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



A kind of converse to the Hopf theorem on ergodicity of geodesic flow in negative curvature
It was proved by [Donnay](https://mathscinet.ams.org/mathscinet-getitem?mr=970551) that any compact orientable surface can be given a Riemannian metric for which the geodesic flow is ergodic. See [Theorem 1 of this article.](https://link.springer.com/chapter/10.1007%2FBFb0082827) On the other hand, there are no negatively curved metrics on a sphere or a torus.


