A question about Riemann curvature tensor and metric tensor
Solution 1:
A simple counterexample is a flat Euclidean space of any dimension. The Riemann curvature is uniformly zero. A diffeomorphism of the space back to itself stretching/shrinking/shearing the space can be thought of as a change of metric on the original space, but the curvature remains zero. This argument should generalize to any manifold with constant fixed curvature (n-spheres & hyperbolic geometries), but when the curvature is not constant an automorphism may not preserve the curvature at some points.