paternain g.p. geodesic flows.pdf

Progress in MathematicsVolume 180

Series EditorsHyman BassJoseph OesterleAlan Weinstein

Gabriel P. Paternain

Geodesic Flows

BirkhauserBoston Basel Berlin

Gabriel P. PatemainCentro de MatemdticaFacultad de Ciencias11400 Montevideo. Uruguay

Library of Congress Cataloging-in-Publication DataPatemain, Gabriel P. (Gabriel Pedro), 1964-

Geodesic flows / Gabriel P. Paternain.p. cm. - (Progress in mathematics : v. 180)

Includes bibliographical references and index.ISBN 0-8176-4144-0 (alk. paper). -ISBN 3.7643-41440 (alk.

paper)1. Geodesic flows. I. Title. 11. Series: Progress in

mathematics (Boston, Mass.) : vol. 180.QA614.82.P38 1999514'.74-dc21 99-38332

CIP

AMS Subject Classifications: 58F17, 58F05, 54C70, 58FI5, 58F11, 53022

Printed on acid-free paper.1999 Birkhauser Boston Birkhi user qlh}'

All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Birkhauser Boston, do Springer-Verlag New York. Inc., 175 Fifth Avenue,New York, NY 10010, USA). except for brief excerpts in connection with reviews or scholarly analysis.Use in connection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the formerare not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marksand Merchandise Marks Act, may accordingly be used freely by anyone.

ISBN 0-8176-4144-0ISBN 3-7643-4144-0

Reformatted from author's disk by TTXniques, Inc., Cambridge, MA.Printed and bound by Hamilton Printing, Rensselaer, NY.Printed in the United States of America.

987654321

To Graciela

Contents

Preface xi

0 Introduction 1

1 Introduction to Geodesic Flows 71.1 Geodesic flow of a complete Riemannian manifold . . . . . . . . 8

1.1.1 Euler-Lagrange flows ................ .... 81.2 Symplectic and contact manifolds . . . . . . . . . . . . . . . . . 9

1.2.1 Sympiectic manifolds ................... 91.2.2 Contact manifolds ..................... 10

1.3 The geometry of the tangent bundle . . . . . . . . . . . . . . . . 111.3.1 Vertical and horizontal subbundles . . . . . . . . . . . . . 111.3.2 The symplectic structure of TM . . . . . . . . . . . . . . 141.3.3 The contact form . . . . . . . . . . . . . . . . . . . . . . 15

1.4 The cotangent bundle T'M .......... ........... 191.5 Jacobi fields and the differential

of the geodesic flow . . . . . . . . . . . . . . . . . . . . . . . . . 201.6 The asymptotic cycle and the stable norm ............. 21

1.6.1 The asymptotic cycle of an invariant measure . . . . . . . 211.6.2 The stable norm and the Schwartzman ball . . .. . . . . 26

2 The Geodesic Flow Acting on Lagrangian Subspaces 312.1 Twist properties ........................... 322.2 Riccati equations . . . . . . . . . . . . . . . . . . . . . . . . . . 37

viii Contents

2.3 The Grassmannian bundle of Lagrangian subspaces . . . . . . . . 382.4 The Maslov index . . . . . . . . . . . . . . . .. . . . . . . . . . 39

2.4.1 The Maslov class of a pair (X, E) . . . . . . . . . . . . . 412.4.2 Hyperbolic sets ....................... 422.4.3 Lagrangian submanifolds . .. . . . . . . . . . . . . . . . 43

2.5 The geodesic flow acting at the level of Lagrangian subspaces . . 442.5.1 The Maslov index of a pseudo-geodesic and recurrence . . 45

2.6 Continuous invariant Lagrangian subbundles in SM . . . . .. . . 482.7 Birkhoff's second theorem for geodesic flows . . . . . . . . . . . 50

3 Geodesic Arcs, Counting Functions and Topological Entropy 513.1 The counting functions . . . . . . .. . .. . ... . . . . . . . . 51

3.1.1 Growth of n(T) for naturally reductivehomogeneous spaces . . . . . . . . . .. . . . . . . . . . 56

3.2 Entropies and Yomdin's theorem . . . .. . . . . .. . . . .. . . 583.2.1 Topological entropy .................... 583.2.2 Yomdin's theorem . . . . .. . . . . .. . . . . . . . . . 603.2.3 Entropy of an invariant measure . . . .. .. . . . . . . . 613.2.4 Lyapunov exponents and entropy . . . . . . . . . . . . . 623.2.5 Examples of geodesic flows with positive entropy . . . . . 63

3.3 Geodesic arcs and topological entropy . . . . .. . . . . . . . . . 643.4 Manning's inequality ........................ 693.5 A uniform version of Yomdin's theorem . . . . .. . . . . . . . . 73

3.5.1 Another proof of Theorem 3.32 using Theorem 3.44 . . . 75

4 Mane's Formula for Geodesic Flows and Convex Billiards 774.1 Time shifts that avoid the vertical .. . . . .. . . . . . . . . . . . 774.2 Mane's formula for geodesic flows . . . .. . . . . . . . . . . . . 82

4.2.1 Changes of variables .................... 834.2.2 Proof of the Main Theorem ................. 88

4.3 Manifolds without conjugate points . . . . . . . . . .. . . . . . 904.4 A formula for the topological entropy for manifolds of positive

sectional curvature . . . . . . . . . . . . .. . . . . . .. . . . . 924.5 Mane's formula for convex billiards . . . . . . .. . . . . . . . . 93

4.5.1 Proof of Theorem 4.30 ................... 974.6 Further results and problems on the subject ............ 102

4.6.1 Topological pressure . . .. . .. . . . . .. . . . . . . . 104

5 Topological Entropy and Loop Space Homology 1095.1 Rationally elliptic and rationally hyperbolic manifolds . . . . . . 109

5.1.1 The characteristic zero homology of H-spaces . . . . . . . 1125.1.2 The radius of convergence .. . . . . . . . . . . . . .. . 114

5.2 Morse theory of the loop space . . . . . . . . . . . . . . . . . . . 1155.2.1 Serre's theorem . . . . . . . . . . . . . . . . . . . . . . . 1165.2.2 Gromov's theorem . . . . . . . . . . . . . . . . . . . . . 117

Contents ix

5.3 Topological conditions that ensure positive entropy ........ 1195.3.1 Growth of finitely generated groups . . . .. . . . . . . . 1195.3.2 Dinaburg's Theorem . . .. . . . . . . . . . . . . . . . . 1205.3.3 Arbitrary fundamental group . . . . . . . . . . . . . . . . 1215.3.4 Proof of Theorem 5.20 ................... 122

5.4 Entropies of manifolds ....................... 1265.4.1 Simplicial volume ..................... 1265.4.2 Minimal volume . . . . .. . . . . . . . . . .. . . . . . 127

5.5 Further results and problems on the subject . . . . . . . . . . . . 130

Hints and Answers 133

References 139

Index 147

Preface

The aim of this book is to present the fundamental concepts and properties ofthe geodesic flow of a closed Riemannian manifold. The topics covered are closeto my research interests. An important goal here is to describe properties of thegeodesic flow which do not require curvature assumptions. A typical example ofsuch a property and a central result in this work is Mafid's formula that relates thetopological entropy of the geodesic flow with the exponential growth rate of theaverage numbers of geodesic arcs between two points in the manifold.

The material here can be reasonably covered in a one-semester course. I havein mind an audience with prior exposure to the fundamentals of Riemanniangeometry and dynamical systems.

I am very grateful for the assistance and criticism of several people in preparingthe text. In particular, I wish to thank Leonardo Macarini and Nelson Moller whohelped me with the writing of the first two chapters and the figures. GonzaloTornaria caught several errors and contributed with helpful suggestions. PabloSpallan7ani wrote solutions to several of the exercises. I have used his solutionsto write many of the hints and answers. I also wish to thank the referee for avery careful reading of the manuscript and for a large number of comments withcorrections and suggestions for improvement.

This book grew out of lectures which I gave at Bah(a Blanca and Cordoba(Argentina) in 1996, IMPA (Rio de Janeiro, Brasil) in 1997 and Montevideo(Uruguay) in 1998. Part of the text was written while I was visiting IMPA duringthe second semester of 1997 and the ICTP in Trieste during the first two monthsof 1998. 1 wish to thank them for their hospitality.

xii Preface

Finally, my thanks go to Professor Alan Weinstein for his kind interest in themanuscript and to Ann Kostant and Tom Grasso at Birkhauser for their helpthroughout the various stages of publication.

Gabriel Pedro PaternainMontevideo, March 1999

Geodesic Flows

0Introduction

Let M be a closed connected manifold and g a Riemannian metric on M. Lety(x,v)(t) be the unique geodesic with the following initial conditions:

J Y(x.u)(0) = x;

l Y(x,v)(0) = v.

For a given t e IR, we define a diffeomorphism of the tangent bundle TM

of:TM -p TM,as follows

O,(x, v) := (Y(x.v)(t), Y(x,v)(t)) .The family of diffeomorphisms Of is in fact aglow, that is, it satisfies of+s = OroosThis last property is an easy consequence of the uniqueness of the geodesic withrespect to the initial conditions. Let SM be the unit tangent bundle of M, thatis, the subset of TM given by those pairs (x, v) such that v has norm one. Sincegeodesics travel with constant speed we see that Of leaves SM invariant, that is,given (x, v) a SM, for all t E IR we have that , (x, v) a SM. The restriction ofOf to SM is called the geodesic flow of g.

Chapter 1 describes the basic properties of the geodesic flow. We begin by re-calling that geodesics can be obtained as solutions of the Euler-Lagrange equationof a Lagrangian given by the kinetic energy. We define symplectic and contactmanifolds and we set up the basic geometry of the tangent bundle: we introducethe connection map, horizontal and vertical subbundles, the Sasaki metric, the

2 0. Introduction

symplectic form and the contact form. We describe the main properties of theseobjects and we show that the geodesic flow is a Hamiltonian flow. Also, when werestrict the geodesic flow to the unit sphere bundle of the manifold, we obtain acontact flow. The contact form naturally induces a probability measure that is in-variant under the geodesic flow and is called the Liouville measure. Next we writethe differential of the geodesic flow in terms of Jacobi fields. In the last sectionof the chapter we define the asymptotic cycle of an invariant probability measureand the stable norm. We show that the asymptotic cycle of the Liouville measurevanishes and that the same holds for the measure of maximal entropy if the latteris unique. Finally we show that the unit ball of the stable norm coincides with theset of asymptotic cycles of all invariant probability measures.

Chapter 2 describes how the geodesic flow acts on Lagrangian subspaces. Weintroduce Lagrangian subspaces and Lagrangian submanifolds and we show animportant property of the vertical subbundle that we call the twist property of thevertical subbundle. This property reflects the fact that the geodesic flow arisesfrom a second order differential equation on TM. Next we derive the Riccatiequations, after which we introduce the Grassmannian bundle of Lagrangian sub-spaces and we show how to attach an index, the Maslov index, to every closedcurve of Lagrangian subspaces. The definition we use of the Maslov index fol-lows Mane in [Mall and it is particularly adapted to the Riccati equations. Thisallows us to show that the lift of the geodesic flow to the Grassmannian bundle ofLagrangian subspaces is transverse to the Maslov cycle. This important propertyreflects the convexity of the unit spheres in tangent spaces.

