Conditions for Subjective Smooth Submersion from Riemannian Manifold to its Orbit Space to be Riemannian Submersion

Theorem

Let $\struct {\tilde M, \tilde g}$ be a Riemannian manifold.

Let $\pi : \tilde M \to M$ be a surjective smooth submersion.

Let $G$ be a Lie group acting on $\tilde M$.

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

Let $M = \tilde M \setminus G$ be the orbit space.

Then $M$ has a unique smooth manifold structure and Riemannian metric $g$ such that $\pi$ is a Riemannian submersion.

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