Quotient manifold (for a free proper action)

Let a Lie group $G$ act smoothly, freely, and properly on a smooth manifold $M$ . Then the orbit space $M/G$ admits a unique smooth manifold structure for which the quotient map $\pi:M\to M/G$ is a smooth submersion. Moreover:

The fibers of $\pi$ are the orbits, and $\dim(M/G)=\dim(M)-\dim(G)$ .
With this structure, $\pi:M\to M/G$ is a principal $G$ -bundle, after converting a given left action to the corresponding right action if necessary.

In particular, local differential geometry on $M/G$ can be studied through $G$ -invariant data on $M$ .

Examples

Hopf fibration. The free proper $S^1$ -action on $S^{2n+1}$ by scalar multiplication yields the quotient manifold $S^{2n+1}/S^1 \cong \mathbb{CP}^n$ .
Positive scalings. For $n\ge1$ , the action of $\mathbb{R}_{>0}$ on $\mathbb{R}^n\setminus\{0\}$ by $t\cdot x = tx$ is free and proper; the quotient is diffeomorphic to $S^{n-1}$ .
Covering space quotient. The free proper action of $\mathbb{Z}$ on $\mathbb{R}$ by translations gives the quotient manifold $\mathbb{R}/\mathbb{Z}\cong S^1$ .