Free smooth Lie group action - Imported Knowlpedia Content

Let $G$ be a Lie group acting smoothly on a manifold $M$ via a smooth action $G\times M\to M$ , $(g,p)\mapsto g\cdot p$ .

Definition (Free action). The action is free if for every $p\in M$ , the stabilizer

G_p=\{g\in G: g\cdot p=p\}

is trivial, i.e. $G_p=\{e\}$ .

Equivalent characterizations

Equivalently, for each $p\in M$ , the orbit map $G\to M$ , $g\mapsto g\cdot p$ , is injective. It is then an injective immersion onto the orbit; it is an embedding, and hence identifies the orbit diffeomorphically with $G$ , when the action is also proper.

Remarks

Motivation. Free actions are the infinitesimal starting point for principal bundles: when an action is free and proper, the orbit space $M/G$ is a manifold and $M\to M/G$ becomes a principal $G$ -bundle; in the special case when the action is also transitive, $M$ is a principal homogeneous space.