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 $\pi : \tilde M \to \tilde M ~/~ \Gamma$ be a quotient map.

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

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

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