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Geodesic Flows / by Gabriel P. Paternain.

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The aim of this book is to present the fundamental concepts and properties of the geodesic flow of a closed Riemannian manifold. The topics covered are close to my research interests. An important goal here is to describe properties of the geodesic flow which do not require curvature assumptions. A...

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Bibliographic Details
Online Access: Full Text (via Springer)
Main Author: Paternain, Gabriel P.
Format: eBook
Language:English
Published: Boston, MA : Birkhäuser Boston : Imprint : Birkhäuser, 1999.
Series:Progress in mathematics (Boston, Mass.) ; v. 180.
Subjects:
Mathematics.
Differentiable dynamical systems.
Global analysis.
Topology.
Table of Contents:
  • 0 Introduction
  • 1 Introduction to Geodesic Flows
  • 1.1 Geodesic flow of a complete Riemannian manifold
  • 1.2 Symplectic and contact manifolds
  • 1.3 The geometry of the tangent bundle
  • 1.4 The cotangent bundle T*M
  • 1.5 Jacobi fields and the differential of the geodesic flow
  • 1.6 The asymptotic cycle and the stable norm
  • 2 The Geodesic Flow Acting on Lagrangian Subspaces
  • 2.1 Twist properties
  • 2.2 Riccati equations
  • 2.3 The Grassmannian bundle of Lagrangian subspaces
  • 2.4 The Maslov index
  • 2.5 The geodesic flow acting at the level of Lagrangian subspaces
  • 2.6 Continuous invariant Lagrangian subbundles in SM
  • 2.7 Birkhoff's second theorem for geodesic flows
  • 3 Geodesic Arcs, Counting Functions and Topological Entropy
  • 3.1 The counting functions
  • 3.2 Entropies and Yomdin's theorem
  • 3.3 Geodesic arcs and topological entropy
  • 3.4 Manning's inequality
  • 3.5 A uniform version of Yomdin's theorem
  • 4 Mañé's Formula for Geodesic Flows and Convex Billiards
  • 4.1 Time shifts that avoid the vertical
  • 4.2 Mañé's formula for geodesic flows
  • 4.3 Manifolds without conjugate points
  • 4.4 A formula for the topological entropy for manifolds of positive sectional curvature
  • 4.5 Mañé's formula for convex billiards
  • 4.6 Further results and problems on the subject
  • 5 Topological Entropy and Loop Space Homology
  • 5.1 Rationally elliptic and rationally hyperbolic manifolds
  • 5.2 Morse theory of the loop space
  • 5.3 Topological conditions that ensure positive entropy
  • 5.4 Entropies of manifolds
  • 5.5 Further results and problems on the subject
  • Hints and Answers
  • References.

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