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

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  • $\begingroup$ @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? $\endgroup$ Jul 8 '18 at 14:28
  • $\begingroup$ @RW What is the argument for ergodicity? May I ask you to read my previous comment? $\endgroup$ Jul 8 '18 at 14:29
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    $\begingroup$ @Ali Taghavi - Oups - sorry - of course the geodesic flow on the torus is not ergodic - since the slope is preserved $\endgroup$
    – R W
    Jul 8 '18 at 14:45
  • $\begingroup$ 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. $\endgroup$
    – Lee Mosher
    Jul 8 '18 at 15:03
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    $\begingroup$ @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. $\endgroup$
    – thedude
    Jul 8 '18 at 19:19
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It was proved by Donnay that any compact orientable surface can be given a Riemannian metric for which the geodesic flow is ergodic. See Theorem 1 of this article. On the other hand, there are no negatively curved metrics on a sphere or a torus.

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Anosov flows are ergodic and a geodesic flow can be Anosov even if the curvature is not strictly negative. This was studied by Eberlein in the seventies, in an article from 1973 entitled when is a geodesic flow of Anosov type? and other authors (Klingenberg etc).

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.

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  • $\begingroup$ Thank you very much for this very great answer. I am sorry that I can not accept the two answers simultaneously. meta.mathoverflow.net/questions/1491/… $\endgroup$ Jul 8 '18 at 19:49
  • $\begingroup$ But a slight perturbation of a compact surface with strictly negative curvature is stile a surface with strictly negative curvature. Am I right? But i think that there is modification of your argument. I try to find the valuable reference you mentioned in your answer and learn them. thanks again for your attention to my question and these very helpful references $\endgroup$ Jul 9 '18 at 7:54
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    $\begingroup$ The curvature is zero at a point, the flow is Anosov, you make a perturbation such that the curvature becomes positive around the point and the flow stays Anosov. $\endgroup$
    – coudy
    Jul 9 '18 at 16:26

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