\begin{definition}[Definition:Local Isometry]
Let $\struct {M, g}$ and $\struct {\tilde M, \tilde g}$ be Riemannian manifolds.
Let $p \in M$ be a point.
Let $\phi : M \to \tilde M$ be a mapping such that for each $p$ there is a neighborhood $U$ such that $\valueat \phi U$ is an isometry onto an open subset of $\tilde M$.
Then $\phi$ is called the '''local isometry'''.
\end{definition}