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 $\tilde M ~/~\Gamma$.

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics