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

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

1. 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 $\{0\}$.
2. Conjugacy classes. For the conjugation action of $G$ on itself, $g\cdot h := ghg^{-1}$, the orbit of $h$ is its conjugacy class.
3. 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 $\{0\}$.