\begin{definition}[Definition:Riemannian Covering]
Let $\struct {M, g}$ and $\struct {\tilde M, \tilde g}$ be Riemannian manifolds.
Let $\pi : \tilde M \to M$ be a smooth covering map.
Suppose $\pi$ is a local isometry.
Then $\pi$ is called a '''Riemannian covering'''.
\end{definition}