Proper action

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

The action is proper if the map

is a proper map (that is, the preimage of every compact subset of $M\times M$ is compact in $G\times M$).

Properness is a topological finiteness condition on how group elements can move points around. In particular, for a proper action the orbit space quotient space $M/G$ is Hausdorff, and all stabilizers are compact.

Examples

1. Compact groups act properly. If $G$ is compact, then any continuous (hence any smooth) action of $G$ on a Hausdorff space is proper.
2. Translations are proper. The translation action of $\mathbb{R}^n$ on $\mathbb{R}^n$, $a\cdot x=x+a$, is proper.
3. Non-example (periodic stabilizer). The action of $\mathbb{R}$ on $S^1$ by rotations $t\cdot e^{i\theta}=e^{i(\theta+t)}$ is not proper: the preimage of the diagonal $\{(x,x)\}$ contains $(2\pi\mathbb{Z})\times S^1$, which is not compact.