Geodesic Flows

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.