Principal action

Let $G$ act smoothly on a manifold $M$.

A principal action is an action that is simultaneously a free action and a proper action .

Equivalently, a principal action is precisely the hypothesis under which the orbit space carries a canonical smooth structure making the projection $M\to M/G$ into a principal G-bundle (see quotient manifold ).

Examples

1. The defining action of a principal bundle. If $\pi:P\to B$ is a principal $G$-bundle, then the right $G$-action on $P$ is principal.
2. Integer translations. $\mathbb{Z}$ acts on $\mathbb{R}$ by $n\cdot x=x+n$; the action is free and proper, hence principal.
3. Hopf action. The standard $S^1$-action on $S^{2n+1}\subset\mathbb{C}^{n+1}$ by scalar multiplication is principal; the quotient is complex projective space.