Author | Paternain, Gabriel P. author |
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Title | Geodesic Flows [electronic resource] / by Gabriel P. Paternain |
Imprint | Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 1999 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1600-1 |
Descript | XIII, 149 p. online resource |
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 Birkhoffโs second theorem for geodesic flows -- 3 Geodesic Arcs, Counting Functions and Topological Entropy -- 3.1 The counting functions -- 3.2 Entropies and Yomdinโs theorem -- 3.3 Geodesic arcs and topological entropy -- 3.4 Manningโs inequality -- 3.5 A uniform version of Yomdinโs theorem -- 4 Maรฑรฉโs Formula for Geodesic Flows and Convex Billiards -- 4.1 Time shifts that avoid the vertical -- 4.2 Maรฑรฉโs 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 Maรฑรฉโs 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