Using these tools we show two results, one motivated by hyperbolic sets andthe other by KAM tori. We show a theorem of Mane [Mal] that states that ifthere exists a continuous invariant Lagrangian subbundle E defined on SM, thenE is transverse to the vertical subbundle and M does not have conjugate points.When the geodesic flow of M is Anosov, the stable and unstable bundles are con-tinuous invariant Lagrangian subbundles, hence we deduce that if M is a closedRiemannian manifold whose geodesic flow is Anosov, then M does not have con-jugate points. This last result was first proved by W. Klingenberg using differenttechniques [Kll].

Finally we show the following result due to L. Polterovich [Poll] (see also[BP]) which can be seen as the higher dimensional autonomous version of a re-sult of G. Birkhoff that asserts that an invariant circle of a twist map in the cylin-der T*SI which is homologous to the zero section must be a graph. Consider aRiemannian metric on the n-torus T. Suppose that P is a Lagrangian torus con-tained in S1" which is homologous to the zero section of T. Suppose that theset of non-wandering points of the geodesic flow restricted to P coincides withall P. Then P is a graph, that is, the restriction to P of the natural projectionn : ST" -> it is a diffeomorphism.

In Chapter 3 we introduce the counting functions and we relate them to thetopological entropy hrop(g) of the geodesic flow of g. Given x and y in M andT > 0, define nT(x, y) as the number of geodesic arcs joining x and y withlength < T. Already in 1962 M. Berger and R. Bott [BB] observed that there are

0. Introduction 3

significant relationships between integrals of this function and Jacobi fields. Weshow that, for each T > 0, the counting function nT(x, y) is finite and locallyconstant on an open full measure subset of M x M, and integrable on M x M.More generally, if N is a compact smooth submanifold of M, define nT(N, y) tobe the number of geodesic arcs with length < T that join a point in N to y andare initially orthogonal to N. The function nT(N, y) enjoys properties similar tothose of nT(x, y). If g is CO, it was shown in [PP] that Yomdin's theorem [Y]can be used to prove that

l o glim sup< hrop(g)T-.oo T

It was also shown in [PP] that when N is the diagonal in the product manifoldM x M, the above inequality reduces to

IimsupIlogJ nT(x, y)dxdy (2)T-+,c T M Vol B(z, T), (6)

with equality if M has no conjugate points. Manning showed that, for any x E M,T-IVol B(z, T) converges to a Iimit A that is independent of x. From (3) and(6), it is easy to obtain the inequality h,op(g) > A for any Riemannian manifold,which was first proved by Manning in [Manl]. All this material is covered inChapter 3.

Maffe gives in [Ma2] another proof of (2) based on a uniform version ofYomdin's theorem. We explain this at the end of Chapter 3. This uniform version

4 0. Introduction

of Yomdin's theorem is useful since it can be applied to other situations, as for in-stance, convex billiards. Chapter 3 also contains a section with a brief descriptionof the concepts of topological entropy, entropy of an invariant probability measureand their relationship.

Chapter 4 presents a proof of Maid's formula for geodesic flows and convexbilliards. Maid shows in [Ma2]rthat for any C' Riemannian metric (r > 3),

lim inf1 log J nT(x, y) dxdy ? hrop(g) (7)T-oo T MxMMaid thereby obtains the first purely Riemannian characterization of hrop(g)

for an arbitrary CO Riemannian metric. Combining (2) and (7) gives Ma>{e'sformula

lim 1 log f nT(x, y) dxdy = h,op(g). (8)T-*oo T MxM

For metrics with no conjugate points we shall prove that

lim 1 lognT(x, y) = h,op(g) = A for all x, y e M.T-.oo T

The equality h,op(g) =1l if M has no conjugate points was first proved by Freireand Maid in [FM]. Chapter 4 also contains a formula for h,op(g) in terms ofthe horizontal subbundle when M has positive sectional curvature and a proof ofMaid's formula for convex billiards. The chapter concludes with a section aboutother related results and problems on the subject.

Besides the natural appeal of a formula like (8), there are other reasons to beinterested in relations between the topological entropy of the geodesic flow andthe growth rate of the average number of geodesic arcs between two points on themanifold. The function nT(x, y) also counts the number of critical points of theenergy functional on the path space c2T (x, y) given by all the curves joining xand y with length < T. Therefore using Morse theory, nT(x, y) can be boundedfrom below by the sum of the Betti numbers of S2T (x, y) (provided of course thatx and y are not conjugate). By averaging over M, one can obtain in this fashionremarkable relations between the topology of M and h,op(g). This is the contentof Chapter 5.

The chapter begins with the definitions of rationally elliptic and rationallyhyperbolic manifolds and with a summary of various properties and characteriza-tions of rationally elliptic manifolds. A manifold X is said to be rationally ellipticif the total rational homotopy,r.(X) Q is finite dimensional, i.e., there exists apositive integer io such that for all i > io, r (X) Q = 0. The manifold X is saidto be rationally hyperbolic if it is not rationally elliptic (see [FHT, FHT2, GHI]and references therein and the first section of Chapter 5). Afterwards, we discussresults of J.P. Serre [Sel] and M. Gromov [Grl] which allows us to relate thegrowth of nT (x, y) with the topology of M via Morse theory. Using these ideaswe show the result in [P3] that says that if M is a closed manifold that fibres over

0. Introduction 5

a closed simply connected rationally hyperbolic manifold, then for any C Rie-mannian metric g on M, ht p(g) > 0. The more classical result of E. Dinaburg[D] which says that if iri (M) grows exponentially, then 0 for any g, isalso discussed here. The chapter concludes with various definitions of entropiesof manifolds. We connect them with other notions like Gromov's minimal volumeand simplicial volume and we propose various related problems.

There are all too many other topics which we have not mentioned here. Mostnotably, the notes cover almost no material on the vast theory of geodesic flows onmanifolds of nonpositive curvature, particularly the rigidity theory of negativelycurved manifolds. Fortunately, there are various lectures notes and surveys on thistopic (e.g., [Ba, BGS, BCG2, El, E2, EHS]).

1Introduction to Geodesic Flows

Our aim in this chapter is to introduce the geodesic flow on the tangent bundleof a complete Riemannian manifold from several points of view. Geodesic flowshave the remarkable property of being at the intersection of various branches inmathematics; this gives them a rich structure and makes them an exciting subjectof research with a long tradition.

In Section 1.1 we define the geodesic flow of a complete Riemannian manifold.We also recall that geodesics can be obtained as solutions of the Euler-Lagrangeequation of a Lagrangian given by the kinetic energy. In Section 1.2 we definesymplectic and contact manifolds. In Section 1.3 we set up the basic geometryof the tangent bundle: we introduce the connection map, horizontal and verticalsubbundles, the Sasaki metric, the symplectic form and the contact form. We de-scribe the main properties of these objects, and we show that the geodesic flowis a Hamiltonian flow and that when we restrict it to the unit sphere bundle ofthe manifold we then obtain a contact flow. The contact form naturally inducesa probability measure that is invariant under the geodesic flow and is called theLiouville measure. In Section 1.4 we describe the canonical symplectic form ofthe cotangent bundle and, using the musical isomorphisms, we shall describe itsrelations with the symplectic form defined in Section 1.3. In Section 1.5 we writethe differential of the geodesic flow in terms of Jacobi fields. In the last sectionwe define the asymptotic cycle of an invariant probability measure and the stablenorm. We show that the asymptotic cycle of the Liouville measure vanishes andthat the same holds for the measure of maximal entropy if the latter is unique.Finally we show that the unit ball of the stable norm coincides with the set ofasymptotic cycles of all invariant probability measures.

8 1. Introduction to Geodesic Flows

Chapter I of Besse's book [Be], Chapter 3 of Klingenberg's book [Kl I], Chap-ter II of Sakai's book [Sal and Chapter IV in Ballmann's lecture notes [Ba] alsocontain introductions to the geometry of the tangent and unit tangent bundles aswell as some basic facts about geodesic flows.

1.1 Geodesic flow of a complete Riemannian manifoldLet M be a complete Riemannian manifold and let y(x.,)(t) be the unique geodesicwith initial conditions as follows:

Y(x.u)(0) = x;1 Y(x.u)(0) = V.

Definition I.I. For a given t E R, we define a diffeomorphism of the tangentbundle TM

0,: TM -TM,as follows

Of (X, v) := (Y(x.u)(t), Y(x.u)(t))The family of diffeomorphisms 0, is in fact a flow, that is, it satisfies 0,+s =

Ot.os. This last property is an easy consequence of the uniqueness of the geodesicwith respect to the initial conditions. Let SM be the unit tangent bundle of M,that is, the subset of TM given by those pairs (x, v) such that v has norm one.Since geodesics travel with constant speed, we see that Ot leaves SM invariant,that is, given (x, v) E SM, for all t E R we have that /t (x, v) E SM.

1.1.1 Euler-Lagrange flowsLet L : TM -+ R be a smooth function and let Stxy be the space

Stay :_ (u : [0, 1] -+ M, piecewise differentiable and u(0) = x, u(1) = y}.The action A of L over a path from x to y is the map,

: s R, A(u) := L(u(t), ti(t))dt.fALet us try to find the critical points or extremals of A. Consider a variation

s 1+ us E Stxy with s E (-e, e) such that ULA(s)Is=o = 0. If we set W(t)a (t)is=o, then a computation in local coordinates shows that u is a critical pointif and only if

1 8f0 1 (u, u) - Id L (u, u)I (W)dt = 0.7x TV

1.2 Symplectic and contact manifolds 9

If we assume that this equation is satisfied for all variational vector fields W(t)arising from variations with u = up, we have

a(u,u)-d aU(u.u)=0.This is known as the Euler-Lagrange equation.

There is a class of Lagrangians that has received lots of attention in recentyears. We shall say that a Lagrangian L is convex and superlinear if the followingtwo properties are satisfied.

1. Convexity. We require that LITM : TM -+ R has positive definite Hes-sian for all x E M. This condition is usually known as Legendre's condition.In local coordinates this means that

a2Lis positive definite.

auiau;

2. Superlinearity. There exists a Riemannian metric such that

L(x, u)lim = +oo,lvl-+. lviuniformly on x.

If M is compact, the extremals of A give rise to a complete flow Or : TM --TM called the Euler-Lagrange flow of the Lagrangian. A very interesting aspectof the dynamics of the Euler-Lagrange flows is given by those orbits or invariantmeasures that satisfy some global variational properties. Research on the dynam-ics of these special orbits and measures goes back to M. Morse and G.A. Hedlundand has reappeared in recent years in the work of J. Mather, trying to generalizeto higher dimensions the theory of twist maps on the annulus. See [Mat, Fa] andreferences therein for an account of this theory.

It is well-known that geodesics can be seen as the solutions of the Euler-Lagrange equation of the following convex and superlinear Lagrangian:

IL(x, v) := 2-g' (V, v)

where g denotes the Riemannian metric of M.

1.2 Symplectic and contact manifolds

1.2.1 Symplectic manifoldsDefinition 1.2. A 2-form w is said to be symplectic if w is:


closed, dw = 0;

nondegenerate, that is, if wp(X, Y) = 0 for all Y E TpM then X = 0.A pair (M, w) of a smooth manifold and a symplectic form is called a symplec-

tic manifold.

Remark 1.3. The existence of a symplectic form in a manifold M implies thatM is even dimensional.

