπitz1, e

λπitz2, t0 − t).

Here (ϕ, θ) denotes spherical coordinates on S2, and λ denotes an irrational real number. Thequotient, (S2 × C2) ×R R = (S2 × C2 × R)/R, is diffeomorphic to S2 × C2, and this identificationprovides a new nonnegatively curved metric g on S2×C2 under which the quotient map S2×C2×R → (S2 × C2)×R R ≈ S2 × C2 becomes a Riemannian submersion.

Let g denote the product metric on S2×C2, and let V denote the Killing vector field associatedto the R action on S2 × C2; namely, V ((ϕ, θ), z1, z2) = (θ, iz1, λiz2). According to [5, Example 2],the new metric g on S2 × C2 is obtained from g simply by, at each point, rescaling the norms ofvectors parallel to V by a factor of 1/(1 + |V |2)1/2.

It is easy to see that the soul of M := (S2 ×C2, g) will still be the zero section, Σ = S2 × (0, 0),and that the metric projection π : M → Σ will still be the projection (q, z1, z2) 7→ q. It isstraightforward to show that the horizontal distribution H of π can be described as follows:

H((ϕ,θ),z1,z2) = span

(ϕ, 0, 0),

(

θ,|θ|2

1 + |θ|2iz1,

|θ|2

1 + |θ|2λiz2

)

The two vectors in this expression correspond to the horizontal lifts of the spherical coordinatevectors ϕ and θ in T(ϕ,θ)S

2.If σ(t) = (ϕ(t), θ(t)), t ∈ [0, 1], is a loop in S2 based at q := (ϕ(0), θ(0)), it follows that the

horizontal lift σ of σ to the point (q, z1, z2) in M will end at the point σ(1) = (q, eπit0z1, eλπit0z2)

where t0 =∫ 10

|Ýθ|2

1+|Ýθ|2θ′(t)dt.

In particular, the set of points in π−1(q) which can be achieved as endpoints of horizontal liftsto (q, z1, z2) of loops in S2 at q is exactly (q, eπit0z1, eλπit0z2) | t0 ∈ R. It follows from this thatthe holonomy group Φ of the normal bundle of Σ is exactly:

Φ :=

(

eπito 00 eλπit0

)

∣

∣

∣

∣

∣

t0 ∈ R

,

which is isomorphic to R.

31

We next wish to compute the ideal boundary M(∞) of M = (S2 × C2, g). Let p ∈ Σ be eitherthe north or south pole; that is, one of the two zeros of the vector field V . The fiber π−1(p) istotally geodesic because it is a connected component of the fixed point set of the following isometryof M :

((ϕ, θ), z1, z2) 7→ ((ϕ, θ + π/2), z1, z2).

Let Sp denote the sphere of unit-vectors normal to the soul at p. Think of Sp = (z1, z2) ∈ C2 ||z1|2 + |z2|2 = 1. For any angle α ∈ [0, π/2], let Tα := (z1, z2) ∈ Sp | |z1| = cos(α). This isthe standard decomposition of the 3-sphere into a family of tori. Notice that each torus Tα is theclosure of an orbit of the action of Φ on Sp. For any two rays γ1, γ2 from p in M with initialtangent vectors V1 ∈ Tα1 and V2 ∈ Tα2 , the distance in M(∞) between the rays is easily seen tobe d∞([γ1], [γ2]) = |α1 − α2|. In other words, the tori into which Sp decomposes collapse to singlepoints in the ideal boundary. We conclude that M(∞) = [0, π/2].

4.4 Measuring size in the holonomy of a vector bundle

In this section we wish to show that, in a vector bundle with a connection, the “size” of the holonomyelement associated to a loop can be controlled linearly in terms of the length of the loop. Later, insection 6.2, we will generalize this result to the holonomy group of a Riemannian submersion withbounded tensors. The main lemma of this section, which says that a nulhomotopy of a loop canalways be found with derivatives controlled in terms of the length of the loop, is central not onlyto our vector bundle result, but also to all of the main results of Chapter 6.

Let Rk → E → B be an Riemannian vector bundle. Assume that B is compact and simplyconnected. Let ∇ be a connection which is compatible with the inner products on the fibers. LetR∇ denote the curvature tensor of ∇. Fix p ∈ B. Let Φ denote the holonomy group, and G itsLie algebra. Notice that Φ is a Lie subgroup of the orthogonal group, since it acts naturally byisometries on the unit sphere E1

p of the fiber Ep at p. For V ∈ G and w ∈ E1p , denote by V (w)

the value at w of the vector field on E1p associated to V . We call a left-invariant metric, m, on Φ

acceptable if for all V ∈ G, |V |m ≤ supw∈E1p|V (w)|. Notice that any left invariant metric on Φ

can be made acceptable by rescaling. For g ∈ Φ, we define |g| as the supremum over all acceptablemetrics on Φ of the distance between g and id. This provides a natural notion of “size” in theholonomy group.

Proposition 4.4.1. There is a constant C = C(B) such that for any piecewise smooth loopα in B, |Pα| ≤ C · CR · length(α), where CR denotes a bound on |R∇|.

This proposition is particularly interesting when the holonomy group of the vector bundle isnoncompact, in which case (as in the example of Section 4.3), arbitrarily long loops in the basespace may be needed to represent the whole holonomy group.

The proof of this proposition turns on the following lemma, which will be of central importanceto later results as well:

Lemma 4.4.2. Let B be a compact and simply connected Riemannian manifold. There existsa constant Q = Q(B) such that for any piecewise smooth loop α : [0, 1] → B (parameterizedproportional to arclength), there exists a piecewise smooth nulhomotopy H : [0, 1]× [0, 1] → Bof α (that is, H(0, t) = p = α(0), H(1, t) = α(t)) for which the natural coordinate vectorfields along the image of H are everywhere bounded in norm as follows: | ∂

∂sH| ≤ Q and

| ∂∂tH| ≤ Q · length(α). In particular, this implies that the area, A(H), of the image of H is≤ Q2 · length(α).

Proof. Let α : [0, 1] → B be a piecewise smooth loop in B. Assume that α is parameterizedproportional to arclength. Denote p := α(0) = α(1), and l := length(α). To start, we willassume that l ≤ 1

2 inj(B), in which case it is easy to construct a nulhomotopy of α with derivative

32

information controlled linearly in terms of l. Since l ≤ 12 inj(B), α lifts to a loop α = exp−1

p αat 0 in TpB. The natural nulhomotopy of α is H(s, t) := s · α(t) (s ∈ [0, 1], t ∈ [0, 1]). Clearly

| ∂∂tH|, | ∂∂sH| ≤ l. Letting H := expp H, which is a piecewise smooth nulhomotopy of α, we see

that | ∂∂tH|, | ∂∂sH| ≤ Q1 · l for a properly chosen Q1 = Q1(B).

Next we assume only that l ≤ D := 2 · diam(B) + 1. More precisely, we seek a constant Q2

such that for any piecewise smooth loop α in B at p of length ≤ D, there exists a piecewise smoothnulhomotopy H of α, with | ∂∂tH|, | ∂

∂sH| ≤ Q2.Suppose no such Q2 exists. Then there must be a sequence αi : [0, 1] → B of piecewise

smooth loops, each with length ≤ D, such that the minimal derivative bounds of piecewise smoothnulhomotopies of the loops αi go to infinity. By restricting to a subsequence, we can assumethat αi converges in the sup norm; in particular αi is a Cauchy sequence. For i, j large enoughthat ε := dsup(αi, αj) ≤ 1

2 inj(B), the natural piecewise smooth homotopy H : [0, ε] × [0, 1] → Bbetween H0(t) = H(0, t) = αi(t) and Hε(t) = H(ε, t) = αj(t), which retracts along shortestgeodesics between corresponding points of the two curves, is well defined. More precisely, defineH(s, t) = ct(s), where ct is the minimal path between ct(0) = αi(t) and ct(ε) = αj(t), parameterizedso as to have the constant speed d(αi(t), αj(t))/ε. Clearly | ∂

∂sH| ≤ 1. Further, | ∂∂tH| ≤ Kfor an appropriate constant K = K(B). To see this, notice that for fixed t, the vector fieldJt(s) = ∂

∂tH(s, t) along the geodesic ct(s) is a Jacobi field because it is the variational vectorfield of the family of geodesics which defines the homotopy. Jt(s) is determined by its endpointsV1 := Jt(0) = α′

i(t) and V2 := Jt(ε) = α′j(t), each of whose norm is ≤ D. We can thus take K

as the supremum (over all pairs of vectors V1 ∈ Tp1B, V2 ∈ Tp2B with d(p1, p2) ≤ 12 inj(B) and

|V1|, |V2| ≤ D) of the maximal norm of the Jacobi field along the shortest geodesic between p1 andp2 with end values V1 and V2.

It follows that any piecewise smooth nulhomotopy Hi of αi can be extended to a piecewisesmooth nulhomotopy Hj of αj , with similar derivative bounds. More precisely, if | ∂

∂sHi| ≤ Qs and

| ∂∂tHi| ≤ Qt, then | ∂∂sHj | ≤ Qs + E(ε) and | ∂∂tHj | ≤ maxQt,K, where limε→0 E(ε) = 0. This

provides a contradiction. Therefore, such a constant Q2 exists.Finally, we handle the case where l = length(α) is arbitrary. It is possible to find l loops,

α1, ..., αl, each of length ≤ D, such that Pα = Pα1 · · · Pαl, where l denotes the smallest integer

which is ≥ l. This is done exactly as in the proof of Lemma 4.1.3, by defining αi to be thecomposition of a minimal path in B from p to α(i− 1), followed by α|[i−1,i], followed by a minimalpath from α(i) to p. Let γ : [0, 1] → B denote the composition of the loops αi, re-parameterizedproportional to arclength. Notice that length(γ) ≤ l ·D. It is straightforward to define a piecewisesmooth homotopy H : [0, 1

2 ]× [0, 1] → B between H(0, t) = α(t) and H( 12 , t) = γ(t) with | ∂∂sH| ≤

2 ·diam(B) and | ∂∂tH| ≤ l ·D. Next extend H by defining H : [ 12 , 1]× [0, 1] → B as the nulhomotopyof γ which simultaneously performs nulhomotopies of each loop αi. H is clearly a piecewise smoothnulhomotopy of α for which | ∂

∂sH| ≤ max2 · diam(B), 2Q2 and | ∂∂tH| ≤ lQ2 ≤ (l+ 1)Q2 ≤ 2lQ2.The final inequality above assumes that l ≥ 1, but the case l ≤ 1 can be handled as follows:

If l ≤ 12 inj(B) then there exists a homotopy with | ∂

∂sH| ≤ Q1 · l ≤ Q1 and | ∂∂tH| ≤ Q1 · l.On the other hand, if 1

2 inj(B) ≤ l ≤ 1 then there exists a homotopy with | ∂∂sH| ≤ Q2 and

| ∂∂tH| ≤ Q2 = Q2ll ≤ 2Q2

inj(B) l. Here Q2 is derived similarly to Q2, but for loops of length ≤ 1 rather

than loops of length ≤ D. In all cases, | ∂∂tH| is bounded linearly in terms of l, and | ∂∂s | is bounded

absolutely, which completes the proof.

Next we prove Proposition 4.4.1

Proof. Let α : [0, 1] → B be a unit-speed piecewise smooth loop in B at p. By Lemma 4.4.2,there exists a piecewise smooth nulhomotopy H : [0, 1]× [0, 1] → B of α such whose area A(H) isbounded linearly in terms of length(α). We now describe how to control |Pα| linearly in terms ofA(H). Let g(s) = Pt→Hs(t), which is a piecewise smooth path in Φ between the identity and Pα.

33

For any vector w in E1p , we define w(s, t) as the parallel transport of w along t 7→ Hs(t). Then,

|Pα| ≤∫ 1

0sup|g′(s)|m : m is acceptableds

≤∫ 1

0supw∈E1

p

∣

∣

∣

∣

D

dS

∣

∣

∣

S=sg(S)(w)

∣

∣

∣

∣

ds

=

∫ 1

0supw∈E1

p

∣

∣

∣

∣

D

dS

∣

∣

∣

S=sw(S, l)

∣

∣

∣

∣

ds

≤∫ 1

0supw∈E1

p

∫ 1

0

∣

∣

∣

∣

D

dt

D

dsw(s, t)

∣

∣

∣

∣

dt ds

=

∫ 1

0supw∈E1

p

∫ 1

0

∣

∣

∣

∣

R∇(

∂

∂tH(s, t),

∂

∂sH(s, t)

)

w(s, t)

∣

∣

∣

∣

dt ds

≤∫ 1

0

∫ 1

0supw∈E1

p

∣

∣

∣

∣

R∇(

∂

∂tH(s, t),

∂

∂sH(s, t)

)

w(s, t)

∣

∣

∣

∣

dt ds

≤∣

∣R∇∣

∣

∫ 1

0

∫ 1

0

∣

∣

∣

∣

∂

∂sH(s, t) ∧ ∂

∂tH(s, t)

∣

∣

∣

∣

dt ds

= CRA(H).

This completes the proof.

34

Chapter 5

Volume growth bounds

The purpose of this chapter is to use Perelman’s Theorem to study the volume growth, VG(M), ofM . In particular, VG(M) is proven to be bounded above by the difference between the codimensionof the soul and the maximal dimension, m, of an orbit of the action of Φ on νp(Σ). Additionally,if M satisfies an upper curvature bound, then we prove that VG(M) is bounded below by one plusthe dimension of M(∞). These two inequalities placed together read:

dim(M(∞)) + 1 ≤ VG(M) ≤ codim(Σ)−m. (5.0.1)

It was first observed by Shioya in [30] that the first term of Equation 5.0.5.1 is ≤ the last term(Shioya mistakenly defined m to be the dimension of Φ rather than the maximal dimension of anorbit of the action of Φ). Our result is therefore that V G(M) is sandwiched between these twoterms, at least when M satisfies an upper curvature bound.

