Orbit map - Knowlpedia

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.

Image and stabilizer

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

The fiber $(\Phi^x)^{-1}(x)$ is the stabilizer subgroup $G_x$ . The orbit map is constant on cosets $gG_x$ , because $\Phi^x(gh)=g\cdot x$ for $h\in G_x$ , and hence factors through $G/G_x$ .

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 $h$ .