Definition 1.4. Let (M, w) be a symplectic manifold and H : M -- R a givenC' function. The vector field XH determined by the condition

w (X H , Y) = d H (Y) or equivalently ixx w = d H

is called the Hamiltonian vector field associated with H or the symplectic gradientof H. The flow apt of the vector field X H is called the Hamiltonian flow of H.

The nondegeneracy of w ensures that XH exists and that it is a C'-1 vectorfield. In the next lemma we shall see that the Hamiltonian flow of H preserves thesympletic form w. Let us denote by Lx,, w the Lie derivative of w with respect toXH.

Lemma1.5.

Proof. Using Cartan's formula

Lxyw = dixxw +

and the fact that d w = 0 and d i x w = ddH = 0, we get LX, w = 0.

Exercise 1.6. Show that Lxs w = 0 if and only if for all t e R, v, *w = w, whererp, is the flow of XH.

1.2.2 Contact manifoldsDefinition 1.7. A 1-form a on a (2n - 1)-dimensional orientable manifold Mis called a contact form if the (2n - 1)-form a n (da)"-t never vanishes. Apair (M, a) of a smooth odd-dimensional manifold and a contact form is called acontact manifold. A contact flow is a flow on M that preserves the contact formon M.

Unlike the symplectic manifolds, which admit a variety of Hamiltonian vectorfields, a contact manifold comes with a canonical vector field X which is definedby the conditions ixa = 1 and ixda = 0. The first condition says that X pointsalong the unique null direction of the form dot and the second condition normal-izes X (see for example [MS]).

1.3 The geometry of the tangent bundle I 1

Exercise 1.8. Show that X is unique. Show that a 1-form a on a (2n - 1)-dimensional orientable manifold M is a contact form if for all x E M the re-striction of dax to the kernel of a at x is nondegenerate.

Definition 1.9. The vector field X is called the characteristic vector field (alsocalled the Reeb vector field) and its flow is called the characteristic flow. Notethat this flow preserves a since Lxa = 0 by the definition of X.

In the next section we shall describe the basic geometry of TM and SM andwe shall prove that the geodesic flow is a Hamiltonian flow and a characteristicflow in TM and SM respectively.

1.3 The geometry of the tangent bundle

1.3.1 Vertical and horizontal subbundlesLet n : TM -+ M be the canonical projection, i.e., if 0 = (x, v) E TM thena(9) = X.Definition 1.10. There exists a canonical subbundle of TTM called the verticalsubbundle whose fiber at 0 is given by the tangent vectors of curves a : (-e, e) ->TM of the form: a(t) = (x, v + tw), where 9 = (x, v) E TM and W E TxM. Inother words,

V(O) = ker(dotr).Geometrically, V(9) is the tangent space to the fiber TM C TM at the point

0 (see Figure 1.1).Remark 1.11. Note that there is no canonical complementary "horizontal" sub-bundle. In fact to construct such a subbundle, we shall use a Riemannian metricon M.

Definition 1.12. Suppose that we endow M with a Riemannian metric. We shalldefine the connection map

K: TTM -+TM,as follows. Let $ E TOTM and z : (-e, e) -+ TM be a curve adapted to l:, thatis, with initial conditions as follows:

Z(O) = 8;

i(0) = t.Such a curve gives rise to a curve a : (-e, e) -+ M, a := troz, and a vector fieldZ along a, equivalently, z(t) = (a(t), Z(t)).

Define

KB (D) := (DaZ) (0).


TM v(e)

e

M 1t

den

x

Figure 1.1: The vertical subbundle

The horizontal subbundle is the subbundle of TTM whose fibre at 9 is given byH(9) := ker KB.Lemma 1.13. K9 has the following properties:

1. KB is well defined;

2. KB is linear.

Exercise 1.14. Prove the lemma.

Another equivalent way of constructing the horizontal subbundle is by meansof the horizontal lift

L9: TxM -+TBTM,

which is defined as follows (9 = (x, v)). Given V' E TM and a : (-e, e) - Man adapted curve to v', let Z(t) be the parallel transport of v along a. Let a(-e, e) -+ TM be the curve a(t) = (a(t), Z(t)). Then

LB(v') := 0(0) E TBTM.

It is immediate from the definition of parallel transport that Ke(L9(v')) = 0,for all v' E T, M.

1.3 The geometry of the tangent bundle 13

Lemma 1.15. Lo has the following properties:

1. LB is well defined;

2. LB is linear;

3. ker(KB) = im(Lo);4. do,r,Le = IdT,M;

5. The maps d1trlH(B) : H(9) - TxM and KOIV(B) : V(9) - TxM arelinear isomorphisms.

Exercise 1.16. Prove the lemma.

From the lemma we conclude that

TBTM = H(9) V(B),and that the map je : TBTM - TxM x TxM given by

je(t) = (den(4), K9(4)),is a linear isomorphism.

Figure 1.2 on the next page summarizes in a diagram the information given byLemma 1.15.

From now on, whenever we write _ (l h, i;,) we mean that we identify $ withje(s), where h = den(y) and v = KO (y).Definition 1.17. Using the decomposition TO TM = H(6) V(9), we can de-fine in a natural way a Riemannian metric on TM that makes H(9) and V(8)orthogonal. This metric is called the Sasaki metric and is given by

((s. n))B :_ (deny, denn),i(e) + Ke(n)),r(e) .

Exercise 1.18. Show that if we endow TM with the Sasaki metric, the map nbecomes a Riemannian submersion from TM into M with totally geodesic fibres.

To finish this section, we show that the geodesic vector field has a very simpleexpression in terms of the identification JB. The geodesic vector field G : TM -TTM is given by

G(9) := a L-0 Of(6) = a I (ye(t)(t)),at at r=O

where ye is, as usual, the unique geodesic with initial condition 6 = (x, v). But,note that t Y6(t) is the parallel transport of v along ye. Therefore, G(B) _Le(v), or equivalently, G(B) = LB(v) = (v, 0) using the identification je.


TM

denM

V (0)

0 H(9) Ke

I I

Id LO ZO

x

x

Figure 1.2: Splitting of ToTM

1.3.2 The symplectic structure of TMBased on the splitting TeTM = H(9) V(O) and the identification Je given inthe previous section, we can introduce an almost complex structure in TM. Given0 E TM we define

JO : TBT M - TeTM,

by setting

Definition 1.19. Using the Sasaki metric and the almost complex structure, wecan define the symplectic form by

2e($, h) := ((Jet, n))s = Ke(p)) - (Ke(y), dojr(t!)).

Exercise 1.20. Show that SZB is antisymmetric and nondegenerate. Show that foreach 9, Jo is a linear isometry of the Sasaki metric. Show that JO is skew symmet-ric relative to the Sasaki metric, JB = -Id and that Je interchanges the subspacesH(9) and V(B).

M

In the next subsection we shall see that S2 is exact; in particular it is closed andtherefore it defines a symplectic form.


The following result shows that the geodesic vector field is the Hamiltonianvector field of the energy function H(x, v) (v, v)5 with respect to the sym-plectic form n.

Proposition 1.21. d H = iG S2 or equivalently, for all e E TM and all l; E TB T Mwe have

no (G (0). ).Proof. Let z : (-e,e) - TM be a curve adapted to , and write z(t) _(a(t), Z(t)). Then

at 1t=o H(z(t)) = iii lr=o (Z(t), Z(t))(t)= ((VaZ)(O), Z(O)) = (K9(t'), v)

On the other hand,

Qo(G(B), t) = (de,r(G(B)), Ke(y))= (detr(Le(v)), Ke(y)) = (v, Ke(y)).

Corollary 1.22. The geodesic flow preserves the symplectic form n.

1.3.3 The contact formIn many situations, it is more convenient to work with the geodesic flow restrictedto the unit tangent bundle SM since the latter is compact when M is compact.Thus, it is natural to ask if there exists, like the symplectic form in TM, somestructure in SM preserved by the geodesic flow. We shall define in this section aform a in TM such that, when restricted to SM, it becomes a contact form whosecharacteristic flow is the geodesic flow restricted to SM.

Definition 1.23. The one-form a of TM is defined as

G(B))) = v),.

Observe that a annihilates V (O). The form a and the symplectic form are re-lated by:

Proposition 1.24.

9 = -da.

To prove this proposition we will use the following lemma that has its owninterest.


Lemma 1.25. Let V/ denote the Riemannian connection of the Sasaki metric.Then for all ?7 E H(8) we have that V/,,G E V (O).

Proof. Let (E,,... , be an orthonormal frame field that is geodesic at x =7r (0) and defined in a neighborhood U of x. This means VE,Ej(x) = 0, for alli and j. Let Xi (y, w) := (Ei(y), 0) be the horizontal lift of the vector field Ei,in other words, X-(y, w) = L(y. ) (Ei(y)). The vector fields {X1..... X} areorthonormal relative to the Sasaki metric and they span the horizontal subbundle(where defined) in T M. Note that it suffices to show that V/x, G belongs to V (8)for all j. The geodesic vector field can be written as

G(y, w) _ (Ei(y), w) Xi,i=1

hence

V/x,G =n n

JXj (Ei(y),w)Xi+ E(Ei(y).w) V/,xfXi.i=1 i=1

(i) (ii)Since n : TM -> M is a Riemannian submersion, the horizontal componentof V/X, Xi (0) equals DES E; (x) = 0, therefore the expression (ii) is vertical. Tocomplete the proof of the lemma we need to show that (i) is vertical. In fact weshall prove that it vanishes.

Let aj : (-e, e) -+ M be a curve adapted to E j (x) and let Z j be the paralleltransport of v along a. We have

Xj(Ei('),-)= d I _ (Ei(aj),Zj)=(VajE1,v)=0.dtt-o

Proof of Proposition 1.24. Recall that we can write

da (t1. 2) = 1a(2) - 2a(i) - (Y([1, 2})Using the definition of a and the symmetry of the connection V/ we obtain,

da(41, 2) = 6 ( ( 2 , G)) - $2 (($1, G)) - (([I, '2} , G)) (1.1)

= ((1;2, 0/h, G)) - (('1, V/f2G)) (1.2)

We keep the notation of the previous lemma. Let us consider the vector fieldsY1(y, w) := J(y,w)(Xi(y, w)). For each (y, w), the vector fields

(X1,..., Xn, Y1,..., Yn}


form an orthonormal basis of T(y,,,,)TM. Note that [Yi, Y1] is vertical since eachY, is tangent to the fibres of TM. Therefore using equation (1.1) and the fact thatG is horizontal we get

da01V(0)XV(0) =0.From the lemma and equation (1.2) we obtain

dOI H(0)xH(0) = 0To end the proof it suffices to show that, for all i and j, we have

dae(X1, Yj) = -n0(Xi, Yj).Exercise 1.26. Show that for alI i and j, [Xi, Yj] = 0.

Using the exercise we get

dcro(Xi, Y j) = -Yjct(Xi)(B) = -Yj ((Xi, G)) (0);since t H (x, Ej (x)t + v) is an integral curve of Yj, we get

da0(Xi, Yj) dt (Ei(x), Ej(x)t + v) = -dij.r=0On the other hand, from the definition of n we obtain

n0(Xi, Yj) = Y1)) = ((Y1, Yj)) = 3ij.From now on we shall consider a only restricted to SM.