5.1 Previous results

For a Riemannian manifold Mn with nonnegative sectional curvature, the volume of a ball BΣ(r)of radius r about a compact totally geodesic submanifold Σn−k can be estimated by a generalizedBishop-Gromov type inequality which states that

V (r) :=vol(BΣ(r))

rk

is a monotonically non-increasing function of r [9]. If M is open, this implies that the volumegrowth of M is not greater than the codimension k of Σ, where the volume growth, VG(M), of Mis defined as follows:

VG(M) = inf

x ∈ R∣

∣

∣

∣

limr→∞

vol(Bp(r))

rx= 0

Note that this definition is independent of the choice of p ∈ M .The case where Σ is a soul of M and the volume growth of M is maximal in the sense that

VG(M) = k was studied by Schroeder and Strake in [27]. Their main result states that in this casethe reduced holonomy group Φ0 of the normal bundle ν(Σ) of Σ in M is trivial. Hence, in this caseTheorem 2.2.1 implies that M splits locally isometrically over Σ, and the splitting is global if B issimply connected.

Our main theorem is the following generalization of Schroeder and Strake’s result: V G(M) ≤k − m, where m denotes the maximal dimension of an orbit of the action of Φ0 on νp(Σ). Thepossibility that Φ0 might be a non-closed subgroup of SO(k) introduces certain technical difficultiesin our proofs. The previously constructed example in Section 4.3 justifies these worries.

35

It is worth mentioning that Perelman’s theorem provides an elementary proof of the previouslymentioned fact that VG(M) ≤ k. For, if ν(Σ) is endowed with the connection metric with flatfibers, then exp⊥ becomes a distance non-increasing map. To see this, take any v ∈ ν(Σ) and let

x := exp⊥(v) ∈ M . Let ˜Hv and ˜Vv denote the horizontal and vertical spaces of ν(Σ) at v determined

by the natural connection. Perelman’s Theorem implies that d(exp⊥)v maps ˜Hv isometrically onto

Hx. Also, Rauch’s theorem implies that d(exp⊥)v maps ˜Vv in a distance non-increasing way intoVx. Since V G(ν(Σ)) = k, it follows that VG(M) ≤ k.

5.2 An upper bound on volume growth

In this section, we prove that VG(M) ≤ k−m. Since Φ0 might be non-compact, most of the theoryof transformation groups does not apply to the action of Φ0 on a fiber νp(Σ). We therefore requirethe following two lemmas about Lie subgroups of SO(k).

Lemma 5.2.1. Let G be a (possibly non-compact) connected Lie subgroup of SO(k). Theunion, Ω, of all orbits of the action of G on Sk−1(1) which have maximal dimension is openand dense.

Proof. Ω is trivially open. To establish that Ω is dense, decompose the Lie algebra G of G asG = Z ⊕ A, where Z is the center of G, and A is semi-simple. Let Z and A denote the Liesubgroups of G associated with Z and A respectively. Z is Abelian and A is compact. Z isa subgroup of a maximal torus of SO(k), and with respect to a properly chosen basis of Rk, Zembeds in SO(k) as follows:

Z :=

es1(v)i

. . .

esm(v)i

I

∣

∣

∣

∣

∣

v ∈ Z

Here, each si is an linear function from Z to R. Write Rk = V1⊕· · ·⊕Vm⊕W for the correspondingorthogonal decomposition of Rk; dim(Vi) = 2 and dim(W ) = k − 2m.

For any v ∈ Sk−1(1), dim(G(v)) ≤ dim(Z(v)) + dim(A(v)), and we claim that equality holdson an open dense subset E of Sk−1(1). To construct E, consider a basis X1, ..., Xa of Z andY1, ..., Yb of A, and regard the elements Xi and Yi as Killing vector fields on Rk. For eachi = 1, ..., b, the set Ki := w ∈ W | Yi(w) = 0 is a linear subspace of W , which is not necessarilyproper. Define:

E := v1 + · · ·+ vm + w ∈ Sk−1(1) | each vi 6= 0 and if w ∈ Ki then Ki = W.

E is clearly open and dense. For every v ∈ E and any i = 1, ..., b, Yi(v) /∈ spanX1(v), ..., Xa(v).If i is such that Ki 6= W , this is clear from construction; otherwise, Yi(v) ∈ spanX1(v), ..., Xa(v)implies Yi ∈ Z, which is impossible. Therefore, dim(G(v)) = dim(Z(v)) + dim(A(v)) for everyv ∈ E.

From the previous description of Z, one can easily see that the union, ΩZ , of all orbits of theaction of Z on Sk−1(1) which have maximal dimension is open and dense in Sk−1(1). Further,ΩA (defined analogously) is open and dense by the theory of compact group actions. Therefore,ΩZ ∩ ΩA ∩ E is an open dense subset of Sk−1(1) which is contained in Ω. It follows that Ω isdense.

Lemma 5.2.2. Let G be a (possibly non-compact) Lie subgroup of SO(k), and C be a closedneighborhood of the identity in G. Let m denote the maximal dimension of an orbit of theaction of G on Rk. Then there exists a finite union, W =

⋃

Wi, of (k − m)-dimensionalsubspaces Wi of Rk such that for any u ∈ Rk, there exists some g ∈ C with g(u) ∈ W .

36

Proof. The proof is by induction on k. The statement is trivial in dimensions k = 1 or 2. Assumethat the lemma is true in dimensions less than k. Let C ⊂ G ⊂ SO(k), as in the statement of thelemma. For every v ∈ Sk−1(1) ⊂ Rk, we will construct an open neighborhood U of v in Sk−1(1)and a finite union, W =

⋃

Wi, of (k −m)-dimensional subspaces Wi of Rk which contain v, suchthat for every u ∈ U , there exists some g ∈ C with g(u) ∈ W . Since Sk−1(1) is compact, this willcomplete the proof.

Fix v ∈ Sk−1(1). Let mv denote the dimension of the orbit G(v). Decompose the Lie algebraG of G as G = Gv ⊕ spanX1, ..., Xmv, where Gv denotes the Lie algebra of the stabilizer Gv, andthe elements Xi, when considered as Killing vector fields on Sk−1(1), form a basis of TvG(v) at v.Define Y to be the orthogonal compliment of TvG(v) in TvRk ≈ Rk. Y is a (k −mv)-dimensionalsubspace of Rk containing v, and we claim that the set g(u) | g ∈ C, u ∈ Sk−1(1) ∩ Y containsan open neighborhood U of v in Sk−1(1). This follows from the fact that the derivative at (id, v)of the map Ψ : G× (Y ∩ Sk−1(1)) → Sk−1(1) defined as Ψ(g, u) := g(u) is surjective.

If the orbit G(v) has maximal dimension, that is if mv = m, then the single subspace W := Yand the open set U will be as required for the construction. Otherwise, the dimension of Y(= k −mv) is too large, so Y cannot serve as the required subspace. In this case, we will chooseW to be a union of smaller subspaces of Y , as follows.

The stabilizer Gv acts by isometries on the subspace Y ′ consisting of those vectors in Y whichare orthogonal to v; we claim that the maximal dimension of an orbit of this action is m−mv. Tosee this, choose a neighborhood E of v in Y ∩ Sk−1(1) small enough so that Xi(w) is linearlyindependent and transverse to Y for all w ∈ E. Then, for any w ∈ E, Gv(w) ⊂ Y ∩ Sk−1(1),and TwG(w) = TwGv(w) ⊕ spanXi(w). Therefore, dim(G(w)) = dim(Gv(w)) + mv. Fromlemma 5.2.1 and the fact that G(E) contains an open neighborhood of v in Sk−1(1), it followsthat E must contain a vector w for which dim(G(w)) = m. For this w, dim(Gv(w)) = m −mv =maxdim(Gv(w

′)) | w′ ∈ Y ′.By the inductive hypothesis, there exists a finite union, V :=

⋃

Vi, of (k −m− 1)-dimensionalsubspaces Vi of Y

′ such that for any u ∈ Y ′ there exist an element g ∈ Gv ∩C such that g(u) ∈ V .Define Wi := spanv, Vi, and W :=

⋃

Wi. Each subspace Wi has dimension k − m, and byconstruction, g(u) | g ∈ Gv ∩ C, u ∈ W = Y . It follows that the set g(u) | g ∈ C, u ∈Sk−1(1) ∩ W will contain the open neighborhood U of v defined above. This completes theproof.

Theorem 5.2.3. V G(M) ≤ k −m

Proof. Fix p ∈ Σ. Lemma 4.1.1 states that if Φ0 is compact, then all of Φ0 can be represented usingloops at p in Σ of bounded length. Even when the compactness assumption is dropped, the proofstill establishes the following: there exist a constant b1 and a closed neighborhood C of the identityin Φ0 such that any g ∈ C can be represented as parallel translation along a loop at p in Σ withlength ≤ b1. By lemma 5.2.2, we can then find a finite union, W :=

⋃

Wi, of (k −m)-dimensionalsubspaces Wi of νp(Σ) which meets every orbit of C in the sense of the lemma.

Next we argue that all of M is contained in a ball of radius b2 := b1+diam(Σ) about exp⊥(W ).To see this, take any point x ∈ M , and write x = exp⊥(v) for some v ∈ ν(Σ). There exists a pathα in Σ of length less than b2 such that Pα(v) ∈ W (namely, take any minimal path from q := π(v)to p, followed by a loop at p which represents the proper holonomy element). The horizontal lift αof α to x is then a path in M of the same length connecting x to y := exp⊥(Pα(v)) ∈ exp⊥(W ).Therefore Bexp⊥(W )(b2) = M . Additionally, observe that if v was chosen so that the radial geodesic

t 7→ exp⊥(tv) provides a minimal connection from q to x, it follows that the radial geodesic t 7→exp⊥(tPα(v)) will provide a minimal connection from p to y.

We conclude with an argument similar to the proof of [27, Thm. 2], by showing that there is aconstant K such that for large real numbers r > 0,

Vol(BΣ(r)−BΣ(r − 1)) ≤ Krk−m−1

37

This clearly implies the statement of the theorem.Let W 1 be the set of unit-length vectors in W . Let r be a positive real number. Take a net

of points vi in W 1 such that every vector in W 1 makes an angle of less than 1/r with some vi.Since W 1 is a finite union of round spheres of dimension k−m−1, we need no more than b3r

k−m−1

points vi, where b3 is a constant which does not depend on r.Now we claim that

BΣ(r)−BΣ(r − 1) ⊂⋃

i

Bexp⊥(rvi)(b2 + 2).

To see this, take any point x ∈ M with r − 1 ≤ dist(x,Σ) ≤ r. As described above, x has distance≤ b2 from some point y = exp⊥(v), where v ∈ W , and γ(t) := exp⊥(tv) provides a minimalconnection between p and y. Let vi be a point of the net which makes angle less than 1/r withv. Since γ is minimizing, we can apply Toponogov to conclude that d(y, exp⊥(rvi)) ≤ 2. Thusdist(x, exp⊥(rvi)) ≤ b2 + 2.

Finally, since the Bishop-Gromov inequality implies that the volume of each ball Bexp⊥(tvi)(b2+2) is not greater than the volume b4 of a ball of the same radius in the Euclidean space of the samedimension, we have

Vol(BΣ(r)−BΣ(r − 1)) ≤ b4b3rk−m−1.

So choosing K := b4b3 is as required to conclude the proof.

5.3 Bounded vertical Jacobi fields

In this section we prove the existence of many bounded vertical Jacobi fields along radial geodesics(that is, geodesics which intersect the soul orthogonally). Since the existence of bounded verticalJacobi fields restricts volume growth, this has the same flavor as Theorem 5.2.3.

Suppose that p ∈ Σ, V ∈ νp(Σ), and γ(t) := exp⊥(tV ). Choose vectors X,Y ∈ TpΣ, and letJ(t) be the Jacobi field along γ with J(0) = 0 and J ′(0) = R∇(X,Y )V . Lemma 3.1.1 implies thatJ(t) is an everywhere vertical Jacobi field which is bounded in norm because the A-tensor of π isbounded in norm. In other words, along γ we can find a number of linearly independent boundedvertical Jacobi fields equal to the dimension of the subspace R∇(X,Y )V | X,Y ∈ TpΣ. In fact,we can do slightly better:

Proposition 5.3.1. Along almost every radial geodesic in M , there exist at least m linearlyindependent bounded vertical Jacobi fields, where m denotes the maximal dimension of anorbit of the action of Φ on νp(Σ).

Proof. Begin by choosing a collection τ1, ..., τa of paths in Σ beginning at p, and collectionsX1, ..., Xa, and Y1, ..., Ya of vectors in TpΣ such that the set Vi := P−1

τi R(Pτi(Xi), Pτi(Yi)) Pτi is a basis of the Lie algebra of Φ0. This is possible by a theorem of Ambrose and Singer;see for example [21, Thm. 8.1]. Consider each Vi to be a vector field on the unit sphere ν1p(Σ) ofνp(Σ). Let Ω be the union of all orbits of the action of Φ0 on ν1p(Σ) which have maximal dimension.By Lemma 5.2.1, Ω has full measure. Fix w ∈ Ω. Vi(w) spans an m-dimensional subspaceof Tw(ν

1p(Σ)). To simplify notation, re-index so that V1(w), ..., Vm(w) is linearly independent.

