Free action

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

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

Equivalently, for each $p\in M$, the orbit map $G\to M$, $g\mapsto g\cdot p$ is injective, so each orbit is diffeomorphic to $G$ as a set. (For finer geometric conclusions, freeness is often paired with properness .)

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 .