Definition 1.27. For 0 E SM we define S(0) as the subspace of TOSM givenby kera0. Equivalently, S(0) is the orthogonal complement with respect to theSasaki metric of the one dimensional subspace spanned by G(0).Exercise 1.28. Let 0 = (x, v). Show that

1. a vector E TOT M lies in TOSM if and only if (KO(i ), v) = 0;2. no(4,G(0)) =0forallt E TOSM;3. S20(4, J0G(0)) = 0 for all t E S(0) C TOSM;4. the orthogonal complement of S(0) in TOT M is given by the subspace

spanned by G(0) and JOG(0) and therefore S(0) and its orthogonal com-plement are JO-invariant;

5. show that no IS(O) is nondegenerate. More generally, show that the restric-tion of 00 to any subspace of TO SM complementary to the subspace spannedby G(O) is nondegenerate.

18 I . Introduction to Geodesic Flows

Corollary 1.29. The form a is a contact form in SM.

Proof. By Exercise 1.8 it suffices to show that dae restricted to S(O) is nonde-generate. This a consequence of the last proposition and the last exercise.

Lemma 1.30. The geodesic flow in SM is the characteristic flow of a.

Proof. Take 0 E SM. From the definitions, we obtain

ae(G(9)) = (dotr(G(0)), v) = (v, v) = 1.and, for E TOSM, we have

iGdae(f) = dao(G(0), $) _ - 79(G(0), t;) _ -doH(4) = 0,because H is constant on SM.

Corollary 1.31. The geodesic flow in SM preserves the contact form a.

Exercise 1.32. Using that r : SM -a M is a Riemannian submersion showthat the volume of SM equals the volume of M times the volume of the n - 1dimensional sphere in R" with the canonical metric.

Exercise 1.33. Show that volume form on SM induced by the Sasaki metric co-incides (up to sign) with "17a A (da)"-1, where n = dim M.

When M has finite volume, the last two exercises show that the volume forma A (da)"-t has finite integral over SM and hence it gives rise to a probabilitymeasure t defined on SM called the Liouville measure.

Lemma 1.34. Let 1 : SM -+ SM be the flip given by I (x, v) = (x, -v). Then1'a = -a and I preserves the Liouville measure.

Proof. Note that no1 = n. Hence

1

(d(x.-v)n(d(x,v)I -v)= (d(x.u)nol (l; ), -v)= (d(x.v) r( ), -v)= -a(x.v)( )-

Therefore

1'(a A (dot)"-t) = (-1)"a A (da)"-(,which implies that the Jacobian of I is I and hence I preserves the Liouvillemeasure.

1.4 The cotangent bundle T * M 19

1.4 The cotangent bundle T*MWe showed that TM has a symplectic structure and SM a contact structure buttheir definitions depend on the Riemannian metric of M. In this section, we willshow that the cotangent bundle T*M has an intrinsic symplectic structure, thatmeans that it does not depend on any metric. When M has a metric, we will provethat there exists a canonical relation between the corresponding structures in TMand T' M.

Definition 1.35. Let n : T*M i M be the canonical projection. Given (x, p) ET * M and 4 E T(x, p) T * M we define the canonical one form A in T * M by:

The symplectic form in T* M is cv := -d A.

Definition 1.36. Let M be a Riemannian manifold. The musical isomorphisms,

TM - T*M, T'M -+TM

are defined as

(v, ) , (v. u0) := u(v)The maps b and # are called musical isomorphisms because in classical notationthey lower and raise indices.

Lemma 1.37. There exist the following relations:

1. a = b*A;

2. SZ=b*w.Proof Let 9 = (x, v) and b(9) = (x, p) be points in TM and T*M respectively.Then,

A(deb(4)) = p(dn(e)iradeb(4))= p(den(t)) = (den(y), v) = se(t),

where in the third equality we used that nib = n.Finally, using that the exterior derivative commutes with the pullback, we ob-

tain

b*w = b*(-dA) = -d(b'A) = -dot = 2,concluding the proof of 2. 0

It follows right away from the last lemma and the fact that SZ is a symplecticform that w is also a symplectic form.


1.5 Jacobi fields and the differentialof the geodesic flow

In this section we shall describe an isomorphism between the tangent space TOT Mand the Jacobi fields along the geodesic yo. Using the decomposition of TOT M invertical and horizontal subspaces, we shall give a very simple expression for thedifferential of the geodesic flow in terms of Jacobi fields.

Recall that a Jacobi vector field along the geodesic Ye is a vector field alongyo that is obtained as the variational vector field of a variation of yo throughgeodesics. It is well known that J is a Jacobi vector field along yO if and only itsatisfies the Jacobi equation

J+R(y9,J)ye=0, (1.3)where R is the Riemann curvature tensor of M and dots denote covariant deriva-tives along yo. We recall that R is given by

R(X, Y)Z = VyVXZ - VXDyZ + V1X.YIZ.Let E To TM and z : (-e, e) - TM be an adapted curve to . Then the

map (s, t) H tr4,(z(s)) gives rise to a variation of ye = ,r ,(O). The curvest H Jr0 ,(z(s)) are geodesics and therefore the corresponding variational vectorfield J4 (t) TIJ_osro,(z(s)) is a Jacobi vector field with initial conditionsgiven by

Jt(0) = 1-ls=on4)r(z(s)) =den(y)Jt (0) = a Ir=O 4Is=o n^Or(Z(S))

U= as r=o 'F1 Ir=o a, I.f=0 Z(s) = KO()

Let us denote by J(y9) the vector space of all solutions of the Jacobi equation(1.3). It is a 2 dim M dimensional vector space. Let us consider the map iOTOTM -, J(ye) given by

ie( ) = JtIt is obvious that iO is a linear isomorphism.

Definition 1.38. We shall say that a Jacobi field J is normal to the geodesic ye if(J(t), yg(t)) = 0 for all t E R.Exercise 1.39. Take 0 E SM. Show that iO restricted to S(0) gives an isomor-phism between S(0) and all the normal Jacobi fields to ye.

The next lemma describes the differential of the geodesic flow in terms ofJacobi fields and the splitting of T T M into horizontal and vertical subbundles.

Lemma 1.40. Given 0 E T M, E TOT M and t E R, we have

do 0, (t 4 (t)) .

1.6 The asymptotic cycle and the stable norm 21

Proof.

t(t) = s I-o de(n0r)(l;) = dmr(e)aadoOt(l:).

Jt (t) = 6 a I s=o n.0t(z(s)) _ (s=o n-Or(z(s)).I,00t(z(s)) =

By means of this identification we can write

fmte(do0t(k),dort(rl)) = (-Jew, Jn(t))+(Jt(t), Jn(t)).Since Q is invariant under 0,, the right hand side should be a constant function oft. This can be checked by differentiation and using the Jacobi equation:

((-Jt. ill) + (Jt, 4Y = - (Jt, in) - (Jt, in) + (Jt, Jn) + (Jt. Jn)= (Jn, R(Y, Jt))) - R(y, Jn)Y) = 0.

Exercise 1.41. Show that 0, : TM -> TM is an isometry of the Sasaki metricfor all t E R if and only if M has constant sectional curvature 1.

1.6 The asymptotic cycle and the stable normWe shall assume in this section that M is compact.

1.6.1 The asymptotic cycle of an invariant measureLet A be a probability measure defined on the Borel a-algebra of SM.

Definition 1.42. The measure is said to be invariant under the geodesic flow if,for any Borel set B e SM, we have (01(B)) = (B) for all t E R. An invariantmeasure is said to be ergodic if any Borel invariant set has measure zero or one.We shall denote by M(O) the set of all Borel invariant probability measures. Theset M () is a compact metrizable convex set with the weak topology of measures[W].Exercise 1.43. Show that the Liouville measure ut is invariant under the geodesicflow.

The asymptotic cycle of an invariant measure will be an element p() EHi (SM, R). Suppose that c is a cohomology class in H t (SM, R). Take a smoothclosed one-form rI in SM such that [A] = c. The asymptotic cycle p(A) is definedas the unique element in Ht (SM, R) such that

(P(), c) = fsM A(G) d.Lemma 1.44. p() is well-defined.


Proof We need to show that our definition does not depend on the choice of theclosed one-form A. Equivalently, we need to show that if f is a smooth functionon SM, then

SMdf(G)dp=0.

Since is invariant, Birkhoff's ergodic theorem implies that

JSM df (G) du = JSM J,lim t (f (0,(8)) - f (8)) } d,4 = 0.

Suppose now that it is ergodic. By Birkhoff's ergodic theorem

fr Jlim I1

A(G(,,,0)) ds = A(G) d;c,1 00 t 0 SM

for u-a.e. 8 E SM. Let j,(8) be the oriented orbit segment from 0 to O,8. Bydefinition of integration of /forms

1 r A(G(0,0)) ds =J

A.o fir(e)

Pick a family of arcs 69i.e2 of bounded length connecting points 01 and 02, forexample, a shortest geodesic with respect to the Sasaki metric. Then there existsa constant C > 0 such that

< C.

Replace the orbit segment 6,(8) by the closed loop A(8) obtained by joiningfit (0) with fi e. We obtain

lim 1 1_ A = J A(G) d.r-+oo ' SMHence

P(A) = rllimoo

where denotes homology class. This gives a geometric interpretation of theasymptotic cycle when is ergodic. Asymptotic cycles were first introduced byR. Schwartzman [Sch].

The map n : SM -+ M induces a map in cohomology n' : H 1(M, R) --H I (SM, R).Lemma 1.45. Suppose that M" is orientable and different from the two-torus.Then the map it* is an isomorphism.


Proof. Using the Gysin sequence (cf. [BT, Proposition 14.33]) of the unit spherebundle n : SM -+ M, we obtain

0 - Ht (M. R) '* Ht (SM, R) HZ-"(M, R) HZ(MR) - ,where a. is integration along the fibre and Ae is multiplication by the Euler class.Then if n > 3, H2-"(M, R) vanishes and n` is an isomorphism. If n = 2,since M is not a two-torus, the Euler class e does not vanish and the map Ae :H0(M, R) = R -+ H2(M, R) = R is an isomorphism and hence ,r' is also anisomorphism.

Lemma 1.46. If f3 is any one form in M, then

,r f(G(0)) = P. (v),where B = (x, v).Proof.

n'P(G(B)) = Pa(e)(detr(G(B))) =18 (v). 0

Exercise 1.47. Let M be diffeomorphic to the two-torus. Show that there exists abasis (Ch c2, c3} of H 1 (SM, R) such that ct and c2 can be represented by the pullback of closed one-forms in M and c3 can be represented by a closed one-form Ain SM with the following property. There exists a one-form f on M such that

).(G(0)) = 8. (v),where 0 = (x, v). Conclude that if a probability measure is invariant under theflip (x, v) H (x, -v), then

ISMA(G) d = 0.

Definition 1.48. We set

PM (9) n.(p(u)) E H,(M,R),where n.: Ht (SM, R) -+ H, (M, R) is the map induced in real homology by 7r.

The homology class pM () is also called the asymptotic cycle of A. By Lemma1.45 and Exercise 1.47 we do not lose essentially any information by looking atthe cycle in M rather than in SM, and as the next lemma shows, there is a verysimple and natural expression for it which does not involve the geodesic vectorfield explicitly.


Lemma 1.49. pM () is the unique element in HI (M, R) such that for any closedone form w in M we have

(PM(14), [w]) = fM

wd,S

where we think of w as a function w : TM -+ 1R.