For each i between 1 and m, consider the Jacobi field Ji(t) along γ(t) := exp⊥(tw) with initialconditions Ji(0) = 0 and J ′

i(0) = Vi(w). We claim that each Ji is a bounded vertical Jacobi field.To see this, let Ji(t) be the Jacobi field along γi(t) := exp⊥(tPτi(w)) with initial conditions

Ji(0) = 0 and J ′i(0) = Pτi(Vi(w)). Lemma 3.1.1 implies that Ji is a bounded vertical Jacobi field.

Next observe that Ji = (hτi)∗Ji, where hτi : π−1(p) → π−1(τi(1)) is the diffeomorphism between

the fibers naturally associated to the path τi. This observation is a re-wording of [17, Prop. 1.1.1].Also, each hτi is a biLipschitz map according to Lemma 4.1.2. It follows that Ji is also a boundedvertical Jacobi field.

38

It would be interesting to find an alternative proof of Theorem 5.2.3 which is based on Propo-sition 5.3.1.

5.4 A lower bound on volume growth

The only known lower bound for the volume growth of an open manifoldM of nonnegative curvatureis due to Calabi, who proved that VG(M) ≥ 1. Calabi’s result appeared before Cheeger andGromoll’s Soul Theorem. Today the technology is in place to say much more. In this section weprove the following result:

Theorem 5.4.1. If M satisfies an upper curvature bound, then the VG(M) ≥ 1+dim(M(∞)).

We believe that this theorem is true even without the assumption that M satisfies an uppercurvature bound. See [8] for an example of an open manifold of nonnegative curvature which doesnot satisfy an upper curvature bound.

We begin by stating a useful version of Toponogov’s theorem, which can be found in [32, Lemma19]:

Lemma 5.4.2 (Toponogov). Let M be a Riemannian manifold of nonnegative curvature.Let αi : [0, li] → M (i = 1, 2) be two distance minimizing geodesics starting at the same point.Let θ(t1, t2) denote the “comparison angleÔ; that is, the angle opposite the third side of thetriangle in flat R2 with side lengths t1, t2, dM (α1(t1), α2(t2)). Then θ is monotonic in t1 andt2. That is θ(t1, t2) ≤ θ(s1, s2) whenever s1 ≤ t1 ≤ l1 and s2 ≤ t2 ≤ l2.

Corollary 5.4.3. Suppose that αi (i = 1, 2) are two unit speed rays from the same point and

that a, b are positive real numbers. Define g(t) := dM (α1(at),α2(bt))t . Then g(t) is monotonically

decreasing in t, and

limt→∞

g(t) =√

a2 + b2 − 2ab cosmind∞([α1], [α2]), π. (5.4.1)

Proof. The proof of Equation 5.4.5.1 is found in [31, Proposition 2.2]. The monotonicity claimfollows immediately from Lemma 5.4.2 simply because g(t) can be re-described as

g(t) =√

a2 + b2 − 2ab cos θ(at, bt),

where θ is as in the lemma.

The metric cone, C(X), over a metric space X is defined as the quotient space C(X) :=X × [0,∞)/ ∼, where (x, 0) ∼ (y, 0) for any x, y ∈ X. The metric of C(X) is defined by the law ofcosines formula:

d((x, t), (y, s)) :=√

s2 + t2 − 2st cosmind(x, y), π.

Since M(∞) is an Alexandrov space with curvature bounded below by 1, it follows thatC(M(∞)) is an Alexandrov space with curvature bounded below by zero; see [4, Proposition4.2.3]. The dimension of C(M(∞)) is clearly one greater than the dimension of M(∞). A rephras-ing of Theorem 5.4.1 is thus that the volume growth of M is bounded below by the dimension ofC(M(∞)); this is probably the best way to think about the claim. The following well known result,which provides motivation for Theorem 5.4.1, is due to Shiohama [29]:

Proposition 5.4.4 (Shiohama). Let M be an open manifold of nonnegative curvature. Fixp ∈ M . Denote by λM the result of rescaling the metric on M by a factor of λ. Denote by othe vertex (cone point) of C(M(∞)). Then the Gromov-Hausdorff limit (as λ goes to zero)of the pointed spaces (λM, p) is equal to (C(M(∞)), o)

39

Idea of the proof of Proposition 5.4.4. We sketch the proof found in [15, Lemma 3.4]. Let Bdenote the ball about o of radius 1. Fix p ∈ M . For any R, we wish to define a map fR : B → Bp(R).An arbitrary point q of C(M(∞)) can be denoted as q = ([γ], t), where γ is a unit-speed ray in Mfrom p and t ∈ [0,∞). The map fR is then defined as follows:

fR(([γ], t)) := γ(tR).

This definition uses implicitly the axiom of choice; for every point q = ([γ], t) we are asked torandomly choose a ray γ from p representing the equivalence class [γ]. The map is thus non-canonical. Also it is not clear whether the choices can always be made so as to make fR continuous.In any case, we can consider fR as a map from B into the ball of radius 1 about p in 1

RM . AsR → ∞, it follows essentially from Equation 5.4.5.1 that fR becomes a better and better Hausdorffapproximation (notice that the right hand side of Equation 5.4.5.1 can be interpreted as the distancein C(M(∞)) between ([α1], a) and ([α2], b)). The analogous construction holds when B is definedto be a ball of arbitrary radius in C(M(∞)), and the proposition follows.

Although we did not need this observation in the previous proof, we mention now that fR(interpreted as a map from B to the ball of radius 1 about p in 1

RM) is distance non-decreasing.This follows from the monotonicity claim of Corollary 5.4.3. This observation is central to thefollowing proof.

Proof of Theorem 5.4.1. Fix p ∈ M . Let B denote the ball of radius 1 about the vertex o ∈C(M(∞)) and let fR : B → Bp(R) be defined as in the proof of Proposition 5.4.4. Let a denote anypositive real number smaller than the dimension of C(M(∞)). By the definition of the dimensionof an Alexandrov space ([4, Section 6]),

limε→0

εa · capε(B) = ∞,

where capε(B) denotes the maximal number of disjoint ε-balls which can be packed into B.As mentioned previously, fR : B → 1

RBp(R) is distance non-decreasing. This means thatdM (fR(q1), fR(q1)) ≥ R · d(q1, q2). Therefore, any disjoint packing of 1

R -balls into B induces adisjoint packing of the same number of 1-balls into Bp(R) via the map fR.

Since M satisfies an upper curvature bound, it is shown in [44] that inj(M) ≥ C > 0 for some Cdepending on the upper curvature bound and on the injectivity radius of the soul of M . Thereforeit follows from a well known result of Croke that the volume of any ball of radius 1 on M is boundedbelow by a constant, C, depending on C and on the dimension of M [7].

Putting together the above observations,

vol(Bp(R))

Ra≥

C · cap 1R(B)

Ra

R→∞−−−−→ ∞,

which proves that V G(M) ≥ a. This completes the proof.

40

Chapter 6

Bounded Riemannian submersions

In this chapter we generalize away from nonnegative curvature in order to study Riemanniansubmersion whose A and T tensors are both bounded. Since the metric projection onto a soul,π : M → Σ, is an example of such a Riemannian submersion, the relevance of this study tononnegative curvature is obvious. One consequence of the study will be that when the soul issimply connected, the ideal boundary of M can be determined completely from a single fiber of π.We also explore consequences outside of the field of nonnegative curvature; in particular, we provethat there are only finitely many isomorphism types among the class of Riemannian submersionswhose base space and total spaces both satisfy fixed geometric bounds.

6.1 A bound on the folding of the fibers

Let π : Mn+k → Bn denote a Riemannian submersion. Assume that B is compact. Let A and Tdenote the fundamental tensors of π. Assume that |A| ≤ CA and |T | ≤ CT . The main purposeof this chapter is to explore consequences of these bounds. For p ∈ B, denote by dFp the intrinsicdistance function of the fiber Fp = π−1(p), and by dM the distance function of M restricted to Fp.In this section we establish the following global metric property of the fibers of π:

Theorem 6.1.1. If B is simply connected then:

1. There exists a constant C1 = C1(B,CA, CT , k) such that for any p ∈ B, dFp ≤ C1 · dM .

2. If π has compact holonomy then there exists a constant C2 = C2(π) such that for anytwo points x, y ∈ Fp between which there exists a piecewise smooth horizontal path,dFp(x, y) ≤ C2.

We begin by proving two lemmas which provide technical bounds on a Riemannian submersionwith bounded tensors. Hereafter we denote by Y V and Y H the vertical and horizontal componentsof a vector Y ∈ TM .

Lemma 6.1.2.

1. Along any horizontal path σ(t) in M it is possible to construct an orthonormal verticalframe V1(t), ..., Vk(t) with |V ′

i (t)| ≤ 4k · k! · CA · |σ′(t)|.

2. If Y (t) is any vector field along any horizontal path σ(t) in M , then

d

dt|Y (t)V | ≤ k|(Y ′(t))V |+ k · 4k · k! · CA · |σ′(t)| · |Y (t)|.

41

Proof. To establish part 1, let σ(t) be a horizontal path in M . Let Yi(t), i = 0..k, denote theparallel transport along σ(t) of an orthonormal basis Yi(0) of the vertical space at σ(0). Denoteby Yi(t) = Xi(t)+Vi(t) the decomposition of Yi(t) into horizontal and vertical components. Noticethat

V ′i (t) = V ′

i (t)H + V ′

i (t)V = V ′

i (t)H −X ′

i(t)V = A(σ′(t), V (t))−A(σ′(t), X(t)).

Therefore, |V ′i (t)| ≤ 2CA|σ′(t)|. We next define the frame Vi(t) as the Gram-Schmidt or-

thonormalization of the (ordered) frame Vi(t). For example, V1(t) = 〈V1(t), V1(t)〉−12V1(t). Dif-

ferentiating this expression gives: |V ′1(0)| ≤ 2|V ′

1(0)| ≤ 4CA|σ′(0)|. Continuing the Gram-Schmidtprocess gives:

Vl(t) =

(

Vl(t)−l−1∑

i=1

〈Vi(t), Vl(t)〉Vi(t)

)

normalized

Differentiating this expression gives:

|V ′l (0)| ≤ 2

(

|V ′l (0)|+

l−1∑

i=1

(|V ′l (0)|+ 2|V ′

i (0)|)

)

≤ 2

(

2lCA|σ′(0)|+ 2l−1∑

i=1

|V ′i (0)|

)

≤ a(l)CA|σ′(0)|,

where a(l) is the solution to the following recurance relation: a(0) = 0 ; a(l) = 4(

l +∑l−1

i=1 a(i))

.

It is easy to see that a(l) ≤ 4l · l!, which proves part 1 of the lemma.

Part 2 of the lemma follows by writing |Y (t)V |2 =∑k

i=1〈Y (t), Vi(t)〉2 for the frame Vi(t) givenin part 1, and then differentiating with respect to t.

We use the previous Lemma to establish the following bound, which will be central to our proofof Theorem 6.1.1:

Lemma 6.1.3. Let αs(t) = α(s, t) (s ∈ [0, ε), t ∈ [0, 1]) denote a family of piecewise-smoothpaths in B with fixed endpoints: αs(0) = p, αs(1) = q. Assume |α′

0(t)| ≤ C1. Assume for thevariational vector field V (t) := ∂

∂sα(0, t) along α0 that |V (t)| ≤ C2. Let x ∈ Fp. For each fixeds, let t 7→ αs(t) = α(s, t) denote the horizontal lift the path t 7→ αs(t) with αs(0) = x. Thenτ(s) := αs(1) is a path in the fiber Fq, and |τ ′(0)| ≤ ρ(1), where ρ(t) denotes the solution tothe following differential equation:

ρ′(t) = kCAC1C2(1 + 4kk!) + kC1(CT + 4kk!CA)ρ(t) ; ρ(0) = 0. (6.1.1)

Later, in section 7.3, we will be interested in the case CT = 0. In anticipation of this we mentionnow that when CT = 0, differential equation 6.1.6.1 simplifies greatly, and it is easy to show that:

ρ(1) ≤ 2C2ek4k(k+1)!C1CA (6.1.2)

Proof. Let ∂∂t α(s, t) and

∂∂s α(s, t) denote the natural coordinate vector fields along the parameter-

ized surface α. Notice that ∂∂t α is everywhere horizontal. Also, | ∂∂t α(0, t)| = | ∂∂tα(0, t)| ≤ C1 and

|( ∂∂s α(0, t))

H| = | ∂∂sα(0, t)| ≤ C2.

Applying part 2 of Lemma 6.1.2 to the vector field ∂∂s α(0, t) along the horizontal curve t 7→ α(0, t)

gives:

42

d

dt

∣

∣

∣

( ∂

∂sα(0, t)

)V ∣∣

∣ ≤ k∣

∣

∣

(D

dt

∂

∂sα(0, t)

)V ∣∣

∣+ k4kk!CAC1

∣

∣

∣

∂

∂sα(0, t)

∣

∣

∣

= k∣

∣

∣

(D

ds

∂

∂tα(0, t)

)V ∣∣

∣+ k4kk!CAC1

∣

∣

∣

∂

∂sα(0, t)

∣

∣

∣

≤ k

∣

∣

∣

∣

A

(

( ∂

∂sα(0, t)

)H,∂

∂tα(0, t)

)

+ T

(

( ∂

∂sα(0, t)

)V,∂

∂tα(0, t)

)∣

∣

∣

∣

+ k4kk!CAC1

(

∣

∣

∣

( ∂

∂sα(0, t)

)H∣

∣

∣+∣

∣

∣

( ∂

∂sα(0, t)

)V ∣∣

∣

)

≤ kCAC1C2 + kCTC1

∣

∣

∣

( ∂

∂sα(0, t)

)V ∣∣

∣

+ k4kk!CAC1

(

C2 +∣

∣

∣

( ∂

∂sα(0, t)

)V ∣∣

∣

)

.

