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OverviewThis 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. Full Product DetailsAuthor: Gabriel P. PaternainPublisher: Birkhauser Verlag AG Imprint: Birkhauser Verlag AG Volume: v. 180 ISBN: 9783764341442ISBN 10: 3764341440 Pages: 168 Publication Date: September 1999 Audience: College/higher education , Professional and scholarly , Postgraduate, Research & Scholarly , Professional & Vocational Replaced By: 9780817641443 Format: Hardback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsIntroduction 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.ReviewsAuthor InformationTab Content 6Author Website:Countries AvailableAll regions |
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