Hyperbolic Closed Geodesics

TL;DR: In this article, Eberlein has investigated the geometry of complete riemannian manifolds M for which the universal covering admits such a compactification by points at infinity and showed that such a manifold has many properties in common with a manifold of negative curvature.

Abstract: Publisher Summary This chapter describes hyperbolic closed geodesies. It discusses that the riemannian metric determines a diffeomorphism. The chapter also discusses the characterization of the flow-invariant vector field, Lagrangian bundles, index theorem for closed geodesies, the structure of manifolds with geodesic flow of Anosov type, and triangle theorem for manifolds with geodesic flow of Anosov type. The chapter considers compact riemannian manifolds M for which the geodesic flow is of Anosov type. The chapter presents a theorem which shows that such a manifold has many properties in common with a manifold of negative curvature. Eberlein has investigated the geometry of complete riemannian manifolds M for which the universal covering admits such a compactification by points at infinity. He then asks that certain axioms be satisfied. In particular, he has the axiom that any two points at infinity can be joined by exactly one geodesic. One can show that this axiom is satisfied in the case where is the universal covering of a compact manifold M for which the geodesic flow is of Anosov type. The chapter formulates a comparison theorem between sufficiently large triangles on M and triangles on an appropriately chosen hyperbolic space 0 of constant curvature.