This work begins with an introduction to the geodesic flow of a complete Riemannian manifold, emphasizing its sympletic properties and culminating with various applications such as the non-existence of continuous invariant Lagrangian sub-bundles for manifolds with conjugate points. Subsequent chapters develop the relationship between the exponential growth rate of the average number of geodesic arcs between two points.
Author(s): Gabriel P. Paternain
Series: Progress in Mathematics 180
Edition: 1
Publisher: Birkhäuser Boston
Year: 1999
Language: English
Pages: 169
Contents......Page 8
Preface......Page 12
0 Introduction ......Page 16
1 Introduction to Geodesic Flows ......Page 22
1.1.1 Euler-Lagrange flows ......Page 23
1.2.1 Sympiectic manifolds ......Page 24
1.2.2 Contact manifolds ......Page 25
1.3.1 Vertical and horizontal subbundles ......Page 26
1.3.2 The symplectic structure of TM ......Page 29
1.3.3 The contact form ......Page 30
1.4 The cotangent bundle T'M ......Page 34
1.5 Jacobi fields and the differential of the geodesic flow ......Page 35
1.6.1 The asymptotic cycle of an invariant measure ......Page 36
1.6.2 The stable norm and the Schwartzman ball ......Page 41
2 The Geodesic Flow Acting on Lagrangian Subspaces ......Page 46
2.1 Twist properties ......Page 47
2.2 Riccati equations ......Page 52
2.3 The Grassmannian bundle of Lagrangian subspaces ......Page 53
2.4 The Maslov index ......Page 54
2.4.1 The Maslov class of a pair (X,E)......Page 56
2.4.2 Hyperbolic sets ......Page 57
2.4.3 Lagrangian submanifolds ......Page 58
2.5 The geodesic flow acting at the level of Lagrangian subspaces ......Page 59
2.5.1 The Maslov index of a pseudo-geodesic and recurrence ......Page 60
2.6 Continuous invariant Lagrangian subbundles in SM ......Page 63
2.7 Birkhoff's second theorem for geodesic flows ......Page 65
3.1 The counting functions ......Page 66
3.1.1 Growth of n(T) for naturally reductive homogeneous spaces ......Page 71
3.2.1 Topological entropy ......Page 73
3.2.2 Yomdin's theorem ......Page 75
3.2.3 Entropy of an invariant measure ......Page 76
3.2.4 Lyapunov exponents and entropy ......Page 77
3.2.5 Examples of geodesic flows with positive entropy ......Page 78
3.3 Geodesic arcs and topological entropy ......Page 79
3.4 Manning's inequality ......Page 84
3.5 A uniform version of Yomdin's theorem ......Page 88
3.5.1 Another proof of Theorem 3.32 using Theorem 3.44 ......Page 90
4.1 Time shifts that avoid the vertical ......Page 92
4.2 Mane's formula for geodesic flows ......Page 97
4.2.1 Changes of variables ......Page 98
4.2.2 Proof of the Main Theorem ......Page 103
4.3 Manifolds without conjugate points ......Page 105
4.4 A formula for the topological entropy for manifolds of positive sectional curvature ......Page 107
4.5 Mane's formula for convex billiards ......Page 108
4.5.1 Proof of Theorem 4.30 ......Page 112
4.6 Further results and problems on the subject ......Page 117
4.6.1 Topological pressure ......Page 119
5.1 Rationally elliptic and rationally hyperbolic manifolds ......Page 124
5.1.1 The characteristic zero homology of H-spaces ......Page 127
5.1.2 The radius of convergence ......Page 129
5.2 Morse theory of the loop space ......Page 130
5.2.1 Serre's theorem ......Page 131
5.2.2 Gromov's theorem ......Page 132
5.3.1 Growth of finitely generated groups ......Page 134
5.3.2 Dinaburg's Theorem ......Page 135
5.3.3 Arbitrary fundamental group ......Page 136
5.3.4 Proof of Theorem 5.20 ......Page 137
5.4.1 Simplicial volume ......Page 141
5.4.2 Minimal volume ......Page 142
5.5 Further results and problems on the subject ......Page 145
Hints and Answers ......Page 148
References ......Page 154
Index ......Page 162