Proof. Note that

,r w(G)d,(n.P(t), [w]) = (P(,s), (n*w]) = fand the lemma follows from Lemma 1.46.

Lemma 1.50. If it is invariant under the flip (x, v) (x, -v), thenPM(IL) = 0.

Proof. By Lemma 1.49 we have

(PM(), [w]) = J wd.SMSince wJI = -w and p is invariant under I we have

wd = 0,SM

0

and hence pM(it) = 0.Lemma 1.51. p(t) = 0, where t is the Liouville measure.Proof Suppose that M is orientable. Since the Liouville measure is invariant un-der the flip (cf. Lemma 1.34), it follows from Lemma 1.50 that pM(t) = 0.Using Lemma 1.45 and Exercise 1.47 we conclude that p(t) = 0. When M isnonorientable, by passing to an orientable double cover, we can obtain right awaythat p(Al) = 0. We now give another proof that also holds for arbitrary contactflows.

It suffices to show that if x is any closed one-form in SM, then

X (G)a A (da)"-t = 0.SM

But note that

and that

X(G)CI A (da)"-t = iG(a A (da)"-t) AX

iG(a A (da)"-t) = (da)'.


Hence

'SM k(G)a n (da)"= 'SM n x.

sing that (da)'- 1 A A = d(a A (da)"-2 AX) and Stokes theorem, we obtainU

(dot)"-t n x = 0.JSM

Another interesting case of vanishing asymptotic cycle occurs when the mea-sure of maximal entropy is unique. See Section 3.2 for the definition of measureof maximal entropy.

Lemma 1.52. If there exists a unique measure of maximal entropy to, thenPM (uo) = 0.Proof. Let I be the flip (x, v) r* (x, -v). Since ,01 = 1.0-1 and uo is unique,then AO is invariant under I. Now apply Lemma 1.50.

Definition 1.53. We call the set

13 := pM(M(O)) C Ht (M, llt)

the Schwartzman ball.

The set B contains all the homologies in M arising from invariant probabilitymeasures, and hence it packs a lot of geometric information. We shall see that B isnothing but the unit ball of the stable norm that we define in the next subsection.

Lemma 1.54. The map pM : M(0) -+ Hi (M, R) is affine and continuous.

Proof. Clearly pM is an affine map. Take a sequence of measures un -+ it. Forany closed one-form w,

ISMILJSMSince Hi (M, R) is a finite dimensional vector space this implies that pM is con-tinuous.

Proposition 1.55. The Schwartzman ball B is a compact convex set symmetricabout the origin. Moreover, the origin lies in its interior.

Proof. We follow D. Massart's thesis [Mst]. Since pM : M(O) -+ Ht (M, R) isof lne and continuous, it is clear that B is compact and convex. Let us show that Bis symmetric about the origin. As before, let I be the flip (x, v) t-- (x, -v). Since

26 I. Introduction to Geodesic Flows

4>to1 = J4_t it follows that 1 maps invariant measures to invariant measures.Hence

(PM(l.), [w]) = J wd(4p)SMwol dlt

ISM

wd'SM

= - (PH(IL). [w])Hence pM(I /z) = -pM(,u) which shows that 8 is symmetric about the origin.

Let us prove now that the origin belongs to the interior of B. Recall that everynontrivial homology class in H 1 (M, Z) contains a closed geodesic. Hence thereexist closed geodesics with unit speed yi, ... , y k such that (h i , ... , hk ) is a basisof Hi (M, R), where h; = [yi ]. Let ti be the length of yi. Let i be the probabilitymeasure uniformly distributed along yi. That is, i satisfies

/'t;Yi(t))dt,SM f Ili = fi I f

ti

for any continuous function f : SM -* R. Clearly p.i is invariant and

PM(ILI) = Pi thi,since

(PM(Iti), [w]) = 'SM wdi = 1J w = ('hi[w]) .;

Consequently, B contains the convex envelope of

{t 'hi}which is a convex set containing the origin in its interior.

1.6.2 The stable norm and the Schwartzman ballIn this subsection we define the stable norm and we show that the unit ball ofthe stable norm coincides with the Schwartzman ball. Our presentation followsD. Massart's thesis [Mst]. Massart mentions in his thesis that this identificationis due to A. Fathi. There is also a proof of this fact for the n-torus in [BIK].The study of the stable norm has attracted great attention in recent years, see forexample [Ban, Bu, BIK]. One of the main questions in the subject is: what normscan arise as stable norms of Riemannian metrics?

Let (M, g) be a closed Riemannian manifold and let r be the quotient ofHi (M, Z) by its torsion subgroup; r is a cocompact lattice in Ht (M, R).


Definition 1.56. For It E r we set

f (h) := inf (y),where the infimum is taken over all closed piecewise differentiable curves with[y] = h, where (y) denotes the length of y.

Exercise 1.57. Show that lim,-,, i"h1 exists by showing first that

f(ht + h2) < f(ht)+ f(h2)+2dwhere d is the diameter of M.

For h E r, we let

lim f (nh)n-,00 n

It is immediate to check that Ih1,, has the following properties.

1. Ihls < f(h);2. for all A E Z and all h E 17, we have

Ixhls = IAIIhIs;

3. forallht,h2 E r we have

I h t +h2ls < IhI Is +Ih21s-

Definition 1.58. A homology class h E Ht (M, R) is said to be rational if thereexists a positive real number r such that rh E r.

We extend h i-i Ih1, to all rational points in HI(M, R) by homogeneity. Sincethe rational points are dense in Ht (M, IR), we can extend Ih Is by uniform continu-ity to a function defined in all Ht (M, R). One can easily show that this extensiondefines a norm in Hi (M, R) called the stable norm (see [Ban, Appendix] for thedetails). We shall denote by I[w]I5 the dual norm induced in Ht (M, R).

Theorem 1.59. The Schwartzman ball is the unit ball of the stable norm.

Proof. Let Bs be the unit ball of the stable norm. Let us prove first that B C_ Bs.Let pM() E B and let w be a closed one-form in M. For any closed piecewisedifferentiable curve a, we have

I ([a], [w]) 1

28 I . Introduction to Geodesic Flows

Given (x, u) E SM and t > 0, let a be the closed curve obtained following thegeodesic defined by (x, v) up to time t and then returning to x by a curve of length< d where d is the diameter of M. We have

f t w(0r(x, v))dt < ([a], [w]) +dm x Iwx I0 XCM

I [w] Is (t + d) + d m ax Iwx 1.

Now observe that by Fubini's theorem and the invariance of pt, we have

f fw(cbr(xv))dtdtt=t fM 0 SMHence

(PM(IA),[wl)I t-t f {I[w]Is(t+d)+dm xlwxl} d.XCM

11SM

If we let t - oo we obtain

I (PM(u), [w)) 1:5 I[wlls,which shows that IPM () Is < I

Now, let us prove that Bs C B. By compactness of B and the definition of thestable norm, it suffices to show that if h 96 0 is a rational point with Ih l < 1, thenh E B. Since h is a rational point there exists a positive real number r such thatrh E 1. Let yn be unit speed closed geodesics such that t(yn) = f (nrh) for allnonnegative integers n. As in Proposition 1.55, let n be the probability measureuniformly distributed along y,,. We have

PM(An) = f(yn)-t nrh = nrhf(nrh)*

Let tt be an accumulation point of {n 1. By the continuity of PM we haverh h

PM (A) = =IrhI5 IhIsBut since IhIs < 1, Proposition 1.55 implies that h E B. El

Corollary 1.60.

I[w]Is = max f wd s.SM


Proof. Using the theorem we have

(w] 1, = (h, [w])

max (P(u), [w])tcEM(m)

max wd./LEM(O) SM El

2The Geodesic Flow Acting onLagrangian Subspaces

This chapter describes how the geodesic flow acts on Lagrangian subspaces. Weintroduce Lagrangian subspaces and Lagrangian submanifolds and we show animportant property of the vertical subbundle which we call the twist property ofthe vertical subbundle. This property reflects the fact that the geodesic flow arisesfrom a second order differential equation on TM. Next we derive the Riccatiequations, after which we introduce the Grassmannian bundle of Lagrangian sub-spaces and show how to attach an index, the Maslov index, to every closed curveof Lagrangian subspaces. The definition used of the Maslov index follows Maidin [Ma I] and it is particularly adapted to the Riccati equations. This allows us toshow that the lift of the geodesic flow to the Grassmannian bundle of Lagrangiansubspaces is transverse to the Maslov cycle. This important property reflects theconvexity of the unit spheres in tangent spaces.

Using these tools we show two results, one motivated by hyperbolic sets andthe other by KAM tori. We show a theorem of Mane [Ma I ] that says that if thereexists a continuous invariant Lagrangian subbundle E defined on SM, then E istransverse to the vertical subbundle and M does not have conjugate points. Whenthe geodesic flow of M is Anosov, the stable and unstable bundles are continuousinvariant Lagrangian subbundles; hence we deduce that if M is a closed Rieman-nian manifold whose geodesic flow is Anosov, then M does not have conjugatepoints. This last result was first proved by W. Klingenberg using different tech-niques [Kll].

Finally we show the following result due to L. Polterovich [Poll ] (see also[BP]) which can be seen as the higher dimensional autonomous version of a re-sult of G. Birkhoff that asserts that an invariant circle of a twist map in the cylin-der T*SI which is homologous to the zero section must be a graph. Consider a

32 2. The Geodesic Flow Acting on Lagrangian Subspaces

Riemannian metric on the n-torus T. Suppose that P is a Lagrangian torus con-tained in SV which is homologous to the zero section of TIT". Suppose that theset of nonwandering points of the geodesic flow restricted to P coincides withall P. Then P is a graph, that is, the restriction to P of the natural projectionit : ST -+ T" is a diffeomorphism.

2.1 Twist properties

Definition 2.1. Let (V, S2) be a symplectic vector space. Since V must be evendimensional, we write dim V = 2n for some positive integer n. A subspace E CV is said to be Lagrangian if its dimension is n and the symplectic form satisfiesS2I Ex E = 0. A submanifold P of a symplectic manifold is said to be Lagrangianif the tangent space T, P is a Lagrangian subspace for all x E P.

It is straightforward to check from the definition of the symplectic form S2 ofTM that the subspaces H(O) and V(O) are Lagrangian.Remark 2.2. More generally, a subspace S C V is said to be isotropic if S2ISXS =0. Because of the nondegeneracy of n, an isotropic subspace has at most dimen-sion n, so Lagrangian subspaces are maximal isotropic subspaces.

Exercise 2.3. Let V be a 2n dimensional linear space and S2 a symplectic formon V. Prove that there exists a basis lei, en+i } 1:5i5, of V such that 12 (ei , e j) _S2(en+i, en+j) = 0 and S2(ei, Sij for I < i, j < n.

The basis of the Exercise 2.3 gives a decomposition of V as the direct sum oftwo Lagrangian subspaces.

Exercise 2.4. A subspace E C ToTM is Lagrangian if JoE = E.Exercise 2.5. Let V be a 2n dimensional real vector space, and let n be a nonde-generate two-form in V. Define an action of GL(2n, R) on the set of nondegen-erate two-forms on V by (ac2)(v, w) = S2(av, aw) for all vectors v and w in V.Using Exercise 2.3 prove that GL(2n, R) acts transitively on the set of nondegen-erate two-forms on V.

