On the ergodicity of geodesic flows

Abstract

AbstractIn this paper we study the ergodic properties of the geodesic flows on compact manifolds of non-positive curvature. We prove that the geodesic flow is ergodic and Bernoulli if there exists a geodesic γ such that there is no parallel Jacobi field along γ orthogonal to γ. In particular, this is true if there exists a tangent vector v such that the sectional curvature is strictly negative for all two-planes containing v, or if there exists a tangent vector v such that the second fundamental form of the horosphere determined by v is definite at the support of v.