I am studying for the qualifying exams of my PhD, and I found the following exercise in the reference book: If $M$ is a compact Riemannian manifold, $N$ is a connected Riemannian manifold and $M$ is locally isometric to $N$, then $N$ is locally isometric to $M$.
I can't figure out how to use the book definition of local isometry to get the result. Namely, the book defines local isometry as follows: $M$ is locally isometric to $N$ if for all $p\in M$ there exists an isometry between an open neighborhood of $p$ to an open set of $N$.
Thanks.