On the ergodicity of geodesic flows on surfaces without focal points

In this paper, we study the ergodicity of the geodesic flows on surfaces with no focal points. Let M be a smooth connected and closed surface equipped with a C-infinity Riemannian metric g, whose genus g >= 2. Suppose that (M, g) has no focal points. We prove that the geodesic flow on the unit tangent bundle of M is ergodic with respect to the Liouville measure, under the assumption that the set of points on M with negative curvature has at most finitely many connected components.