\begin{definition}[Definition:Isometry (Riemannian Manifolds)]
Let $\struct {M, g}$ and $\struct {\tilde M, \tilde g}$ be Riemannian manifolds with Riemannian metrics $g$ and $\tilde g$ respectively.
Let the mapping $\phi : M \to \tilde M$ be a diffeomorphism such that:
:$\phi^* \tilde g = g$
Then $\phi$ is called an '''isometry (from $\struct {M, g}$ to $\struct {\tilde M, \tilde g}$)'''.
\end{definition}