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