Conditions for Connected Riemannian Manifold to be Isometric to Quotient of Connected Riemannian Manifold by Covering Automorphism Group

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

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

Let $\Gamma = \map {\operatorname {Aut}_\pi} {\tilde M}$ be the covering automorphism group.

Then $M$ is isometric to $\tilde M ~/~\Gamma$.