Orbit of a Lie group action

The set of points reachable from x under the action; it is an immersed homogeneous space.

Orbit of a Lie group action

Let $G$ be a Lie group acting smoothly on a manifold $M$ ; see smooth Lie group actions . Fix $x\in M$ .

Definition

The orbit of $x$ is

G\cdot x=\{g\cdot x \mid g\in G\}\subseteq M.

The orbit map is $\Phi_x:G\to M$ , $\Phi_x(g)=g\cdot x$ .

The stabilizer (isotropy group) is

G_x=\{g\in G\mid g\cdot x=x\},

a subgroup of $G$ ; it is a Lie subgroup because it is closed and closed subgroups are Lie subgroups .

Theorem (orbit as a homogeneous space)

The orbit map $\Phi_x$ descends to a smooth map

\overline{\Phi}_x: G/G_x \to M

from the coset space $G/G_x$ , whose image is $G\cdot x$ . Moreover:

$\overline{\Phi}_x$ is an immersion, and $G\cdot x$ is an immersed submanifold of $M$ .
If the action is proper , then $\overline{\Phi}_x$ is an embedding, so each orbit is an embedded submanifold.

In particular, each orbit carries a canonical structure of a homogeneous space for $G$ .

Tangent space description

The differential of the orbit map at the identity gives the infinitesimal action map

d(\Phi_x)_e:\mathfrak g \to T_xM,

and its image equals $T_x(G\cdot x)$ . This identifies tangent vectors to the orbit with values at $x$ of the fundamental vector fields generated by elements of $\mathfrak g$ .

Context

Orbit geometry is the local building block of the orbit space $M/G$ , and transitivity of the action is exactly the condition that there is a single orbit (see transitive actions ).