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?
 A: 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.
A: 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.
