Riemannian manifolds with Anosov geodesic flow do not have conjugate points

TL;DR

This paper proves that complete non-compact Riemannian manifolds with bounded below curvature and Anosov geodesic flow cannot have conjugate points, resolving an open problem in differential geometry.

Contribution

It establishes a new link between Anosov geodesic flows and the absence of conjugate points in certain Riemannian manifolds, solving a previously open problem.

Findings

Manifolds with Anosov geodesic flow lack conjugate points

Curvature bounds are crucial for the result

Addresses an open problem in geometric dynamics

Abstract

This paper establishes a significant result concerning the absence of conjugate points in certain complete Riemannian manifolds. Specifically, we demonstrate that any complete non-compact manifold with curvature bounded below and an Anosov geodesic flow does not possess conjugate points. This resolves an open problem left by R. Ma\~n\'e in [9] and subsequently highlighted by [7].