Smooth action of a Lie group

Free: if g ⋅ p = p g\cdot p=p g ⋅ p = p for some p ∈ P p\in P p ∈ P , then g = e g=e g = e .
Transitive: for any p , q ∈ P p,q\in P p , q ∈ P , there exists g ∈ G g\in G g ∈ G with g ⋅ p = q g\cdot p=q g ⋅ p = q .

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

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

such that:

Associated constructions

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

(X_M)_m = \left.\frac{d}{dt}\right|_{t=0}\exp(tX)\cdot m,

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.