Proper action

Let a Lie group $G$ act smoothly on a manifold $M$ (see smooth Lie group action). The action is proper if the map

is a proper map (preimages of compact sets are compact).

A frequently used equivalent criterion is: for every pair of compact subsets $K_1,K_2\subset M$, the transporter set

is compact in $G$.

Properness is a key hypothesis for turning orbit spaces into reasonable geometric objects (compare orbit space). Standard consequences include:

• each stabilizer $G_m$ (see stabilizer) is compact;
• each orbit is an embedded, closed submanifold of $M$ (see orbits of Lie group actions);
• the quotient topology on $M/G$ is Hausdorff.

If, in addition, the action is free (see free action), then $M\to M/G$ is a smooth submersion and the quotient inherits a smooth manifold structure; in many settings this is the first step toward principal bundle geometry (compare principal homogeneous spaces).