We talked about isometries /local isometries and linear isometries in the lecture, but unfortunately we did not say when the isometry preserves the curvature tensor or sectional curvature.
First I am wondering: Is preserving the sectional curvature equivalent to preserving the whole curvature tensor?
A linear isometry $L :T_pM \rightarrow T_qN$ is defined as
$$g_q(LV,LV) = g_p(V,V).$$
Such a map does not generally preserve sectional curvature, as we can always find such a map between two tangent spaces of two Riemannian manifolds, but clearly the curvature tensors can be different. So such a map does not generally preserve the curvature tensor.
A local isometry is a local diffeomorphism $\phi: U \subset M \rightarrow V,$ where $V$ is a subset of $N$ such that $D \phi$ is a linear isometry.
Now, one could ask if this mean that $$D \phi(R_p(V,W)(X))= R_{\phi(p)}(D \phi (V),D\phi(W))(D\phi(X))$$ holds? I don't think so cause the polar map $$\phi = exp_{q,N} \circ L \circ \exp_{p,M}^{-1}$$ can be always defined on two manifolds locally with respect to the two appropriate exponential maps. But still two manifolds do not need to have the same curvature tensor.
Hence, my second question is: Is it really true that only global isometries preserve curvature (both sectional curv. and the curvature tensor) on manifolds in general?