In other words,

d

dt

∣

∣

∣

( ∂

∂sα(0, t)

)V ∣∣

∣ ≤ kCAC1C2(1 + 4kk!) + kC1(CT + 4kk!CA)∣

∣

∣

( ∂

∂sα(0, t)

)V ∣∣

∣.

Since τ ′(0) = ∂∂s α(0, 1) = ( ∂

∂s α(0, 1))V , this proves the lemma.

proof of Theorem 6.1.1. Let x, y ∈ Fp and let γ : [0, l] → M be a shortest unit-speed path in Mfrom x to y. Let l denote the smallest integer which is not less than l := length(γ). For each integeri between 0 and l − 1, choose a point xi of Fp closest to γ(i), and choose a shortest unit-speedpath τi from xi to γ(i). Next, for each integer i between 1 and l, define γi as the concatenation ofτi−1 followed by γ|[i−1,i] followed by τ−1

i . By construction, γi is a path in M between xi−1 and xi,whose length, li, is not greater than D := 2 · diam(B) + 1. We describe next how to construct apath βi between xi−1 and xi which remains in the fiber Fp, such that the length of βi can boundedlinearly in terms of li.

Let αi := πγi, which is a loop at p in B whose length is not greater than li. Let zi := hαi+1(xi).We first construct a path β1

i in Fp from xi−1 to zi−1, and then construct a path β2i in Fp from zi−1

to xi.To construct the path β1

i , first re-parameterize αi proportional to arclength, so that αi : [0, 1] →B. Find a piecewise smooth nulhomotopyH : [0, 1]×[0, 1] → B of αi. That is, H(0, t) = p,H(1, t) =αi(t). By Lemma 4.4.2, H can be chosen so that | ∂∂tH| ≤ Qli and | ∂

∂sH| ≤ Q, where Q depends

only on B. Lift the homotopy H to M by defining, for each s ∈ [0, 1], the curve t 7→ H(s, t) to bethe horizontal lift of the curve t 7→ H(s, t) beginning at H(s, 0) = xi−1. Let β1

i (s) := H(s, 1). ByLemma 6.1.3, |(β1

i )′(s)| ≤ ρli(1), where ρli(t) is the solution to the following differential equation:

(ρli)′(t) = kCAQ

2li(1 + 4kk!) + kQli(CT + 4kk!CA)ρli(t) ; ρli(0) = 0.

In particular, length(β1i ) ≤ C · li, where C is a bound on the derivative of the function li 7→ ρli(1)

between li = 0 and l = D.We continue by constructing a path β2

i in Fp between zi−1 and xi, whose length is also con-trolled linearly in terms of li. Let β2

i (s) := hs(γi(s)), where hs : Fαi(s) → Fp is the holonomydiffeomorphism associated to the curve αi|[s,li]. It is clear from construction that β2

i connects zi−1

to xi and that (β2i )

′(s) = dhs(γ′i(s)

V). Thus, length(β2i ) ≤ L · li, where L is a Lipschitz bound on

all holonomy diffeomorphisms associated to curves in B of length ≤ D (such a Lipschitz boundexists by Lemma 4.1.2).

The concatenation, βi, of β1i followed by β2

i satisfies length(βi) ≤ (C + L)li ≤ (C + L)D. Theconcatenation of the paths βi (i = 1, ..., l) is a path from x to y in Fp. Therefore,

dFp(x, y) ≤ l(C + L)D ≤ (l + 1)(C + L)D ≤ 2l(C + L)D.

43

The final inequality is valid only when l ≥ 1, but when l ≤ 1, dFp(x, y) ≤ (C +L) · l. So the choice

C1 := max2(C + L)D,C + L is as required for part 1 of the theorem.To prove part 2, let x, y ∈ Fp be two points between which there exists horizontal path α. Let

α := π α. Notice that hα(x) = y. One can choose a different loop σ at p in B such that hσ = hα

and length(σ) ≤ b1, where b1 is the constant from Lemma 4.1.1. The horizontal lift, σ, of σ providesan alternative horizontal path from x to y. Thus dFp(x, y) ≤ C1 ·dM (x, y) ≤ C1 · length(σ) ≤ C1 ·b1,so the choice C2 := C1b1 is as required for part 2.

Example 6.1.4. The conclusions of parts 1 and 2 Theorem 6.1.1 both fail for π : TS2 → S2 whenTS2 is given the connection metric with flat fibers, which has unbounded A-tensor. It’s easy to seethat two rays from the origin of TpS

2 grow apart linearly in the fiber TpS2, while they maintain a

distance ≤ length(α) in TS2, where α is a loop in S2 which parallel translates the initial tangentvector of the first ray to the initial tangent vector of the second ray.

Example 6.1.5. The conclusions of parts 1 and 2 of Theorem 6.1.1 both fail for the followingexample in which the base space is not simple connected. Let M be the following flat manifold:M = R2 × [0, 1]/(r, θ, 0) ∼ (r, θ + λ, 1). Here λ is any non-zero angle. Guijarro and Petersenstudied this manifold in [16]. The soul of M is the circle 0 × [0, 1]/ ∼, and each fiber Fp of themetric projection onto the soul is isometric to flat R2. Two rays from the same point p of the soulwhich make an angle λ will grow apart linearly in the fiber Fp, but will remain at distance ≤ 1 inM .

Example 6.1.6. In section 4.3 we constructed a metric of nonnegative curvature on S2 × R4 forwhich the holonomy group of the metric projection onto the soul is noncompact. It is straightfor-ward to see that the conclusion of part 2 of Theorem 6.1.1 fails for the metric projection onto thesoul of this manifold.

Example 6.1.7. Let Mn+k be a simply connected open manifold of nonnegative curvature, andlet π : Mn+k → Σn denote the metric projection onto its soul. Theorem 6.1.1 provides the bounddFp ≤ C · dM for each fiber Fp of π. According to the theorem, C depends on Σ, CA, CT , andk, where CA and CT denote bounds on the A and T -tensors of π respectively. But by O’Neill’sformula (2.4.2.1), CA depends only on the maximum curvature of Σ. Further, by Proposition 2.1.4,CT depends only on the injectivity radius of Σ. In other words, C depends only on Σ and k. Thismeans that our bound on the “folding of the fibers” holds uniformly over all open manifolds ofnonnegative curvature of a fixed dimension and with a fixed soul!

6.2 Measuring size in the holonomy of a submersions

In Section 4.4 we proved that, for a vector bundle with a connection, the size of a holonomy elementis bounded in terms of the length of the loop which generates it. In this section we generalize thisresult to bounded Riemannian submersions. Let π : M → B denote a Riemannian submersionwhose fundamental tensors are bounded: |A| ≤ CA and |T | ≤ CT . Assume that the holonomygroup, Φ, of π is a (possibly noncompact) finite dimensional Lie group. We develop a notion of“size” in the holonomy group, and show how to control the size of hα in terms of length(α). Letα0 denote a loop in B at p for which hα0 = id. For example, this is the case when α0 is the trivialloop, but it may also occur for nontrivial loops. Let αs denote a variation of α0. Then hαs definesa path in Φ beginning at id, and V := (hαs)′(0) is an element of the Lie algebra, G, of Φ. Everyvector of G can be described in this way; see the proof of Lemma 4.1.1. It is natural to considerV as a vertical vector field on Fp. We write V (x) for the value of this vector field at x ∈ Fp. It isclear from Lemma 6.1.3 that this vector field has bounded norm. This observation is particularlyinteresting for the metric projection onto a soul. In this setting, one can prove through other routesthat such “holonomic” vertical vector fields are bounded; see for example Proposition 5.3.1.

44

We call a left invariant metric, m, on Φ acceptable if the following condition is satisfied: forall V ∈ G, |V |m ≤ supx∈Fp

|V (x)|. Notice that any left invariant metric can be made acceptableby rescaling. For g ∈ Φ, we denote by |g| the supremum over all acceptable metrics on Φ of thedistance in Φ between g and id. This provides a natural notion of the “size” of a holonomy element.Notice that |g1 · g2| ≤ |g1|+ |g2|. We prove the following analog of Proposition 4.4.1:

Proposition 6.2.1. If B is simply connected and Φ is a finite dimensional Lie group, thenthere exists C = C(B,CA, CT , k) such that for any loop α in B, |hα| ≤ C · length(α).

Proof. By an argument used many times in this paper, it will suffice to find a constant C suchthat |hα| ≤ C · length(α) for all loops α of length ≤ D = 2 · diam(B) + 1. Let α : [0, 1] → Bbe a constant speed loop at p ∈ B with l := length(α) ≤ D. By Lemma 4.4.2, there exists apiecewise smooth nulhomotopy H : [0, 1]× [0, 1] → B of α such | ∂∂tH| ≤ Ql and | ∂

∂sH| ≤ Q. HereH0(t) = H(0, t) = p and H1(t) = H(1, t) = α(t). Let g(s) = hHs , which is a piecewise smooth pathin Φ between the identity and hα. It will suffice to find C such that for any acceptable metric, m,on Φ, |g′(s)|m ≤ C · l. So choose any fixed acceptable metric, m. For fixed s ∈ [0, 1], let σr bethe following family of loops: σr = Hs+r ∗ (Hs)

−1. Notice that hσ0 = id, and hHs hσr = hHs+r .Let V ∈ G be the element V := (hσr )′(0). Since the metric is left invariant, |g′(s)|m = |V |m. ByLemma 6.1.3, for any x ∈ Fp, |V (x)| ≤ ρl(1), where ρl(t) denotes the solution to the followingdifferential equation:

ρ′l(t) = 2kCAQ2l(1 + 4kk!) + 2kQl(CT + 4kk!CA)ρl(t) ; ρl(0) = 0.

In particular, |g′(s)|m ≤ C · l, where C is a bound on the derivative of the function l 7→ ρl(1)between l = 0 and l = D. This completes the proof.

We remark that Proposition 6.2.1 implies a weak version of Proposition 4.4.1. More precisely,if the unit sphere bundle of a Riemannian vector bundle with a connection is endowed with thenatural connection metric, then the projection map becomes a Riemannian submersion. The Ttensor of this submersion vanishes, and the A tensor depends on R∇. This argument produces onlya week version of Proposition 4.4.1, because it does not establish the bound to be linear in CR.

6.3 The ideal boundary is determined by one fiber

We return in this section to our previous set up, in whichM denotes an open manifold of nonnegativesectional curvature with soul Σ ⊂ M , and π : M → Σ denotes the metric projection. A nice wayto rephrase the conclusion of Theorem 6.1.1 is that the inclusion map i : Fp → M is a biLipschiptzmap for any p ∈ Σ. Since any point of M has distance ≤ diam(Σ) from some point of the fiber Fp,i is a quasi-isometric embedding. In other words, M is quasi-isometric to any single fiber of themetric projection onto its soul.

From the above considerations, one expects a fiber Fp to share with M all large-scale geometricproperties. In this section we prove that the ideal boundary of M is determined by a single fiberof π.

Corollary 6.3.1. If π1(Σ) = 1, then the topology of the ideal boundary, M(∞), of M can bedetermined by the pointed manifold (Fp, p) for any p ∈ Σ.

Proof. Choose p ∈ Σ, and use the construction of M(∞) described in section 2.3. Any ray inFp from p is also a ray in M , so we could equally well have defined M(∞) as the set of rays inFp from p. By Theorem 6.1.1, we get the same topology on M(∞) if we replace dM with dFp inequation 2.3.2.1. The statement of the corollary follows.

45

It does not make sense to speak of the “ideal boundary of a fiber”, since the term “idealboundary” has only been defined for open manifolds of nonnegative curvature. However, speakingloosely, the above proof demonstrates that the ideal boundary of M is homeomorphic to the idealboundary of any fiber.

Example 6.3.2. Corollary 6.3.1 fails for the open manifold of nonnegative curvature described inexample 6.1.5 because the soul is not simple connected. In this example, M(∞) = S1 if the angleλ is rational, and M(∞) is a single point if λ is irrational. However, each fiber is isometric to flatR2 regardless of the value of λ.

Example 6.3.3. The manifold M = (S2×R4, g) constructed in Section 4.3 provides a nice illustra-tion of Corollary 6.3.1. Here M(∞) = [0, π/2]. There are two totally geodesic fibers of the metricprojection π : M → Σ; namely the fibers over the north and south poles of the soul Σ = S2× (0, 0).Each of these fibers is itself a manifold of nonnegative curvature whose ideal boundary also equals[0, π/2]. We do not know whether in general the ideal boundary of a totally geodesic fiber isnecessarily isometric (rather than just homeomorphic) to the ideal boundary of the manifold.

We conclude this section by describing an alternative and substantially different way to prove(at least a weak version of) Theorem 6.1.1 in the special case of the metric projection onto a soul.

Proposition 6.3.4 (Special Case of Theorem 6.1.1). Let M be a simply connected openmanifold of nonnegative curvature, and let π : M → Σ denote the metric projection its soul.Then there exists a constant C depending on M such that for any two points x, y in thesame fiber Fp which can be connected by a horizontal path γ in M , dFp(x, y) ≤ C · length(γ).

