Conditions for Smooth Normal Covering Map to be Riemannian Covering

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

Let $\pi : \tilde M \to M$ be a smooth normal covering map.

Let $\tilde g$ be invariant under all covering automorphisms.

Then there exists a unique $g$ such that $\pi$ is a Riemannian covering.