Let $f:M\to N$ be a diffeomorphism, $g^M$ and $g^N$ the metric tensors on $M$ and $N$, and $R^M$ and $R^N$ the curvature tensors of the respective Levi-Civita connections. Let $f$ be such that $R^N = f_* R^M$.
What can we say about $g^M$ and $g^N$, at least locally?
I hope I was clear with the notation.
EDIT: In other words, I am looking for some sort of result that generalizes Spivak's "Test Case" for nonzero curvatures.