This proposition follows immediately from Theorem 6.1.1. In fact, since this proposition doesnot claim that C depends only on Σ, it is quite a bit weaker than Theorem 6.1.1 (see Example 6.1.7).Still, we wish to prove this proposition independently in order to exhibit a nice argument which isspecific to nonnegative curvature. We begin with a lemma. Fix p ∈ Σ and consider each vector Vof the Lie algebra G of the holonomy group Φ of ν(Σ) as a Killing vector field on the unit normalsphere ν1p(Σ). We write V (w) for the value of this Killing field at w ∈ ν1p(Σ). Also, an fix an innerproduct on G, and to consider Φ with the associated left-invariant metric.

Lemma 6.3.5. There exists a constant K, depending only on M and the chosen innerproduct for G, such that for any unit-length vector V ∈ G and any w ∈ ν1p(Σ), the Jacobi

field J(t) along the geodesic t 7→ exp⊥(tw) with J(0) = 0 and J ′(0) = V (w) satisfies |J(t)| < Kfor all t.

Proof. In the proof of Proposition 5.3.1, we constructed a basis V1, ..., Va of G such that for anyw ∈ ν1p(Σ) and any i between 1 and a, the Jacobi field Ji(t) along γ(t) := exp⊥(tw) with initialconditions Ji(0) = 0 and J ′

i(0) = Vi(w) is bounded in norm by a constant which does not dependon the choice of i or w. Since any unit-length vector V ∈ G can be written as V =

∑

λiVi, witheach λi bounded in absolute value by a constant which reflects the extent to which the basis Viof G fails to be orthonormal, the statement of the lemma follows.

Proof of Proposition 6.3.4. Let x, y be two points of the same fiber Fp between which there existsa horizontal path γ. Let α := π γ, which is a loop at p in Σ. Fix an inner product, m, on the Liealgebra G of Φ which is acceptable in the sense of Section 4.4. Consider m as a left-invariant metricon Φ. By Proposition 4.4.1, there exists a constant C depending on Σ and on the connection ofν(Σ) such that

|Pα|m ≤ C · length(α) ≤ C · length(γ).

This means that there is a vector V ∈ G of length |V | ≤ C · length(γ) such that exp(V ) = Pα ∈ Φ.Choose w ∈ ν1p(Σ) such that x = exp⊥(rw), where r := dM (p, x). Define σ(s) := exp(s · V )(w),

which is a path in ν1p(Σ) with σ′(s) = V (σ(s)). Next define β(s) := exp⊥(r · σ(s)), which is a path

46

in the fiber Fp between β(0) = x and β(1) = y. Notice that β′(s) = d(exp⊥)r·σ(s)(r ·σ′(s)) = Js(r),

where Js(t) is the Jacobi field along the radial geodesic t 7→ exp⊥(t · σ(s)) with J(0) = 0 andJ ′(0) = V (σ(s)). Therefore |β′(s)| ≤ K · |V |, where K is the constant given by Lemma 6.3.5. Thismeans that length(β) ≤ K · C · length(γ). This completes the proof.

6.4 Two finiteness theorems for Riemannian submersions

In this section we prove two theorems which say that there are only finitely many equivalence classesof Riemannian submersions whose base space and total space both satisfy fixed geometric bounds.We consider two Riemannian submersions, π1 : M1 → B1 and π2 : M2 → B2, to be Ck-equivalentif there exists a Ck map f : M1 → M2 which maps the fibers of π1 to the fibers of π2. EveryRiemannian submersion is a fiber bundle, and this notion of equivalence just means equivalence upto Ck fiber bundle isomorphism. Our first result is a finiteness theorem for C1-equivalence classesof submersions. Our second result will show that some of the geometric hypotheses of the first canbe weakened if we are willing to settle for proving finiteness only up to C0-equivalence classes.

Theorem 6.4.1 (Finiteness Theorem #1). Let n, k ∈ Z and V,D, λ ∈ R. Then there are onlyfinitely many C1 fiber bundle isomorphism classes in the set of Riemannian submersionsπ : Mn+k → Bn for which:

1. B is simply connected.

2. vol(B) ≥ V,diam(B) ≤ D, |sec(B)| ≤ λ.

3. vol(M) ≥ V, diam(M) ≤ D, |sec(M)| ≤ λ.

Some of the bounds in this theorem are redundant. For example, since Riemannian submersionsare curvature non-decreasing, the lower curvature bound on the base space is redundant to the lowercurvature bound on the total space. Also, the upper diameter bound on B is redundant to theupper diameter bound on M .

This theorem is based on the following result of P. Walczak ([36], as corrected in [37]):

Theorem 6.4.2 (Walczak). Let n, k ∈ Z and V,D, λ, CA, CT ∈ R. Then there are onlyfinitely many C1 fiber bundle isomorphism classes in the set of Riemannian submersionsπ : Mn+k → Bn for which:

1. |A| ≤ CA and |T | ≤ CT .

2. vol(B) ≥ V,diam(B) ≤ D, |sec(B)| ≤ λ.

3. There exists a fiber Fp for which vol(Fp) ≥ V , diam(Fp) ≤ D, |sec(Fp)| ≤ λ.

Proof of Theorem 6.4.1. Let n, k ∈ Z and let V,D, λ ∈ R. Suppose that π : Mn+k → Bn is aRiemannian submersion satisfying conditions 1-3 of Theorem 6.4.1. By O’Neill’s Formula (2.4.2.1),the A-tensor of π is bounded in norm by a constant, CA, depending only on λ. Similarly, byProposition 2.1.4, the T -tensor of π is bounded by a constant CT which depends only on λ andon inj(B). By a well known lemma of Cheeger, inj(B) is in turn bounded below by a constantdepending only on n, V,D, and λ (see [25]).

Let p ∈ B and let Fp := π−1(p). It remains to bound the volume, diameter, and curvature ofFp in terms of n, k, V,D, λ, CA, CT , and then apply Theorem 6.4.2. First, by Gauss’ formula,|sec(Fp)| ≤ λ + 2CT . Second, to control vol(Fp), notice that any two fibers have similar volumes.More precisely, the diffeomorphism hα : Fp → Fq associated to a minimal path, α, in B betweenp and q satisfies the Lipschitz constant eCT ·length(α) ≤ eCT ·D (see Lemma 4.1.2), so vol(Fp) ≤(ekCT ·D) · vol(Fq). But by Fubini’s theorem, vol(M) =

∫

p∈Bvol(Fp) dvolB , which implies that for

47

any p ∈ B, vol(Fp) ≥ vol(M)vol(B) (e

−kCT ·D). By the Bishop-Gromov inequality, vol(B) is bounded

above by a constant depending only on D,λ, and n. This observation completes our argument thatvol(Fp) is bounded below.

Finally, diam(Fp) ≤ C1 ·diam(M) ≤ C1 ·D, where C1 is the constant from Theorem 6.1.1, whichdepends on B,CA, CT , k. In fact, it is clear from the proof of theorem 6.1.1 that C1 really dependsonly on diam(B), Q(B), CA, CT , k, where Q(B) is the constant in Lemma 4.4.2. We argue nowthat Q(B) depend only on the assumed geometric bounds of B. Let M denote the class of all ndimensional Riemannian manifolds for which vol ≥ V , diam ≤ D, and |sec| ≤ λ. By assumption,B ∈ M. M is pre-compact in the Lipschitz topology and contains only finitely many diffeomorphismtypes (see [25]). This means that it is possible to choose a finite set, B1, ..., Bl ⊂ M, and aconstant L = L(n, V,D, λ) such that for any M ∈ M, there exists an L-biLipschitz diffeomorphismbetween M and some Bi. Therefore, for any M ∈ M, Q(M) ≤ L2 ·maxQ(Bi). This proves thatQ(B) satisfies an upper bound depending only on n, V,D, λ, which completes the proof.

Next we discuss the following question: if the upper curvature bound on the base space and/orthe total space is removed from Theorem 6.4.1, is it still true there are only finitely many C0 fiberbundle isomorphism classes in this set of Riemannian submersions? We begin this exploration byrecalling a Theorem of J.Y. Wu [43]:

Theorem 6.4.3 (J.Y. Wu). Let Bn be a compact Riemannian manifold. Let k ∈ Z andV,D, λ ∈ R. Assume k ≥ 4. Then there are only finitely many C0 fiber bundle isomorphismclasses in the set of Riemannian submersions π : Mn+k → Bn for which:

1. sec(M) ≥ λ.

2. For each fiber Fp, vol(Fp) ≥ V , diam(Fp) ≤ D, sec(Fp) ≥ λ.

3. Each fiber Fp is totally convex (that is, dFp = dM).

The advantage of Wu’s theorem is that he requires no upper curvature bounds. The disadvan-tages of Wu’s theorem are as follows:

• The fiber dimension is restricted to k ≥ 4. Wu stated in [43] that he believes the result shouldstill hold when k = 2, 3.

• He fixes the base space rather than fixing geometric bounds on the base space.

• He assumes geometric bounds on the fibers rather than only assuming geometric bounds onthe total space.

• The condition that the fibers are totally convex is very strong. For example, it is muchstronger than assuming that the fibers are totally geodesic.

The following is a fairly immediate application of Theorem 6.1.1:

Theorem 6.4.4. Wu’s Theorem 6.4.3 remains true if hypothesis 3 is removed.

Proof. It follows immediately from Wu’s proof that hypothesis 3 can be replace by the followingweaker assumption:

3’) For any fiber Fp, if x, y ∈ Fp and dM (x, y) ≤ 12 inj(B), then dFp(x, y) ≤ C · dM (x, y),

where C is any fixed constant.Now suppose that π : M → B be a Riemannian submersion satisfying conditions 1 and 2 of

Theorem 6.4.3. As before, the A-tensor is bounded in norm by a constant CA depending only onB and λ, and the T -tensor is bounded in norm by a constant CT depending only on inj(B) and λ.

48

So if the base space B is simply connected then it is immediate from Theorem 6.1.1 thatcondition 3’ is satisfied automatically for a constant C depending only on B, V,D, λ, k. In fact,this is true when B is not simply connected as well. To see this, notice that if x, y ∈ Fp satisfydM (x, y) ≤ 1

2 inj(B), then the projection to B of a minimal path in M between x and y will benulhomotopic, so the proof of Theorem 6.1.1 goes through in this setting.

Wu states in [43, p. 513] that his theorem remains true if “Riemannian submersion” is replacedby “L-Lipschitz map”, where L is any fixed number. It is not clear to us that his proof goes throughin this generality; however, it is obvious that the proof still holds when “Riemannian submersion” isreplaced by “submersion π for which π∗|H is at every point an L-biLipschitz linear map”, where Hdenotes the distribution orthogonal to the fibers. We will call such a submersion an “L-Riemanniansubmersion”. In review, the following generalization of Wu’s Theorem is true:

Theorem 6.4.5 (Generalization of Wu’s Theorem). Let Bn be a compact Riemannianmanifold. Let k ∈ Z and V,D, λ, L ∈ R. Assume k ≥ 4. Then there are only finitely many C0

fiber bundle isomorphism classes in the set of L-Riemannian submersions π : Mn+k → Bn

for which:

1. sec(M) ≥ λ.

2. For each fiber Fp, vol(Fp) ≥ V , diam(Fp) ≤ D, sec(Fp) ≥ λ.

This jump to the generality of L-Riemannian submersions allows us to replace the fixed basespace with a base space which only has fixed geometric bounds. We also replace the geometricbounds on the fibers with only geometric bounds on the total space:

Theorem 6.4.6 (Finiteness Theorem #2). Let n, k ∈ Z and V,D, λ ∈ R. Assume k ≥4. Then there are only finitely many C0 fiber bundle isomorphism classes in the set ofRiemannian submersions π : Mn+k → Bn for which:

1. B is simply connected.

2. vol(B) ≥ V,diam(B) ≤ D, |sec(B)| ≤ λ.

3. vol(M) ≥ V, diam(M) ≤ D, sec(M) ≥ λ.

Proof. Let M denote the set of all candidates for the base space (that is, all Riemannian n-manifoldssatisfying vol ≥ V,diam ≤ D, and |sec| ≤ λ). Choose B0, ..., Bl ⊂ M and L ∈ R such that foreach B ∈ M, there exists an L-biLipschitz diffeomorphism f : B → Bi for some i (as in the proofof Theorem 6.4.1).

Now if π : M → B is a Riemannian submersion satisfying the hypotheses of Theorem 6.4.6,then f π : M → Bi is an L-Riemannian submersion. Further, the two submersions, π and f π,are clearly isomorphic as fiber bundles. By arguments in the proof of Theorem 6.4.1, each fiberof π (equivalently of f π) satisfies a curvature, diameter, and volume bound depending only onV,D, λ, n, k. Therefore, applying Theorem 6.4.5 to each of the base spaces B0, ..., Bl finishes theproof.

This theorem is probably not optimal. For example, the assumption that k ≥ 4 can probably beremoved, as well as the upper curvature bound on the base space. Also, we do not know whetherthe theorem holds without the assumption that the base space is simple connected. This hypothesisis used in the proof only to achieve an upper diameter bound for the fibers, so instead of assumingthat the base space is simply connected, we could instead assume a diameter bound for the fibers.

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Chapter 7

Finiteness theorems for fiberbundles

In this chapter we prove finiteness theorems for vector bundles and principal bundles. For example,there are only finitely many vector bundles of a fixed rank over a fixed compact Riemannianmanifold capable of admitting a connection whose curvature tensor satisfies a fixed bound in norm.An analogous statement holds for principal bundles, and the proof of this analog is based on the“folding of the fibers” bound from Chapter 6.

