Quotient manifold (for a free proper action)

Let Lie group $G$ act smoothly on a smooth manifold $M$.

Theorem (quotient manifold theorem)

Assume the action is a principal action (equivalently, smooth, free, and proper). Then:

1. The orbit space $M/G$ admits a unique smooth manifold structure such that the quotient map $\pi:M\to M/G$ is a smooth map and a submersion.
2. The fibers of $\pi$ are the orbits, and $\dim(M/G)=\dim(M)-\dim(G)$.
3. With this structure, $\pi:M\to M/G$ is a principal G-bundle whose right action is the given action (up to the usual left/right convention).

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

Examples

1. Hopf fibration. The free proper $S^1$-action on $S^{2n+1}$ by scalar multiplication yields the quotient manifold $S^{2n+1}/S^1 \cong \mathbb{CP}^n$.
2. Positive scalings. 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 $S^{n-1}$.
3. Covering space quotient. The free proper action of $\mathbb{Z}$ on $\mathbb{R}$ by translations gives the quotient manifold $\mathbb{R}/\mathbb{Z}\cong S^1$.