Orbit map

The smooth map from a Lie group to a manifold sending a group element to its action on a fixed point.

Orbit map

Let $G$ act on $M$ by a smooth action .

For $x\in M$ , the orbit map at $x$ is the smooth map

\Phi^x: G \longrightarrow M,\qquad \Phi^x(g)=g\cdot x.

Its image is the orbit $G\cdot x$ .

The kernel of $\Phi^x$ (in the sense of elements acting trivially at $x$ ) is the stabilizer subgroup $G_x$ . Consequently, $\Phi^x$ is constant on left cosets of $G_x$ and factors through the quotient $G/G_x$ .

Examples

Left translation on $G$ . For the action of $G$ on itself by left multiplication and a fixed $h\in G$ , the orbit map is $g\mapsto gh$ .
Rotations of a vector. For $SO(2)\curvearrowright \mathbb{R}^2$ and $x\neq 0$ , the orbit map parametrizes the circle of radius $\|x\|$ .
Conjugation. For the conjugation action of $G$ on itself, the orbit map at $h$ is $g\mapsto ghg^{-1}$ , whose image is the conjugacy class of $h$ .