I ask myself, whether the curvature determines the metric.
Concretely: Given a compact Riemannian manifold $M$, are there two metrics $g_1$ and $g_2$, which are not everywhere flat, such that they are not isometric to one another, but that there is a diffeomorphism which preserves the curvature?
If the answer is yes:
Can we chose $M$ to be a compact 2-manifold?
Can we classify manifolds, where the curvature determines the metric?
What happens, if we also allow semi-riemannian manifolds for the above questions?
Thank you for help.