Proper action

A smooth Lie group action is proper if the action graph map is proper; this guarantees good quotient behavior.

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

\alpha: G\times M \longrightarrow M\times M,\qquad (g,m)\mapsto (g\cdot m,\, m)

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

\{g\in G : gK_1\cap K_2\neq \varnothing\}

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 ).