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.

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