Orbit of a group action

Consider a smooth action of a Lie group $G$ on a manifold $M$ .

For $x\in M$ , the orbit of $x$ (under the action of $G$ ) is the subset

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

Orbit maps and equivalence classes

Equivalently, $G\cdot x$ is the image of the orbit map $\Phi^x:G\to M$ , $g\mapsto g\cdot x$ .

Two points $x,y\in M$ lie in the same orbit if and only if there exists $g\in G$ with $y=g\cdot x$ ; this is an equivalence relation whose equivalence classes are precisely the orbits, and the corresponding quotient is the orbit space $M/G$ .

Examples

Rotations in the plane. For the $SO(2)$ -action on $\mathbb{R}^2$ , the orbit of a nonzero vector is a circle of radius $\|x\|$ , while the orbit of $0$ is $\{0\}$ .
Conjugacy classes. For the conjugation action of $G$ on itself, $g\cdot h := ghg^{-1}$ , the orbit of $h$ is its conjugacy class.
Linear actions. For the natural $GL(n,\mathbb{R})$ -action on $\mathbb{R}^n$ , the orbit of any nonzero vector is $\mathbb{R}^n\setminus\{0\}$ , and the orbit of $0$ is $\{0\}$ .