We provide two applications of these finiteness theorems to nonnegative curvature. First, thereare only finitely many vector bundles of a fixed rank over a fixed soul (that is, a fixed compact Rie-mannian manifold of nonnegative curvature) which are capable of admitting nonnegatively curvedmetrics for which the vertical curvatures at the soul satisfy a fixed upper bound. Second, mostvector bundles over Bieberbach manifolds do not admit metrics of nonnegative curvature.

7.1 Introduction

Guijarro and Walschap recently proved that for fixed n, k ∈ Z and Λ ∈ R, if Mn+k is an openmanifold of nonnegative curvature whose soul is isometric to S := Sn(1), and the curvature of Mis bounded above by Λ, then there are only finitely many possibilities for the isomorphism class ofthe normal bundle of the soul in M [18, Theorem A]. The purpose of this chapter is 1) to provethis theorem for an arbitrary soul S, and 2) to use variations on Guijarro and Walschap’s idea toproduce finiteness theorems for vector bundles and principal bundles with connections.

We begin in section 7.2 by proving the following finiteness theorem for vector bundles withconnections:

Theorem 7.1.1. For any set of positive constants V,D, λ,Λ, n, k (n, k ∈ Z), denote by N =N(V,D, λ,Λ, n, k) the number of isomorphism classes of rank k vector bundles Rk → E → Bn

which admit the following structure:

1. A metric on B for which vol(B) ≥ V , diam(B) ≤ D, |sec(B)| ≤ λ.

2. A Euclidean structure (i.e., a smoothly varying inner product on the fibers).

3. A connection ∇ which is compatible with the Euclidean structure, whose associatedcurvature tensor, R∇, satisfies |R∇| ≤ Λ (here |R∇| is defined with respect to themetric on B and the inner products on the fibers).

Then N ≤ C · (1 + Λ)c, where c, C depend only on V,D, λ, n, k.

50

Notice that Cheeger’s Finiteness Theorem implies that there are only finitely many possibilitiesfor the diffeomorphism type of the base space B, which is necessary for this theorem to make sense.Notice also that, for a fixed base B with a fixed metric, a bound on |R∇| implies bounds on thecharacteristic classes of the bundle, which implies that there are only finitely many possibilities forthe isomorphism class of the bundle; see for example [1, Chapter 7]. However, it seems difficult toobtain from this line of reasoning an explicit upper bound for N in terms of Λ as in Theorem 7.1.1.

Using Corollary 3.1.8, we get as an immediate corollary to Theorem 7.1.1 the following versionof Guijarro and Walschap’s result for arbitrary souls:

Corollary 7.1.2. Suppose that Mn+k is an open manifold of nonnegative sectional curvaturesuch that:

1. The soul, Σn, of M satisfies: vol(Σ) ≥ V , diam(Σ) ≤ D, sec(Σ) ≤ λ.

2. The curvature of every vertical 2-plane at every point of Σ is ≤ Λ.

Then the number, N = N(V,D, λ,Λ, n, k), of possibilities for the isomorphism class of thenormal bundle of the soul is ≤ C · (1 + Λ)c, where c, C depend only on V,D, λ, n, k.

By a vertical 2-plane we mean a plane spanned by two vectors which are both normal to thesoul.

Notice that Corollary 7.1.2 is stated somewhat carefully. It would be a stronger statement tosay that “there are only finitely many isomorphism classes of vector bundles which admit a metricof nonnegative curvature with fixed bounds on the geometry of the soul and on the curvatures ofvertical 2-planes at the soul”. This statement would be stronger than Corollary 7.1.2 because thetotal space of a given vector bundle ξ could conceivably admit a metric of nonnegative curvature,but not one for which the soul is the zero section, and hence not one for which the normal bundle ofthe soul is necessarily isomorphic to ξ. The problem is that two vector bundles with diffeomorphictotal spaces need not be isomorphic as vector bundles.

Next, in section 7.3, we prove the following analog of Theorem 7.1.1 for principal bundles:

Theorem 7.1.3. For any compact Lie group G with a fixed bi-invariant metric, and any setof constants V,D, λ,Λ, n (n ∈ Z), let N := N(G,V,D, λ,Λ, n) denote the number of isomor-phism classes of principal G-bundles G → P → Bn which admit the following structure:

1. A metric on B for which vol(B) ≥ V , diam(B) ≤ D, |sec(B)| ≤ λ.

2. A connection, ω, whose curvature form, Ω, satisfies |Ω| ≤ Λ (here |Ω| is defined withrespect to the metrics on B and G).

Then N ≤ ec(1+eCΛ), where c, C depend only on G,V,D, λ, n.

Notice that Theorem 7.1.3 does not quite imply Theorem 7.1.1 because the bound is linear inthe latter theorem, but double exponential in the former. We do not know whether the doubleexponential bound can be improved to exponential or linear.

Our proof of Theorem 7.1.3, viewed properly, provides a solution to the following problem: givena compact manifold Bn and a compact Lie group G, find a bound (depending on B and G) for thenumber of isomorphism classes of principal G-bundles over B which admit a flat connection. Noticethat B and G do not have metrics in this problem, which is why the statement of Theorem 7.1.3 isnot helpful. Nevertheless, the proof of Theorem 7.1.3 allows us to find a bound which depends onfollowing invariants of Bn and G:

1. I(B) := the size of a minimal atlas for B (that is, the minimal number of open sets diffeo-morphic to Rn necessary to cover B).

51

2. P (G,n) := the number of points necessary to form an inj(G)2n+4−8 -net in (G, g0), where g0 denotes

the bi-invariant metric associated to the killing form on G.

Specifically, we prove the following:

Theorem 7.1.4. Let Bn be a compact manifold and let G be compact Lie group. Then thenumber of isomorphism classes of principal G-bundles over B which are capable of admittinga flat connection is finite and is ≤ P (G,n)I(B)2 .

As a consequence of Theorem 7.1.4, we prove that most vector bundles over Bieberbach mani-folds (i.e., compact manifolds which are capable of admitting flat metrics) do not admit metrics ofnonnegative curvature. More precisely,

Corollary 7.1.5. Let Sn be a Bieberbach manifold, and let k ∈ Z. Let N = N(S, k) denotethe number of isomorphism classes of rank k vector bundles over S which admit a metricof nonnegative curvature for which the zero section is the soul. Then N is finite and is infact ≤ P (O(k), n)I(S)2 .

This result is related to a theorem of Walschap and Ozaydin, who proved that no nontrivial ori-entable rank 2 vector bundle over the torus Tn can admit a metric of nonnegative curvature [42]. Asthey showed, a vector bundle over a Bieberbach manifold admits a metric of nonnegative curvaturefor which the zero-section is the soul iff it admits a flat connection. To the best of our knowledge, itis reasonable to ask whether any nontrivial vector bundle over any Bieberbach manifold can admita flat connection (and hence a metric of nonnegative curvature). Corollary 7.1.5 provides evidencethat the answer might be no.

Finally, we address the question of whether Theorem 7.1.3 is valid when G is a noncompact Liegroup with a left-invariant metric.

7.2 A finiteness theorem for vector bundles

In this section we prove Theorem 7.1.1. The proof depends on the following result of Gromov ([10,2.16], [11, 0.5.E]):

Lemma 7.2.1 (Gromov).

1. (Easy version) Let Y be a compact path-connected length space which satisfies thefollowing regularity condition: there exists δ = δ(Y ) such that any two maps f1, f2from the same space into Y with dsup(f1, f2) ≤ δ must be homotopic. Let Xn be acompact Riemannian manifold. Then the number, N , of homotopy classes of mapsX → Y which admit an L-Lipschitz representative is ≤ ecL

n

, where c depends on Xand Y .

2. (More difficult version) If Y is a compact Riemannian manifold with a finite funda-mental group, then N ≤ C ·(1+L)c, where the exponent c depends only on the homotopytypes of X and Y , while the constant C depends on their metrics as well.

Later, in the proof of Theorem 7.1.3, we will need to copy the idea of Gromov’s proof of part 1of Lemma 7.2.1; in anticipation of this, we include here the proof:

Proof of Part 1 of Lemma 7.2.1. Let δ := δ(Y ). Let RY denote a δ4 -net in Y , and let RX denote

a δ4L -net in X. Any map f : X → Y induces a (non-unique) map f : RX → RY , defined so that

f(p) is any point of RY whose distance from f(p) is ≤ δ4 . If f1, f2 : X → Y are two L-Lipschitz

maps for which f1 = f2, it is easy to see that dsup(f1, f2) ≤ δ, so f1 is homotopic to f2. This

52

proves that the number, N(X,Y, L), of distinct homotopy classes of maps X → Y which admit anL-Lipschitz representative is ≤ card(RY )

card(RX).Next we establish a bound for card(RX). Denote by capr(X) the minimal cardinality of an

r-net in X. We prove that capr(X) ≤ K((

1r

)n+ 1), where K depends on X. Notice that without

the “+1” term, this bound would be incorrect when r is very large. Let Bn denote the open ball ofradius 1 in Rn. Choose an atlas for X; that is, an open covering Ui of X, and diffeomorphismsΦi : Bn → Ui, i = 1, .., l. Let L denote a common Lipschitz bound for the maps Φi. Clearly,

capr(X) ≤l

∑

i=1

capr(Ui) ≤ l · cap( rL )

(Bn)(r→0)∼ l · K ·

(

Lr

)n

,

where K depends only on n. In particular, we can choose R small enough so that for any r < R,capr(X) ≤ K ′ ( 1

r

)n, where K ′ := 2 · l · K ·Ln. So defining K := maxK ′, capR(X) will insure that

capr(X) ≤ K((

1r

)n+ 1) for any r.

Thus, N(X,Y, L) ≤ card(RY )K(( 4L

δ )n+1) ≤ ec(Ln+1) for properly chosen c. Since N(X,Y, L) =

1 for L sufficiently close to zero (specifically for L < δdiam(X) ), it is easy to see that c can be

redefined such that N(X,Y, L) ≤ ecLn

.

Proof of Theorem 7.1.1. Let Bn be a compact Riemannian manifold, and let Eπ→ B be a rank

k vector bundle endowed with a Euclidean structure and a compatible connection, ∇. Assumethat |R∇| ≤ Λ. We prove that the number of candidates for the isomorphism class of this bundleis ≤ e(c+CΛ)n , where c, C depend only on B and k. This result implies Theorem 7.1.1 by thefollowing argument: given bounds V,D, λ, n, let M = M(V,D, λ, n) denote the class of Riemanniann-manifolds satisfying vol ≥ V , diam ≤ D, and | sec | ≤ λ. M is precompact in the Lipschitztopology and contains only finitely many diffeomorphism types (see [25]). This means that it ispossible to choose a finite collection M1, ...,Mm ⊂ M and a constant L = L(V,D, λ, n) such thatfor any M ∈ M there exists an L-biLipschitz diffeomorphism from M to some Mi. Therefore, anyvector bundle over any manifold M ∈ M which admits a Euclidean structure and a compatibleconnection satisfying the bound |R∇| ≤ Λ is isomorphic to a vector bundle over some Mi whichadmits a Euclidean structure and a connection satisfying |R∇| ≤ L2 · Λ; namely, the pull-backbundle with the pull back Euclidean structure and connection. From this observation it is easyto see that the above stated “fixed base space” version of the theorem implies the more generalversion of the theorem.

We begin by choosing an explicit atlas of local trivializations for the bundle Eπ→ B. Choose a

value r < 12 mininj(B), π√

λ, where λ denotes an upper curvature bound for B. Notice that any

Jacobi field, J(t), along any unit-speed geodesic of B with J(0) = 0 has monotonically increasingnorm for t ∈ [0, r]; This follows, for example, from the version of Rauch’s theorem stated in [3,Lemma 6.3.5]. Choose a finite collection p1, ..., pl of points of B such that the collection of r-ballsBi := Bpi(r) covers B. Let ηi denote a partition of unity subordinate to Bi.

The connection, ∇, provides a natural collection of local trivializations, Φi : Bi×Rk → π−1(Bi).More precisely, for each i, choose a fixed orthogonal identification si : Rk → Epi := π−1(pi), anddefine Φi(q, v) to be the parallel transport of si(v) along the minimal geodesic, γi

q, between pi and

q. That is, Φi(q, v) := Pγiq(si(v)). Also define hi : π−1(Bi) → Rk as hi := pi Φ−1

i , where pi :

Bi×Rk → Rk denotes the projection onto the second factor. Notice that hi(v) = s−1i (P(γi

π(v))−1(v)).

For any pair (i, j) for which Bij := Bi ∩ Bj is nonempty, we have a transition function gij :Bij → O(k), defined as

gij(q)(v) := hj(h−1i (v) ∩ π−1(q)) = (s−1

j P(γjq)−1 Pγi

q si)(v).

We wish to establish a Lipschitz bound on each gij . Suppose that q ∈ Bij , and let X ∈ TqB with|X| = 1. Let q(s) := exp(sX) and let v ∈ Rk be any unit-length vector. Let τv(s) := gij(q(s))(v),

53

which is a path on the unit sphere of Rk. In order to bound |(gij)∗(X)|, it will clearly suffice tofind a bound on |τ ′v(0)| which does not depend on the choice of v.

For small values of s, let t → σs(t) = σ(s, t) be a minimal unit-speed path between pi andq(s) followed by a minimal unit-speed path between q(s) and pj . By construction, τv(s) = (s−1

j Pσs si)(v). Let area(s0) denote the area of the homotopy between σ0 and σs0 determined bythe family of curves σs, s ∈ [0, s0]. By [3, Proposition 6.2.1], ∠(τv(0), τv(s0)) ≤ Λ · area(s0).Therefore |τ ′v(0)| ≤ Λ · area′(0). But area′(0) =

∫ dist(pi,pj)t=0 | ∂

∂sσ(0, t)|ds ≤ dist(pi, pj) ≤ 2r. To see

the first inequality, notice that the vector field ∂∂sσ is a Jacobi field along γi

q and along γjq , and

that both Jacobi fields have norms which vary monotonically between 0 and |X| = 1. Therefore,|τ ′v(0)| ≤ 2rΛ, which implies that |(gij)∗(X)| ≤ 2krΛ. In other words, each transition function gijis (2krΛ)-Lipschitz.