Exercise 2.6. Let V = C", regarded as a 2n dimensional real vector space, anddefine a two-form 0 on V by S2 (v, w) = Re ((Jv, w)), where v = (v 1, ... , vn)and w = (w1, ... , wn) are arbitrary elements of C",

J(vi,...,vn) = (ivt,...,ivn)

and (v, w) vktUk. Prove1. SZ is a nondegenerate two-form;

2. if g(v, w) = Re ((v, w)) is the usual inner product on R2" = C", thenS2(v, w) = g(Jv, w);

2.1 Twist properties 33

3. if E C C" is a real subspace with real dimension n, then E is Lagrangianif and only if (v, w) ER for allvandwinE;

4. if E C C" is a Lagrangian subspace, then a(E) is a Lagrangian subspacefor all a in the unitary group U(n);

5. U (n) acts transitively on the set of Lagrangian subspaces of C.

Exercise 2.7. Let H := (a E GL(2n, R) : aQ = a). Prove that an elementa E GL(2n, R) lies in H if and only if a'Ja = J, where as denotes the realtranspose of the transformation a relative to the inner product g on R2i. Showthat U(n) is a real subgroup of H of dimension n2.Lemma 2.8. Let N be a submanifold of M and

TN1:=((x,v)ETM : xEN, v1TN)its normal subbundle. The submanifold T N1 is Lagrangian.

Proof. Given rii E T(, .,,)TN', i = 1, 2; define paths t (xi(t), vi(t)) E TN'with initial conditions a; (0) = (x, v) and 6i (0) = >ri.

Using the identification, TBTM - T,,(B)M, we have

Qi (0) = n, = (xi (0), (Vx; vi) (0)) .Extend Xi (0) to a vector field Xi defined in a neighborhood of x such that whenwe restrict Xi to N, Xi is tangent to N for i = 1, 2. Since ai is a curve in TN-L,we have that for all t,

(X I (x2(t)), V2 (0) = 0,

u1(t)) = 0.Differentiating the last two expressions with respect to t and evaluating at t = 0we get

((V X2 X 1WI v) + (X 1(0), V 2 v2 (0)) = 0,

((V1 X2(x), v)+(x2(0), V ,V1(0)) =0.Using the definition of c2 and the last two equalities we obtain

C2(x.u)(2J1 ,'72) _ (x1(0), (Ox2 V2)(0)) - (x2(0), (Or, vl)(0))-(v, (VX2XI)(x))+(v, (VX,X2)(x))

_ (v, (VX1 X2)(x) - (VX2X 1)(x)) = (v, [X I , X2)(x))But (v, [X1, X2](x)) = 0 since [X1, X2](x) E T1N and v c- TXN1.


Exercise 2.9. Given X E N and V E Tx Nl, the shape operator of N at v is thesymmetric linear map

A,,:TN-+TNdefined as follows. Let V be a CO normal vector field in a neighborhood of x in Nwith V (x) = v. Given W E TX N, A, (w) is the orthogonal projection of V , V ontoTxN (see for example [Sa]). Show that T(x,,,) T Nl is given by the set of vectors

E Ttx.,,ITM such that d(x,,,)7r(i;) E TN and K(x,,,)(1;) - ETC Nl. Use the above, together with the fact that A is a symmetric linearmap, togive another proof of the last lemma.

Remark 2.10. Observe that if E is a Lagrangian subspace, then d6 ,(E) CTT,(o) TM is also Lagrangian since the geodesic flow preserves Q.Proposition 2.11 (twist property of the vertical subbundle). Let E be a Lagran-gian subspace of C TeTM. The subset given by

ft E R : doo,(E) n V(4,(9)) 0 {0}}is discrete.

Proof. Suppose that E n v (9) # 10}, we shall prove that there exists a > 0 suchthat for all t 96 0, t E (-a, a) we have:

do0,(E) n v (o,(9)) = (0).Let p : ToTM -- H(9) be the orthogonal projection.

Claim: The subspace p(E) is the orthogonal complement of Je(E n V(9)) inH(O) with respect to the Sasaki metric where, as usual, Je denotes the almostcomplex structure in TM.

Proof of the claim. Take elements S = p(i;) and Jo q with t: E E and 17 E En V (9).Then

((8,J91)) _7e(q,U=0,because and q belong to the Lagrangian subspace E and S and Jeq are horizon-tal. This implies the orthogonality of the subspaces. Their dimensions satisfy

dim p(E) + dim Je(E n V(9))= dim E - dim(E n V (O)) + dim Jo(E n v (o))= dim E = dim H(9),

where in the second equality we used that Jo is an isomorphism. 0

2.1 T\vist properties 35

Let {ill, ... , qm) be a basis of p(E) and let p, : T#,(B)TM -> H(O,(B))be the orthogonal projection. Then if t is near zero, there exists a set of m lin-early independent vectors ( t ) ,. ... , )m(t)} contained in p,(de,(E)). Take anorthonormal basis (l; I, ... , k) of E n V(0). Then from the claim we get thatk + m = dim H(9).

T a k e Jacobi vector fields Ji, i = 1, ... , k with initial conditions:

I(0) = 0;

Ji(0) = wi,where w; := or li = (0, w,) using the identification of TBTM with

From the expression for the differential of the geodesic flow in terms of Jacobivector fields, we get

Pt ((Ji(t), ii (m) = (A (t), 0). (2.1)Since Ji is a nontrivial Jacobi field that vanishes at zero, we can define unitaryvector fields Wi (for ItI small) by

Wi(t) (-Ji(t)/IJi(t)I,0) fort > 0;(Ji(t)/IJi(t)1,0) fort


Definition 2.12. Let 0 = (x, v) be a point in TM. The curvature operator,R(69): (v}1 - (u)1,

is the selfadjoint map given by w i-+ R(v, w)u, where R denotes the curvaturetensor of the Riemannian manifold M.

From now on, we shall restrict the symplectic formS20 to the subspace S(9) (see Exercise 1.28).

The intersection of the vertical and horizontal subspaces with S(6) defines twoLagrangian subspaces of the symplectic vector space (S(0), 0g) which we stilldenote by V(6) and H(e). Recall that the differential of the geodesic flow deb,takes S(0) to S(4 0).Proposition 2.13 (twist property of the horizontal subbundle). Suppose thatR(4,6) is a positive (negative) definite operator for all t E R and let E C S(0)be a Lagrangian subspace. The subset

(t ER: doO1(E)nH(O,(0)):A (0)}is discrete.

Proof. We will indicate the necessary modifications in the proof of Proposition2.11 above leaving the details as an exercise.

Consider the map R(9) : S(O) - S(9) given bylZ(O)(h, o) = (th, R(0)(tv)).

Define an inner product b in S(8) by setting

b(t, n) = ((t, R(B)-t (v)))Let S(0) -- V (O) be the orthogonal projection. As in the proof of Proposition2.11, one can check that p(E) is the b-orthogonal complement of R(9)Ja(E f1H(0)) in V(0).

Now take 1!;1_. , . k } an orthonormal basis (with respect to the Sasaki metric)of 1 (9)JB(E n H(6)). Then i; = (0, R(B)tai), where (w;, 0) E E n H(0). TakeJ;, i = 1,... , k Jacobi fields with initial conditions

Ji(0)=w;;1 J; (0) = 0.

Define the vector fields

_ 0,-J;/IJ;I) fort>0;W' { 0, ii/IJrI) fort

2.2 Riccati equations 37

Use the Jacobi equation to obtain

limW;(t)=t;.t-+0

2.2 Riccati equations

_Let E C S(O) be a subspace with dim E _ dim S(O). Suppose that En V (O){0}. Then it is possible to see E as a graph over the horizontal subspace. We canexpress

E = graph U :_ ((w, Uw), w e H(O)}with U : H(9) -> V (O). If we use the identification of H(9) and V (O) with {v}'L,the map U is defined from {v}l into itself.Lemma 2.14. The subspace E is Lagrangian if the map U is symmetric.

Proof. Let , n be elements in E; if 14 = (w, Uw) and q = (y, Uy) then, becauseof the condition q) = 0, we have

Assume now that E is a Lagrangian subspace of S(O). Let I = (-6, 6) be an in-terval with deb, (E) fl V (O, (8)) = (0) for all t e 1. We set deb, (E) = graph U (t)for all t e 1, with U(t) : {Ye(t)}1 -, {Ye(t)}-L. Let E E; the differential of thegeodesic flow can be written as

de 0, (jt (t), it (r))therefore we can also write

doOt( (Jt(t), UJt(t))which corresponds to the change of variables j4(t) = U(t)JJ(t). Hence,

Jt = U it + U itUJt + U2Jt.

Substituting into the Jacobi equation, we obtain

0 = J4 (t) + R(tbt(0))Jt(t)(U + U2 + R(0,(0))) Jt(t).

Since a;' E E is arbitrary, the family of operators U(t) satisfiesU+U2+R(0,(0))=0.

This is the Riccati equation seen from the horizontal subbundle.


Remark 2.15. The covariant derivative U can be interpreted as follows. Take aparallel basis of {ye(t)}1 and consider the matrix A of U in this basis. Then U isthe linear map whose matrix in the parallel basis coincides with A.

Remark 2.16. When M is a surface, the maps U(t) can be written as U(t)w =u(t)w, with w E (ye(t))-L, where u(t) is a smooth function oft that satisfies theclassical scalar Riccati equation

ti+u2+K(t)=0.where K(t) is the Gaussian curvature of the surface at yo (t). The function u(t) isthe slope of the line determined by do t(E) in the plane S(0,(8)).

Analogously, we can do the same for the vertical subspace when we havedet(E) fl H(,(9)) = (0), for all t E (-e, e). Let U(t) : V (ct(9)) -+ H(O,(8))be a family of linear maps such that

do0t(E) =graph U(t) := ((Uw, w), WE V (t(9))).Let l; E E; we have

d9tdr(4) _ (Jt(t),it (t)) = (U(t)Jl(t),.4(t)).which corresponds to the change of variables J{ = U.4 . Taking the derivative ofJl = UJg and using the Jacobi equation, we obtain

Jl = UJI + UJl = U4 + U(-RJ4)= UJl - URUJI.

The last equality implies that U(t) satisfiesU-URU-Id=0.

This is the Riccati equation as seen from the vertical subbundle.

Remark 2.17. When M is a surface, the maps U(t) can be written as U(t)w =u(t)w, with w E (ye(t))', where u(t) is a smooth function oft that satisfies thescalar equation

ti-K(r)u2-1 =0,where K(t) is the Gaussian curvature of the surface at ye(t).

2.3 The Grassmannian bundle of Lagrangiansubspaces

Suppose that (V, n) is a symplectic vector space.

2.4 The Maslov index 39

Definition 2.18. We shall denote by A the set of all Lagrangian subspaces of(V, 0). The set A has a natural manifold structure and is called the Grassmannianmanifold of Lagrangian subspaces.

Exercise 2.19. Using Exercise 2.6 prove that A is diffeomorphic to the homoge-neous space U(n)/O(n). Hence A is a compact manifold of dimension S ,n ji.1where 2n = dim V.

Definition 2.20. Fix a Lagrangian subspace F E A. For each integer k between0 and n, let us denote by Ak the subset of A given by those Lagrangian subspaceswhose intersection with F is a subspace of dimension k.

