Transitive Lie group action

Let $G$ be a Lie group acting smoothly on a manifold $M$ . The action is transitive if, for every $x,y\in M$ , there exists $g\in G$ such that

g\cdot x = y.

Homogeneous space description

Fix $x_0\in M$ , and let $H=G_{x_0}$ be its stabilizer subgroup. The subgroup $H$ is closed, and the orbit map induces

G/H \to M,\quad gH\mapsto g\cdot x_0.

For a smooth transitive action, this map is a diffeomorphism, so $M$ is the homogeneous space $G/H$ .

Context

Transitive actions encode geometries with a large symmetry group. Many classical manifolds arise as homogeneous spaces, and invariants on $M$ can often be studied through the stabilizer $H$ .

Equivalent characterizations

Equivalently, for some—and hence every— $x\in M$ , the orbit $G\cdot x$ is all of $M$ .