The isomorphism class of the vector bundle Eπ→ B is determined by the homotopy type of

its classifying map f : B → Gk(RN ), where Gk(RN ) denotes the Grassmannian of k-dimensionalsubspaces of RN , and N = k · l. The map f can be described explicitly in terms of our chosenatlas of trivializations of the bundle as follows (see for example [19, Theorem 3.5.5]): think of

RN =∑l

1 Rk, and define

f(p) := (η1(p) · h1(v), ..., ηl(p) · hl(v)) | v ∈ π−1(p).

Our goal is to establish a Lipschitz bound for f which is linear in Λ. For this purpose, the mostuseful description of the homogeneous metric on Gk(RN ) is as follows. Fix orthonormal bases ofRN and Rk. Let Ik denote the k × k identity matrix, and let INk :=

(

Ik0

)

denote the N × k matrixwhose top k rows form Ik, and whose bottom N − k rows are zeros. An arbitrary point, V , ofGk(RN ) can be described as V = im(P · INk ) for some P ∈ O(N). An arbitrary path, γ(t), withγ(0) = V can be described as γ(t) = im(P · A(t)), where A(t) : Rk → RN is a family of linearembeddings with A(0) = INk . Then |γ′(0)|2 = trace(BT ·B), where B := A′(0)− INk · (INk )T · A′(0)(that is, B is obtained from the matrix A′(0) by replacing the top k rows with zeros). In particular,if all components aij(t) of A(t) satisfy |a′ij(0)| ≤ c, then |γ′(0)| ≤ kc.

Now fix q ∈ B1 and X ∈ TqB with |X| = 1, and let q(t) = exp(tX). Let A(t) = (aij(t)) : Rk →RN be defined as

A(t)(v) := (η1(q(t)) · v, η2(q(t)) · g12(q(t))(v), ..., ηl(q(t)) · g1l(q(t))(v)).

It is clear that f(q(t)) = im(A(t)). Choose S = (sij) : Rk → Rk such that the matrix A(0) · S is

an orthogonal map onto its image. That way, P · A(0) · S = INk for some P = (pij) ∈ O(N). SinceA(0) already maps an orthonormal basis of Rk onto an orthogonal set in RN , S can be chosen asa diagonal matrix, and it is clear that the components of S satisfy |sij | ≤ k. f(q(t)) = im(A(t)) =im(P−1P ·A(t) · S). So, by the previous discussion of the homogeneous metric on Gk(RN ):

|f∗(X)| ≤ k ·max |((P ·A · S)′(0))ij |≤ k2N(max |pij |) · (max |a′ij(0)|) · (max |sij |)= k3N(max |a′ij(0)|)≤ k3N [(max |(ηi)∗(X)|) · (max |(g1s)ij(q)|)

+(max |(ηi(q)|) · (max |((g1s)ij)∗(X)|)]≤ k3N(L · 1 + 1 · 2rΛ) = k3N(L+ 2rΛ),

where L denotes a Lipschitz bound on the functions ηi.Thus, f satisfies the Lipschitz constant L := k3N(L + 2rΛ). The homotopy type of this map

classifies the bundle, so the proof is completed by part 2 of Lemma 7.2.1.

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Example 7.2.2. There is a countable collection of rank 2 vector bundles over S2, which can bedescribed as Mk := (S3 × R2)/S1, k ∈ Z+, where S1 acts diagonally as

(p, v)θ7→ (eiθ · p, e−kiθ · v).

Notice that M1 is isomorphic to the normal bundle of CP1 in CP2, and M2 is isomorphic to TS2.The quotient metric on Mk has nonnegative curvature by O’Neill’s formula. Denote by [p, v] theequivalence class in Mk of (p, v) ∈ S3×R2. It is easy to see that the soul Σ of Mk is [p, 0] | p ∈ S3,which is isometric to S3/S1 = S2( 12 ). The metric projection π : Mk → Σ acts as π([p, v]) = [p, 0].Let q = [p, v] ∈ Mk. The distance, r, between q and Σ in Mk is just r = |v|. By studying thedouble Riemannian submersion S3 × R2 → Mk → Σ, it is straightforward to compute that thenorm of the A tensor of π at q is |Aq| = kr√

1+k2r2=: fk(r). Therefore, for any p ∈ Σ, V ∈ νp(Σ)

and X,Y ∈ TpΣ with |X|, |Y |, |V | = 1 and X ⊥ Y , |R∇(X,Y )V | = 2|(DV A)XY | = 2f ′k(0) = 2k.

In other words, |R∇| = 2k. Each Mk has totally geodesic π-fibers. By Corollary 7.1.2, the verticalcurvatures at their souls must also grow proportionally to k, as can be verified computationally.

7.3 A finiteness theorem for principal bundles

In this section we prove Theorem 7.1.3. Our proof begins with a lemma which essentially says thattwo principal G-bundles whose transition functions are sufficiently close must be isomorphic.

Lemma 7.3.1. Let G be a compact Lie group with a bi-invariant metric. Let δ := 12 inj(G).

Let B be a compact manifold. Let B1, ..., Bl be a covering of B by contractible opensets. Let z denote the maximal multiplicity of intersections (that is, no point of B lies inmore than z of the open sets). Let π0 : P 0 → B, π1 : P 1 → B be two principal G-bundlesover B. Let g0ij , g

1ij : Bi ∩ Bj → G denote the transition functions associated to atlases of

trivializations of the two bundles over the sets Bi. If for all pairs (i, j) and for all p ∈ Bi∩Bj,dG(g

0ij(p), g

1ij(p)) ≤ ε := δ

2z−1 , then the two bundles are isomorphic.

Our proof will use only the following properties of (G,dG,δ):

1. dG((a′)−1 · b′, a−1 · b) ≤ dG(a

′, a) + dG(b′, b) for all a, a′, b, b′ ∈ G.

2. For any two points a, b ∈ G with dG(a, b) ≤ δ, there is a canonical path γab : [0, 1] → Gfrom a to b (namely the minimal path) which varies continuously with a and b and for whichdG(γab(t), a), dG(γab(t), b) ≤ dG(a, b).

Proof. We begin by reviewing Milnor’s construction of the classifying space for a principal G-bundle(see [19, Section 4.11]). Define:

EG(l) := (t1g1, ..., tlgl) | gi ∈ G, ti ∈ [0, 1],∑

ti = 1/ ∼,

where two vectors, (t1g1, ..., tlgl) and (t1g1, ..., tlgl), are considered equivalent iff for each i, either1) gi = gi and ti = ti or 2) ti = ti = 0. Then define BG(l) := EG(l)/G, where G acts onEG(l) component-wise on the left. Suppose π : P → B is a principal G-bundle over B, andΦi : Bi × G → π−1(Bi) is an atlas of trivializations for π over the open covering B1, ..., Bl ofB. Let η1, ..., ηl denote a partition of unity subordinate to the open cover B1, ..., Bl. Definehi : π−1(Bi) → G as hi := pi Φ−1

i , where pi : Bi × G → G denotes the projection onto thesecond factor. Let gij : Bi ∩Bj → G denote the transition functions associated to this collection oftrivializations.

Define f : P → EG(l) as follows:

f(z) := (η1(π(z)) · h1(z), ..., ηl(π(z)) · hl(z)).

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f maps fibers to fibers, so it induces a well-defined map f : B → BG(l). f is called the classifyingmap of the bundle. It is useful to describe f in terms only of the transition functions. For example,f restricted to B1 can be described as follows:

f(p) = [η1(p) · e, η2(p) · g12(p), ..., ηl(p) · g1l(p)],

where e denotes the identity element of G. Of course, f satisfies the analogous description whenrestricted to the other balls B2, ..., Bl.

Now let π0 : P 0 → B, π1 : P 1 → B be two principal G-bundles over B. Let g0ij , g1ij : Bi∩Bj → G

denote the transition functions associated to atlases of trivializations of the two bundles over thesame atlas B1, ..., Bl of B. Let f0, f1 : B → BG(l) denote their classifying maps, constructed asabove using the same partition of unity ηi. Assume that for all pairs (i, j) and for all p ∈ Bi∩Bj ,dG(g

0ij(p), g

1ij(p)) ≤ ε. In order to define a homotopy, f t, between f0 and f1, first define for every

pair (i, j) and for every q ∈ Bi ∩Bj , gtij(q) (t ∈ [0, 1]) to be the canonical path in G between g0ij(q)

and g1ij(q). Next define f t restricted to B1 as:

f t(p) = [η1(p) · e, η2(p) · gt12(p), ..., ηl(p) · gt1l(p)]. (7.3.1)

It is tempting to define f t restricted to the other balls B2, ..., Bl analogously; however, this definitionwould only be well defined on the overlaps when t = 0 and t = 1. So instead we must extend f t oneopen set at a time. For example, extend f t to B1∪B2 as follows. First notice that for p ∈ B1∩B2,the following description of f t is equivalent to equation 7.3.7.1:

f t(p) = [η1(p) · (gt12(p))−1, η2(p) · e, ..., ηl(p) · (gt12(p))−1 · gt1l(p)]. (7.3.2)

The formula has just been rewritten so that e appears in the second component, which makes iteasier to compare to the natural definition of f t restricted to B2:

f t(p) = [η1(p) · gt21(p), η2(p) · e, ..., ηl(p) · gt2l(p)]. (7.3.3)

At time t = 0 and t = 1, equations 7.3.7.2 and 7.3.7.3 agree on B1 ∩B2. For t ∈ (0, 1), we redefinef t on B1 ∩B2 as the weighted average of equations 7.3.7.2 and 7.3.7.3. More precisely, for each ofthe l components in these two expressions, the coefficients are equal and the elements of G havedistance ≤ 3ε ≤ δ in G. To see this, consider, for example, the group elements in the lth component:

dG((gt12(p))

−1 · gt1l(p), gt2l(p)) (7.3.4)

≤ dG((gt12(p))

−1 · gt1l(p), g02l(p)) + dG(g02l(p), g

t2l(p))

≤ dG((g012(p), g

t12(p)) + dG(g

01l(p), g

t1l(p)) + dG(g

02l(p), g

t2l(p))

≤ ε+ ε+ ε = 3ε ≤ δ.

So it makes sense to take a weighted average of the two elements of G, using η1η1+η2

and η2

η1+η2as

the weights (by the weighted average of two nearby points a, b ∈ G we mean the the point γab(t0),where t0 is determined in the obvious way by the weights). To make the weights well defined,choose a partition of unity such that ηi(p) > 0 for all p in the interior of Bi.

Finally, extend the definition of f t to the other balls B3, ..., Bl one at a time in exactly thesame way. That is, if f t is already defined on U = B1 ∪ · · · ∪ Bm−1, then describe a component-wise formula for f t on U ∩ Bm with e in the mth position. Then re-define f t on U ∩ Bm as thecomponent-wise weighted average of this formula and the natural definition for f t on Bm. Useη1+···+ηm−1

η1+···+ηmand ηm

η1+···+ηmas the weights. The pairs of elements of G to be averaged will always

agree when t = 0 and t = 1; further, by repeating the argument of equation 7.3.7.4 inductively, itis easy to see that they will have distance ≤ (2z − 1)ε = δ in G for t ∈ (0, 1). The homotopy f t isclearly well-defined and continuous. Since the classifying maps of the two bundles are homotopic,the bundles are isomorphic. This proves the lemma.

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Proof of Theorem 7.1.3. Let Bn be a compact Riemannian manifold, G a compact Lie group withbi-invariant metric, and π : P → B a principal G-bundle with a connection, ω, whose associatedcurvature form, Ω, satisfies |Ω| ≤ Λ. We prove that the number of candidates for the isomorphism

class of this bundle is ≤ ec(1+eCΛ), where c, C depend only on B and G. By the argument whichbegan our proof of Theorem 7.1.1, this “fixed base space” version implies the more general versionof Theorem 7.1.3.

There is a unique Riemannian metric on P for which π is a Riemannian submersion whosehorizontal distribution is defined by ω, and each fiber Gp := π−1(p) is totally geodesic and isometricto G (with the given bi-invariant metric). Since the fibers are totally geodesic, T = 0. Further, theA tensor of π agrees with the curvature form, Ω, so |A| ≤ Λ.

As before, choose r < 12 mininj(B), π√

λ, where λ denotes an upper curvature bound for B,

and choose points p1, ..., pl ⊂ B such that the balls Bi := Bpi(r) cover B. Choose isometriessi : G → Gpi . The natural trivializations, Φi : Bi × G → π−1(Bi), are defined as Φi(q, v) :=

hγiq (si(v)). Remember that γi

q denotes the minimal path from pi to q, and hγiq : Gpi → Gq

denotes the diffeomorphism between the fibers which is naturally associated to this path. Letgij : Bij = Bi ∩Bj → G denote the transition functions associated to this atlas of trivializations ofthe bundle.

