quotient manifold theorem

Statement

Suppose $G$ is a Lie group with an action on a smooth manifold $M$ that is smooth, free, and proper. Then the orbit space $M/G$ is a topological manifold of dimension equal to $\dim M-\dim G$, and has a unique structure with the property that the projection map $\pi :M\to M/G$ is a smooth submersion.

Proof

todo

Notes

The open sets are easy to understand in $M/G$. It is constructed so that $\pi :M\to M/G$ is a quotient map so in the quotient topology, a set $V\subset M/G$ is open if ${\pi }^{-1}(V)\subset M$ is open.

Also, if $G$ is compact, then the proper condition is unnecessary.