Exercise 2.21. Let F E A be a Lagrangian subspace of V, and let GF(k, n) bethe Grassmannian manifold of k dimensional subspaces of F. Define p : Ak -->GF(k, n) by p(E) = E n F. Prove

1. If W E GF(k, n), then E E Ak lies in p-I (W) if and only if E = WW', where W' is a Lagrangian subspace of (W J(W)}l, the orthogonalcomplement of W J(W) in V = C" relative to the standard Hermitianinner product on Cl;

2. Ak is a submanifold of A and p : Ak -+ GF(k, n) is a submersion;3. At has codimension kt- in A.If M is a Riemannian manifold and 0 E SM, we can consider A(9) the Grass-

mannian manifold of Lagrangian subspaces of S(9).Definition 2.22. Given any set X C SM we shall denote by A(X) the set givenby the union of all A(O) for 0 E X. In particular, when X = SM, A(SM) be-comes a fibre bundle over SM called the Grassmannian bundle of Lagrangiansubspaces.

Observe that the vertical and horizontal subbundles can be seen now as sectionsof the bundle A(SM) F-+ SM. We shall denote by Ak(9) the set ofLagrangian subspaces E E A(9) whose intersection with V(9) has dimensionk and by Ak(SM) the union of all Ak(0) for all 9 E SM. Using Exercise 2.21 wehave that

Lemma 2.23. Ak(SM) is a submanifold of A(SM) of codimension k .

2.4 The Maslov indexWe shall describe in this section how to attach an index to every continuous closedcurve a : S1 -+ A(SM).

First we shall define the Maslov cycle.

40 2. The Geodesic Flow Acting on tagrangian Subspaces

Definition 2.24. The Maslov cycle is the set given by

AV = U Ao(SM).k>1

By Lemma 2.23, AV is the union of A 1(SM) with submanifolds of codimen-sion > 3.

Remark 2.25. When M is a surface, the Maslov cycle becomes a closed subman-ifold of codimension one in A(SM).Exercise 2.26. The twist property of the vertical subbundle (Proposition 2.11)says that given a point (0, E) in the Maslov cycle Ao(SM), there exists e =e(8, E) > O such that for all t E (-e, e) with t 54 0, we have that (0,(0), doo,(E))does not belong to the Maslov cycle. Show that e(6, E) > 0 in general is notbounded away from zero even if M is compact. Is e(8, E) bounded away fromzero when M is a closed surface?

Let S' be the unit circle of the complex plane. We define a function mAo(SM) U A1(SM) --), S' as follows. If (0, E) E Ao(SM), by Lemma 2.14we can take a symmetric linear map U : H(0) -). V(6) such that E = graph Uand define

m(8, E) = I - itr(U)I +itr(U)When (0, E) E A1(SM), set

m(8, E) = -1.Lemma 2.27. The map m : Ao(SM) U A1(SM) - S1 is continuous.Proof. Suppose that (8, E) a Ao(SM) and that (8,,, E Ao(SM) is a se-quence converging to (0, E). We can write E = graph U,,, where U : H(8,,) ->V(88) are symmetric linear maps. We claim that there exists a constant C > 0such that for all n the number of eigenvalues of U (counted with multiplicities)larger in absolute value than C is < 1. If this property is false, there exist integersn, < n2 < ... , vectors x j, yj, and two real sequences {A} and (z j } such that

(xj,Y1) = 0, Ixjl = IYjI = 1;

Unjxj = ).jxj, U.jyj = Ajyj.for all j. If S j is the subspace spanned by (x1, A j x j) and (y3, ttj y j ), the prop-erty IA, I - oo implies that the sequence (Sj ) converges to a two dimensionalsubspace of V(8) and since Sj C En,, this two dimensional subspace is con-tained in E. This contradicts the property (0, E) E Ao(SM) and proves the


claim. Moreover the spectral radius of Un goes to 0o when n -+ oo becauseotherwise, since the maps U,, are symmetric, we would have a sequence of in-tegers n i < n2 < such that sups I < oo and then E would be also agraph of a symmetric linear map (obtained as the limit of the subsequence U,,,),thus implying (8, E) E Ak(SM). Then every U,, has exactly one eigenvalue ,lnwith C and IA,, -+ oo. Hence oo and this implieslimn-.ooIn(On. En) = -1.

Exercise 2.28. Show that m cannot be continuously extended to all the bundleA(SM) by showing that if (8, E) E Uk2:2 Ak(SM), then given any real number). there exists, arbitrarily near E, spaces E' E Ak(SM) such that E' is the graphof a symmetric linear map with trace X.

If y : S' -+ Ak(SM) U A i (SM) is a continuous map, we define the integerlc(y) as the degree of the map moy : S' --> S'. Now given a continuous mapy : S' - A(SM) we define s(y) as (P) where P : S' -- Ak(SM) U A l (SM)is a curve homotopic to y.

Lemma 2.29. The integer (y) is well-defined.

Proof Observe that since Uk>2 Ak (SM) is a union of submanifolds of codimen-sion > 3, there always exists a curve P : S' -+ Ak(SM) U Ak(SM) Co-close toy and therefore homotopic to it. The integer (y) does not depend on the choiceof P since given another curve P : S' -+ Ak(SM) U A i (SM) homotopic to ythe curves P and P are homotopic since, again, Uk>2 Ak(SM) is a union of sub-manifolds of codimension > 3 and the homotopy between P and P can be takenavoiding Uk>_2 Ak(SM).

Definition 2.30. The integer (y) is called the Maslov index of the continuousclosed curve y : S' -+ A(SM). It defines a cohomology class in H' (A(SM), Z)called the Maslov class of the Grassmannian bundle of Lagrangian subspaces withrespect to the vertical subbundle.

2.4.1 The Maslov class of a pair (X, E)Let X C SM be any closed connected set and let E be a continuous Lagrangiansubbundle defined on X, that is, E is a continuous section E : X -+ A(X) of thebundle A(X) -> X. Using the Maslov index defined above, we can attach to everypair (X, E) a cohomology class in the singular cohomology group H'(X, Z) asfollows. Given a continuous closed curve a : S' -+ X, we define the Maslovindex of a, (a), as the Maslov index of the continuous closed curve Ea : S' -).A(X) C A(SM). The integer (a) defines a cohomology class in H'(X,Z)which we shall call the Maslov class of the pair (X, E).

There are two main sources of interesting pairs (X. E) that we now describe.


2.4.2 Hyperbolic setsDefinition 2.31. A closed subset X C SM is said to be invariant under thegeodesic flow if t(X) C X for all t e R.Definition 2.32. A closed invariant set X is said to be hyperbolic if there existC > 0 and 0 < x < I such that for all 0 E X, there is a splitting

ToSM = E9 (0) Es(0) E"(0)such that

G(0) E E9 (0), dim E9 (0) = 1;ES "(4,(0)) for all t E R;

IIdoo,IE'(o)II < C>`', NO-,IE-(e)II


But from the definition of the hyperbolic set Ilde0t(q)II -, 0 as t -* +oo,therefore ao(q) = 0 (recall that M is compact) showing that E'(9) C S(9);therefore since E'(9) E"(9) and S(O) have the same dimensions, we haveE''(9) (D E"(9) = S(9). To show that 0 -> are Lagrangian we pro-ceed in a similar way. Take q and C in E'(9). Since , preserves the symplecticform f2, we have

Q9 (q, ) = go,(o)(de0t(q), doOt(C)),and from the definition of the hyperbolic set I Ideot(q)II -+ 0 and I1de0t(C)II --> 0as t - +oo, and thus

2e(q,C)=0.Hence Es(9) and E"(9) are isotropic subspaces such that Es(9) E'(9) = S(9)and therefore they are Lagrangian.

On account of the last lemma, given a hyperbolic set X, we can attach to eachpair (X, E') and (X, E") a Maslov class in H1(X, Z).Exercise 2.37. Are the Maslov classes defined by (X, E') and (X, E") different?

2.4.3 Lagrangian submanifoldsSuppose that P C TM is a Lagrangian submanifold.

Lemma 2.38. If P is in fact contained in SM, then it is invariant under thegeodesic flow.

Proof. Recall from part 2 of Exercise 1.28 that if E TOSM, then

G(9)) = 0.Take 0 E P. If P C SM, then TOP is a subspace of To SM. Since P is a La-

grangian submanifold, the symplectic form 92O restricted to the sum of TOP andthe one dimensional subspace spanned by G(9) must vanish. This implies thatG(9) E TOP and therefore P must be invariant under 01.

Suppose now that P C SM is a Lagrangian submanifold. The intersection ofTOP with S(9) is a Lagrangian subspace of (S(9), S2e1s(o)). Therefore to P wecan attach a Maslov class in HI (P, Z).Exercise 2.39. Show that when M is a surface, any surface P C SM invariantunder the geodesic flow is Lagrangian.

The last exercise gives many examples of Lagrangian submanifolds containedin SM. If we take a surface of revolution, the level sets of the Clairaut integral (cf.[doC]) are Lagrangian submanifolds in SM. KAM tori are the most importantsource of examples of Lagrangian submanifolds in SM.


2.5 The geodesic flow acting at the level ofLagrangian subspaces

Since the differential of the geodesic flow 0, : SM --> SM takes Lagrangiansubspaces to Lagrangian subspaces, it lifts naturally to a flow 0, : A(SM) ->A(SM) by setting

(8. E) _ (4r(e), d,90, (E)).We shall denote by G* the vector field associated to 0,*. In this section we shall

describe a remarkable property of the flow 0; with respect to the Maslov cy-cle Av c A(SM). Recall that the Maslov cycle is the union of AI(SM) withsubmanifolds of codimension > 3. It is not a smooth submanifold since it hassingularities given by Uk>2 Ak(SM) but it is what is called a stratified subman-ifold. When M is a surface the Maslov cycle is an honest closed submanifold ofcodimension one. The remarkable property that we mentioned in the introductionis that 0,* is always transverse to the Maslov cycle. In order to simplify our expo-sition and to avoid the complications arising from the singularities of the Maslovcycle, we shall prove this property in the much simpler case of dim M = 2. Hencefrom now on and until the end of this section, let us suppose that M is a surface.In this case dim A(SM) = 4, dim Ay = 3 and Ay is nothing but the image ofthe section 0 f-s (8, V(0)). Also, the function m from Section 2.3 is defined in allA(SM) and can be written in the following two ways:

J m(B, E) = if E n v(o) = {0} where E = {(w, uw) : w E H(6)};l m(B,E)=-1, ifE=V(B),

and

m(8, E) = u+ , if E n H(8) = {0} where E = {(uw, w) : w E V(8));{ m(8, E) = 1, ifE = H(8).

These two ways of writing m show that m is a smooth function and the Maslovcycle is given by m-I(-I).Proposition 2.40. Consider the differential of m at a point (8, V (8)) E A v, thatis,

d(e.v(e))m : T(e.v(e))A(SM) -' T_1S';then

d(e,v(e))m(G"(8, V(8))) = -2i.This implies that 0, is transverse to the Maslov cycle.