We require a new argument to establish Lipschitz control on the transition functions. As before,let q ∈ Bij , X ∈ TqB with |X| = 1, and q(s) = exp(sX). As before, let t → σs(t) = σ(s, t) bea minimal unit-speed path between pi and q(s) followed by a minimal unit-speed path betweenq(s) and pj . Let τ(s) := gij(q(s)). Notice that τ(s) = (s−1

j hσs si)(e), where e denotes theidentity element of G. Denote k := dim(G). Since C1 := length(σ0) ≤ 2r, and since C2 :=supt∈[0,d(pi,pj)] |

∂∂sσ(0, t)| ≤ 1 as before, Lemma 6.1.3 and Equation 6.1.6.2 provides the following

bound:|τ ′(0)| = |(sj τ)′(0)| ≤ 2e2rk4

k(k+1)!Λ =: L.

This value L provides a Lipschitz bound for the transition functions, gij .The most obvious strategy for completing the proof is the following: 1) describe a natural metric

on the classifying space BG(l), 2) establish a Lipschitz bound for the classifying map f : B → BG(l)depending on Lipschitz bounds for the transition functions and for the partition of unity functions,3) verify that BG(l) satisfies the regularity hypothesis of part 1 of Lemma 7.2.1, and apply thislemma. We believe that this approach works. But in the case where Λ = 0 (the flat bundle case),the bound obtained from this approach for the number of isomorphism class possibilities of thebundle would depend on the metrics of B and G. Therefore, we use a different approach in orderto obtain bounds in the flat bundle case which depend only on the topology of B and the Lie groupstructure of G. Rather than applying Gromov’s lemma, we must use a variation on its proof.

The variation of Gromov’s argument goes as follows. Let δ := 12 inj(G). LetX denote the disjoint

union of all of the nonempty intersections Bij . Let N denote the number of components of X, andlet z denote the maximal multiplicity of intersections for the open cover Bi of B. Let ε := δ

2z−1 .

Let RG denote a ε4 -net in G, and let RX denote a ε

4L -net in X (remember L = 2e2rk4k(k+1)!Λ

is a Lipschitz bound on the transition functions). As argued in our proof of Lemma 7.2.1, the

net RX can be chosen such that card(RX) ≤ K ·(

(

4Lε

)n+ 1

)

, where K = K(X). Think of

the collection gij of transition functions as defining a single L-Lipschitz map g : X → G. ginduces a (non-unique) map g : RX → RG, defined so that g(p) is any point of RG whose distancefrom g(p) is ≤ ε

4 . It is easy to see that if g0, g1 : X → G are any two L-Lipschitz maps, and

g0 = g1, then dsup(g0, g1) ≤ ε. If g0 and g1 happen to be the transition functions associated to two

different bundles, π0 : P 0 → B and π1 : P 1 → B, then Lemma 7.3.1 implies that the two bundlesmust be isomorphic. Thus, the number of candidates for the isomorphism class of our bundle is≤ card(RG)

card(RX) ≤ card(RG)K(( 4L

ε )n+1). Substituting the above value of L and simplifyingcompletes the proof.

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To conclude this section, discuss the case Λ = 0 in the previous proof in order to prove Theo-rem 7.1.4

Proof of Theorem 7.1.4. When Λ = 0, the transition functions satisfy the Lipschitz bound L = 0,so we can use a net RX in X containing exactly one point from each component of X. In thiscase, there is no reason to use a coverings of B by small metric balls; any covering by opensets diffeomorphic to Rn would suffice. For example, if we use a minimal such covering, thencard(RX) ≤ I(B)2. It is easy to choose a minimal open covering for which the maximummultiplicityof intersections is z = n + 1. For convenience, we can work with the bi-invariant metric on Gassociated to the kiling form. RG is defined in the above proof as an r-net in G, where r = ε/4 =(

12 inj(G)2z−1

)

/4 = inj(G)2n+4−8 . By definition, card(RG) = P (G,n). This establishes Theorem 7.1.4.

We end this section by proving Corollary 7.1.5 as a consequence of Theorem 7.1.4.

Proof of Corollary 7.1.5. Suppose that M is an open manifold of nonnegative curvature whosesoul, Σ, is homeomorphic to a Bieberbach manifold. By [6, Corollary 9.5], any metric of non-negative curvature on a Bieberbach manifold is flat, so Σ is flat. It then follows from O’Neill’sformula (2.4.2.1) that the A-tensor of the metric projection π : M → Σ vanishes, which impliesby Lemma 3.1.1 that the connection on the normal bundle of Σ in M is flat. This flat connectioninduces a flat connection in the principal O(k)-bundle associated to the normal bundle of the soul.Theorem 7.1.4 bounds the number of possibilities for the isomorphism type of the principal bundle,and hence also of the normal bundle of the soul.

We end this section by discussing whether Theorem 7.1.3 is still true when G is only a non-compact Lie groups with a left-invariant metric. In this case, we can still prove a finiteness theoremas long as we add the assumption that the base space is simply connected:

Theorem 7.3.2. Let G be a non-compact Lie group with a left-invariant metric, B a compactsimply connected Riemannian manifold, and Λ a constant. Then there exist only finitelymany isomorphism classes of principal G-bundles G → P → B which admit a connection,ω, whose curvature form, Ω, satisfies |Ω| ≤ Λ.

Proof. The compactness assumption for G was used in the proof of Theorem 7.1.3 to insure thatthere exists a finite ε

4 -net RG in G. When G is non-compact, we use the assumption that B issimply connected to insure that the images of the transition functions all lie in a fixed compactsubset G′ of G; thus, we can complete the proof by settling for a finite ε

4 -net in G′ rather than inG.

More precisely, with the setup from the proof of Theorem 7.1.3, we show that the functions sican be contrived so that im(g) lies in the ball of radius R about e in G, where g : X → G denotesthe collection of transition functions, and R depends only on B,Λ, k. The construction is as follows:choose s1 arbitrarily, and for each i = 2, ..., l, define si = hσi s1, where σi is any minimal path in Bfrom p1 to pi. This construction insures that for any pair (i, j), any q ∈ Bij , s1(gij(q)) = hα(s1(e)),where α denotes the path α := σi ∗ γi

q ∗ (γjq)

−1 ∗ (σj)−1. Therefore,

dG(e, gij(q)) = dGp1(s1(e), s1(gij(q)))

≤ C1 · dP (s1(e), s1(gij(q)))≤ C1 · length(α)≤ C1(2r + 2 · diam(B)) =: R,

Where C1 = C1(B,Λ, k) is the constant in Lemma 6.1.1. It follows that im(g) ⊂ G′ := Be(R).

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A second difficulty is that Lemma 7.3.1 is not necessarily valid when G is a noncompact Liegroup with a left-invariant metric. The problem is that, even though any two points of G whosedistance is ≤ δ = 1

2 inj(G) do have a canonical path between them, it is not necessarily true that:

dG((a′)−1 · b′, a−1 · b) ≤ dG(a

′, a) + dG(b′, b) for all a, a′, b, b′ ∈ G.

However, the following is true: There exists C := C(R) such that:

dG((a′)−1 · b′, a−1 · b) ≤ C(dG(a

′, a) + dG(b′, b)) for all a, a′, b, b′ ∈ G′.

To see this inequality, let γ1(t) be a minimal path from a to a′ and γ2(t) a minimal path from bto b′, t ∈ [0, 1]. Let γ := γ−1

1 · γ2, which is a path from a−1 · b to (a′)−1 · b′. By the product rule,|γ′(t)| ≤ |γ′

2(t)| + C|γ′1(t)|, where C := |dRγ1(t)−1 | · |dRγ2(t)|. This weaker statement suffices to

complete the proof.

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Bibliography

[1] I. Belgradek, Counting nonnegatively curved manifolds, Ph.D. Thesis, University of Mary-land, 1998.

[2] A. Besse, Einstein manifolds, Springer-Verlag, 1987.

[3] P. Buser and H. Karcher, Gromov’s almost flat manifolds, Societe Mathematique de France,Asterisque 81, 1981.

[4] Y. Burago, M. Gromov and G. Perelman, A.D. Alexandrov spaces with curvature boundedbelow, Russian Math. Surveys 47 (1992), 1-58.

[5] J. Cheeger, Some Examples of Manifolds of Nonnegative Curvature, J. Differential Geom.8 (1972), 623–628.

[6] J. Cheeger and D. Gromoll, On the structure of compete open manifolds of nonnegativecurvature, Ann. of Math. 96 (1972), 413–443.

[7] C. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. Ecole Norm.Sup. (4) 13 (1980), no. 4, 419–435.

[8] C. Croke and H. Karcher, Volumes of small balls on open manifolds: lower bounds andexamples, Trans. Amer. Math. Soc. 309 (1988), no. 2, 753–762.

[9] J.H. Eschenburg, Comparison theorems and hypersurfaces, Manuscripta Math. 59 (1987),295–323.

[10] M. Gromov, Structures metriques pour les varietes riemanniennes, Cedic-F. Nathan, 1981.

[11] M. Gromov, Carnot-Caratheodory spaces seen from within, Progress in Mathematics, Vol.144, 1996.

[12] L. Guijarro, Improving the Metric in an Open Manifold with Nonnegative Curvature,Proc. Amer. Math. Soc. 126, No. 5 (1998), 1541-1545.

[13] L. Guijarro, On the metric structure of open manifolds with nonnegative curvature,preprint.

[14] L. Guijarro, Rigidity in Open Riemannian Manifolds with Nonnegative Curvature, Doc-toral Thesis, University of Maryland, 1995.

[15] L. Guijarro and V. Kapovitch, Restrictions on the geometry at infinity of nonnegativelycurved manifolds, Duke Math. J. 78, No. 2 (1995), 257–276.

[16] L. Guijarro and P. Petersen, Rigidity in nonnegative curvature, Ann. Sci. Ecole Norm. Sup.(4) 30 (1997), No. 5, 595–603.

60

[17] L. Guijarro and G. Walschap, The metric projection onto the soul, Tans. Amer. Math. Soc.(to appear).

[18] L. Guijarro and G. Walschap, Twisting and nonnegative curvature metrics on vectorbundles over the sphere, preprint.

[19] D. Husemoller, Fiber bundles, McGraw-Hill, 1966.

[20] V. Marenich, The holonomy in open manifolds of nonnegative curvature, Michigan Math.J. 43 (1996), 263–271.

[21] S. Kobayashi and K. Nomizu, Foundations of differential geometry I, Interscience Publishers(1963), J. Wiley and Sons.

[22] S. Kronwith, Convex Manifolds of Nonnegative Curvature, J. Diff. Geom. 14 (1979), 621–628.

[23] T. Moore, Care of the soul: a guide for cultivating depth and sacredness in everydaylife, Harper Perennial, 1992.

[24] G. Perelman, Proof of the soul conjecture of Cheeger and Gromoll, J. Differential Geom.40 (1994), 209-212.

[25] S. Peters, Convergence of Riemannian manifolds, Compositio Mathematica. 62 (1987),3-16.

[26] W.A. Poor, Differential Geometric Structures, McGraw-Hill, 1981.

[27] V. Schroeder and M. Strake, Volume growth of open manifolds with nonnegative curvature,Ann. Global Anal. Geom. 8, no.2 (1990), 159–165.

[28] V. Sharafutdinov, Pogorelov-Klingenberg theorem for manifolds homeomorphic to Rn,Sibirsk. Math. Zh. 18 (1977), 915-925.

[29] K. Shiohama, An Introduction to the Geometry of Alexandrov Spaces, Math. LectureNotes. 8, Pungnam, Seoul, 1993.

[30] T. Shioya, Mass of rays in Alexandrov spaces of nonnegative curvature, Comment. Math.Helv. 69 (1994), 208-228.

[31] T. Shioya, Splitting theorem for nonnegatively curved open manifolds with large idealboundary, Math. Z. 212 (1993), 223-238.

[32] V.A. Toponogov, Riemannian spaces having their curvature bounded below by a positivenumber, Amer. Math. Soc. Transl. Serv. 37 (1964), 291-336.

[33] M. Strake, A splitting theorem for open nonnegatively curved manifolds, ManuscriptaMath. 61, 315-325.

[34] M. Strake and G. Walschap, Σ-flat manifolds and Riemannian submersions, ManuscriptaMath. 64 (1989), 213-226.

[35] M. Strake and G. Walschap, Connection metrics of nonnegative curvature on vector bun-dles, Manuscripta Math. 66 (1990), 309-318.

[36] P. Walczak, A finiteness theorem for Riemannian submersions, Ann. Polon. Math. 57(1992), no. 3, 283-290.

61

[37] P. Walczak, Erratum to the paper “A finiteness theorem for Riemannian submersionsÔ,Ann. Polon. Math. 58 (1993), no. 3, 319.

[38] G. Walschap, Σ-flat manifolds and Riemannian submersion, Manuscripta Math. 64 (1989),213-226.

[39] G. Walschap, Nonnegatively curved manifolds with souls of codimension 2, J. Diff. Geom.27 (1988), 525-537.

[40] G. Walschap, Soul-Preserving Submersions, Michigan Math. J. 41 (1994), 609-617.

[41] G. Walschap, Metric Foliations and Curvature, J. Geom. Anal. 2, no. 4 (1992), 373-381.

[42] G. Walschap and M. Ozaydin, Vector bundles with no soul, Proc. Amer. Math. Soc. 120(1994), no. 2, 565-567.

[43] J.Y. Wu, A parameterized geometric finiteness theorem, Indiana Univ. Math. J. 45, no. 2(1996), 511-528.

[44] J.W. Yim, Distance nonincreasing retraction on a complete open manifold of nonnega-tive sectional curvature, Ann. Global Anal. Geom. 6 (1988), 191-206.

[45] J.W. Yim, Space of souls in a complete open manifold of nonnegative curvature, J.Differential Geom. 32 (1990), 429–455.

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