A smooth map \(F\colon (M,g)\to (\widetilde{M},\widetilde{g})\) is called Riemannian isometry if it is a diffeomorphism that satisfies
\begin{equation*} F^*\widetilde{g}=g. \end{equation*}More generally, \(F\) is called a local isometry if every point \(p\in M\) has a neighbourhood \(U\) such that \(F|_U\) is an isometry on a open subset of \(\widetilde{M}\).
If there exists a (local) isometry between two Riemannian manifolds \(M\) and \(\widetilde{M}\) we call them (locally) isometric.