# A kind of converse to the Hopf theorem on ergodicity of geodesic flow in negative curvature

Is there a 2 dimensional Riemannian manifold $M$ whose curvature is not negative but its geodesic flow is an ergodic flow?

• @thedude but am I mistaken to think that it is not ergodic because the unit tangent bundle possess two disjoint copy of the torus $T^2$ which separate the unit tangent bundle and is invariant under the geodesic flow? Jul 8 '18 at 14:28
• @RW What is the argument for ergodicity? May I ask you to read my previous comment? Jul 8 '18 at 14:29
• @Ali Taghavi - Oups - sorry - of course the geodesic flow on the torus is not ergodic - since the slope is preserved
– R W
Jul 8 '18 at 14:45
• I believe that if $M$ is a closed hyperbolic surface, and if you then alter the metric to be negatively curved except at a single point of zero curvature, the resulting geodesic flow will still be ergodic. I do not have a proof at hand. Jul 8 '18 at 15:03
• @AliTaghavi I thought the geodesic flow was defined on the manifold, but now I see it is defined on the tangent bundle. So I learned something from your question. Thank you. Jul 8 '18 at 19:19

A byproduct of this study is the following: on a nonpositively curved surface, the geodesic flow is of Anosov type as soon as there is a point of negative curvature on all geodesics. So you can start with a surface with negative curvature, make a small perturbation so that the curvature vanishes on a single point $p$, and the geodesic flow is still Anosov and ergodic. Now the Anosov condition is an open one so you can make again a small perturbation so that the curvature becomes positive at the point $p$ and you still have an ergodic Anosov geodesic flow. There are many variations around these ideas.