Contributions to the study of Anosov Geodesic Flows in Non-Compact Manifolds

TL;DR

This paper establishes conditions under which geodesic flows in non-compact manifolds are of Anosov type, linking curvature properties to dynamical behavior and providing new examples of such flows.

Contribution

It proves that negative average sectional curvature implies Anosov geodesic flow in non-compact manifolds and constructs new examples in dimension two.

Findings

Negative average sectional curvature along geodesics implies Anosov flow.

In dimension two, no focal points plus curvature condition suffices for Anosov flow.

New examples of non-compact surfaces with Anosov geodesic flow are constructed.

Abstract

In this paper we prove that if the geodesic flow of a {compact or non-compact} complete manifold without conjugate points is of the Anosov type, then the average of the integral of the sectional curvature along the geodesic is negative and away from zero from a uniform time. Moreover, in dimension two, if the manifold has no focal points, then this condition is sufficient to obtain that the geodesic flow is of Anosov type. This sufficient condition will also be used to construct new examples of non-compact surfaces whose geodesic flow is of the Anosov type.