Geometric conditions to obtain Anosov geodesic flow for non-compact manifolds

Abstract

Let ( M , g ) be a complete Riemannian manifold without focal points and curvature bounded below. We prove that when the average of the sectional curvature in tangent planes along geodesics is negative and uniformly away from zero, then the geodesic flow is of Anosov type. We use this result to construct a non-compact manifold with non-positive curvature, without compact quotient and geodesic flow of Anosov type.