2.5 The geodesic flow acting at the level of Lagrangian subspaces 45

Proof. Note first that t i-+ (r(6), d0 f t(V(6))) is an integral curve of the vectorfield G' such that at t = 0 it passes through (6, V(6)). For tin a neighborhoodof 0 we can write

d04,(V(6)) = ((u(t)w, w) : w E V(,(6))},where u(t) is a function that satisfies (cf. Remark 2.17)

u-K(t)u2-1=0,where K(t) is the Gaussian curvature of the surface at ye(t). The function u(t)has initial condition u(0) = 0, therefore using the differential equation we deducethat u(0) = 1. Hence

d(e.v(o))m(G`(6, V(6))) = d do4t(V (6)))1,=0d u-i 2iiu 0

_

-2i.7It=0 a+i - (u(0)+i)ZO

As we mentioned before, in higher dimensions the Maslov cycle is a stratifiedsubmanifold. In Figure 2.1 we attempt to draw the Maslov cycle with A2(SM) thesingular point of the cone and a suitable orientation. The flow 0, is representedas the flow of vertical lines that crosses the Maslov cycle transversally to all thestrata and always in the 'same direction'

Exercise 2.41. Show that

d(e.H(e))m(G`(6, H(6))) = 2i K(n(O)),where K is the Gaussian curvature of M.

2.5.1 The Maslov index of a pseudo-geodesic and recurrenceDefinition 2.42. Let (X, E) be a pair, where X is a closed connected invariant setand E a continuous Lagrangian subbundle. We shall say that E is invariant if forall 6 E X and t E R we have de,r(E(6)) = E(o1(6)).Definition 2.43. Let (X, E) be a pair with E invariant. A continuous closed curvea : St --> X is a pseudo-geodesic if for all s E Sl for which

E(a(s)) fl V(a(s)) o (0},there exists e > 0 such that for t E (-e, e) we have

a(e"s) = 0r(a(s))Obviously, a closed orbit of the geodesic flow contained in X is a pseudo-

geodesic.


Figure 2.1: The Maslov cycle and the flow !

Lemma 2.44. If a : S' - X is a pseudo-geodesic then (a) > 0 and A(a) > 0if there exists s E St such that

E((Y(s)) fl V(a(s)) # (0}.

Proof Let us denote by P the set of points s E St for which

E(a(s)) fl V(a(s)) 0 {0}.Since a is a pseudo-geodesic and E is invariant, E o a : S' --> A(SM) is differ-entiable in a neighborhood of any points E P. By Proposition 2.40 the derivativeof m o E o a at any s E P is nonzero and preserves the canonical orientation ofSt. Hence the Maslov index (a), which is the degree of m o E o a, is always > 0and it is equal to the cardinality of P if P is not empty. 0

2.5 The geodesic flow acting at the level of Lagrangian subspaces 47

Remark 2.45. We have proved the lemma only for the case of surfaces but itsstatement is valid in any dimensions (cf. [BP, Mal ]).Definition 2.46. We shall say a point 9 E X is a nonwandering point of 01I x ifgiven an open set U of 0, there exists T > I such that OT (U n X) intersects u n Xnontrivially. We shall denote by 0 the set of all nonwandering points of Of Ix.

The set S2 is invariant under Of. Note that the definition also implies that if0 E S2, then given an open set U of 9 there exists a sequence T - oo for which4'T (U n X) intersects U n X nontrivially.Lemma 2.47. Take 9 E S2 and 01, 02 two points in the orbit of 0 with 02 = 0,01for s > 0. Then, given neighborhoods Ut and U2 of 91 and 92 respectively thereexist q E U2 n X and T > 0 such that OT (q) E U, n X.

Proof First note that since n is invariant, 92 E S2. Since O, (Ut) n U2 is anopen set containing 02, there exists q E &,(Ui) n U2 n X and T' > s such thatOT'(q)E,(Ut)nU2nX.Let T=T'-s.Then

OT (U) E 0_,(0S(U,) n U2nx) c U1 nx.

Lemma 2.48. If M has finite volume then the set of nonwandering points of 0, isall SM.

Proof. Take any open set U in SM and suppose by contradiction that for all t > 1,0,(U) does not intersect U. Then given two different real numbers tl and t2 with1t2 - ttl > 1, Of, (U) and 0,2(U) do not intersect. But this is absurd since thegeodesic flow preserves the Liouville measure which is finite when M has finitevolume.

Lemma 2.49. Suppose that X e SM is a locally arcwise connected closed in-variant set and E a continuous invariant Lagrangian subbundle defined on X.If there exists 9 E S2 C X such that E(9) n V(0) 96 (0}, then there exists apseudo-geodesic a with g(a) > 0.Proof. Let 9 E n be a point such that E(0) n V(9) 54 (0). By the twist propertyof the vertical subbundle (Proposition 2.11) there exists e > 0 such that for allt E (-2e, Zr) with t 0 0, we have E(0,(9)) n V (0,(9)) = (0). Let U1 be aneighborhood of 0_E(9) such that for all 0' E Ut n X, E(0') n V(9') = (0).Similarly, let U2 be a neighborhood of such that for all 0' E U2 n X,E(9') n V(01) = (0). Since 0 E S2, Lemma 2.47 says that there exist q E U2 n Xand T > 0 such that OT(q) E U1 n X (cf. Figure 2.2). Connect now OT(n) and0_E(0) by a path yt contained in Ut n X and connect 0, (0) and q by a path y2contained in U2 n X. This is possible since we assumed that X is locally arcwiseconnected.


Figure 2.2: Pseudo-geodesic with positive Maslov index

The path a obtained by joining (see Figure 2.2) the arcs yl, y2and 01(71)I(o,rl is a pseudo-geodesic which has positive Maslov index by Lemma2.44.

2.6 Continuous invariant Lagrangian subbundlesin SM

In this section we shall give a proof, when M is a surface, of the following generalresult obtained by R. Mafld [Ma! ].Theorem 2.50. Let M be a closed Riemannian manifold. If there exists a contin-uous invariant Lagrangian subbundle E defined on SM. then E(8) n V (B) = (0)for all 0 E SM and M does not have conjugate points.

In particular, when the geodesic flow of M is Anosov, the bundles Es and Eare continuous invariant Lagrangian subbundles, and hence we obtain:

Corollary 2.51. Let M be a closed manifold whose geodesic flow is Anosov. ThenM does not have conjugate points.

2.6 Continuous invariant Lagrangian subbundles in SM 49

The corollary was first proved by W. Klingenberg using different techniques[Kill.

For the proof of Theorem 2.50 we shall need the following lemma.

Lemma 2.52. There exists no smooth closed codimension one submanifold inSM transverse to the geodesic flow

Proof. Suppose that E is a smooth closed codimension one submanifold in SMtransverse to the geodesic flow. By part 5 of Exercise 1.28, given 9 E E, therestriction of S4 to TOE is nondegenerate. If i : E -+ SM is the inclusion mapthen (E, i*S2) is a symplectic manifold. Therefore i*S2"-t is a volume form in Eand thus

i*S"-t # 0.E

But up to a sign, n"'t = (da)"-t = d(a A (dot)"-2) and since 8E = 0 Stokestheorem implies that

i*S" =0.JE

This contradiction show that there is no smooth closed codimension one subman-ifold in SM transverse to the geodesic flow.

Exercise 2.53. Show that the symplectic form of a closed symplectic manifoldcannot be exact.

Proof of Theorem 2.50. We begin by noticing that even though the section E :SM -. A(SM) is only continuous, the invariance of E under the geodesic flowimplies that E is differentiable along the orbits of the geodesic flow and

dI E(4,(0)) = d I (Or(9),dechr(E(9))) = G*(9, E(9))dt t=O dt t=O

Approximate now in the C-topology the map E : SM -* A(SM) by a smoothmap E : SM -+ A(SM) such that

1. the maps 9 r-> y,It-OE(4,(9)), 9 r-> G*(9, E(9)) are CO-close.Suppose that there exists a point 9 E SM such that E(9) f1 V (O) 34 (0). This isequivalent to saying that E-t (Ay) # 0. By Lemmas 2.48 and 2.49 there exists apseudo-geodesic a with (a) > 0. If we take k sufficiently close to E so that Eaand La are homot'pic, we deduce that the curve of Lagrangian subspaces Laalso has positive Maslov index and therefore E- I (AV) 76 0. Next observe thatProposition 2.40 and property I above imply that E-I (A v) is a closed nonemptycodimension one submanifold transverse to the geodesic flow. This contradictsLemma 2.52 and therefore for all 9 E SM, E(0) n V (0) = (0). Finally to deducethat M has no conjugate points, note that for every geodesic there exists a Jacobifield orthogonal to it that does not vanish; simply take JO(t) = d9(7r4t)(q) for0 0 q E E(9). Now apply the following exercise.


Exercise 2.54. Let yt and Y2 be two nontrivial solutions of y+K(t)y = 0. Showthat yi (t)Y2(t) - ,yi (t)Y2(t) is constant. Use this fact to prove that if there existsto E (0, oo) such that yt (0) = yt (to) = 0, then Y2 must vanish in [0, to].

2.7 Birkhoff's second theorem for geodesic flowsIn this section we shall give a proof of the following result obtained by L. Pol-terovich in [Poll] (see also [BP]) which can be seen as the higher dimensionalautonomous version of a result of G. Birkhoff that asserts that an invariant circleof a twist map in the cylinder T`S' which is homologous to the zero section mustbe a graph.

Theorem 2.55. Consider a Riemannian metric on the n-torus T". Suppose that Pis a Lagrangian torus contained in ST" which is homologous to the zero sectionof TT". Suppose that the set of nonwandering points of 4r I p coincides with allP. Then P is a graph, that is, the restriction to P of the natural projection rST" -). T" is a diffeomorphism.

The proof of Theorem 2.55 is partially based on the following result of C.Viterbo [V] (see also [Po12]) whose proof is beyond the scope of the present notes.Theorem 2.56. The Maslov class of a Lagrangian torus in TT" that is homolo-gous to the zero section must vanish.

We shall need the following lemma.

Lemma 2.57. Let P be a torus in TT" that is homologous to the zero section.Suppose that the restriction to P of the natural projection r : TT" - T" isa local diffeomorphism. Then the restriction to P of the natural projection is adifeomorphism.

Proof Since P is homologous to the zero section, the map it I p : P -+ 'II" musthave degree one and since tr I p is a local diffeomorphism it must be a global dif-feomorphism.

Proof of Theorem 2.55. Observe first that the map n I p : P -+ T" is a localdiffeomorphism, if for all 9 E P, we have that To P fl v (9) = (0). For 9 E P,let us denote by E(9) the intersection of TOP with S(9). Since P is contained inST", TOP fl V (9) = (0) iff E(9) n V (9) = (0). Therefore, on account of Lemma2.57, to complete the proof of the theorem we need to show that for all 9 E P wehave that E(9) n V(9) = (0). Suppose that this is not case, that is, there existssome 9 E P for which E(9)n V(O) 96 (0). Since we are assuming that every pointof Ot I p is nonwandering, Lemma 2.49 implies that there exists a pseudo-geodesica in P with positive Maslov index. Therefore the Maslov class of P is not zero.On the other hand, by Theorem 2.56, the Maslov class of the Lagrangian torus Pvanishes, thus reaching a contradiction.

3Geodesic Arcs, Counting Functionsand Topological Entropy

In this chapter we introduce the counting functions and we relate them to thetopological entropy h10 (g) of the geodesic flow of g.

In all that follows, unless otherwise stated, M will be a compact manifold en-dowed with a Riemannian metric g and N C M will be a closed submanifold.As usual, we shall denote by exp-'- : T N-L -+ M the normal exponential map,

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