The Ergodicity of Geodesic Flows on Rank One Manifolds of Nonpositive Curvature

Abstract

In this paper, we study the ergodicity of the geodesic flows on closed rank one manifolds of nonpositive sectional curvature. We introduce a subset of the manifold defined by the infinite-order vanishing of the Gaussian curvature in dimension two, or of the fiberwise second moment of the reduced Jacobi determinant in arbitrary dimensions, and derive bounds on the Liouville measure and Hausdorff dimension of the singular set in terms of this subset. In particular, if this subset has zero volume, then the singular set has zero Liouville measure, and the geodesic flow is ergodic with respect to Liouville measure. By extending this method to real analytic metrics, we characterize the singular set as the set of zeros of a nontrivial real analytic determinant constructed from curvature operators, thereby establishing ergodicity in this setting without any additional assumptions.