Conditions for Quotient Map from Riemannian Manifold to its Quotient by Discrete Lie Group to be Normal Riemannian Covering

Theorem
Let $\struct {\tilde M, \tilde g}$ be a Riemannian manifold.

Let $\Gamma$ be a discrete Lie group.

Let $\pi : \tilde M \to \tilde M \setminus \Gamma$ be a quotient map.

Suppose the action of $\Gamma$ is smooth, free, proper, and isometric.

Then $\tilde M \setminus \Gamma$ has a unique Riemannian metric $g$ such that $\pi$ is a normal Riemannian covering.