Smooth action of a Lie group

Let $G$ be a Lie group and $M$ a smooth manifold. A smooth (left) action of $G$ on $M$ is a smooth map

such that:

1. $e\cdot m = m$ for all $m\in M$ (where $e$ is the identity in $G$), and
2. $(g_1g_2)\cdot m = g_1\cdot(g_2\cdot m)$ for all $g_1,g_2\in G$ and $m\in M$.

Associated to any smooth action are the basic orbit-stabilizer constructions: the orbit $G\cdot m$ (see orbit ) and the stabilizer subgroup $G_m$ (see stabilizer ). If the action is transitive, $M$ becomes a homogeneous space ; if it is free, it resembles a principal homogeneous space (compare transitive action and free action ).

Differentiating at the identity produces an infinitesimal action: each $X\in\mathfrak g=\mathrm{Lie}(G)$ defines a vector field $X_M$ on $M$ by

linking smooth actions to one-parameter subgroups and the exponential map . The kernel of the action homomorphism $G\to \mathrm{Diff}(M)$ measures whether the action is effective .