Geodesic Flows

Introduction to geodesic flows: geodesic flow of a complete Riemannian manifold; symplectic and contact manifolds; the geometry of the tangent bundle; the cotangent bundle T*M; Jacobi fields and the differential of the geodesic flow; the asymptotic cycle and the stable norm. The geodesic flow acting on Lagrangian subspaces: twist properties; Riccati equations; the Grassmannian bundle of Lagrangian subspaces; Maslov index; the geodesic flow acting at the level of Lagrangian subspaces; continuous invariant Lagrangian subbundles in SM; Birkhoff's second theorem for geodesic flows. Geodesic arcs, counting functions and topological entropy: the counting functions; entropies and Yomdin's theorem; geodesic arcs and topological entropy; Manning's inequality; a uniform version of Yomdin's theorem. Mane's formula for geodesic flows and convex billiards: time shifts that avoid the vertical; Mane's formula for geodesic flows; manifolds without conjugate points; entropy for positive curvature; mane's formula for convex billiards; further results and problems on the subject. Topological entropy and loop space homology: rationally elliptic and rationally hyperbolic manifolds; Morse theory of the loop space; topological conditions that ensure positive entropy; entropies of manifolds; further results and problems on the subject.