Transitive Lie group action

Definition

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

Equivalently, for some (hence every) $x\in M$, the orbit $G\cdot x$ equals all of $M$.

Homogeneous space description

Fix $x_0\in M$ and let $H=G_{x_0}$ be the stabilizer subgroup . Then $H$ is a Lie subgroup (by the closed subgroup theorem ), and the orbit map induces a smooth surjection

For transitive actions this map is a diffeomorphism under standard hypotheses, so $M$ is a homogeneous space (compare coset spaces ).

Context

Transitive actions encode “geometries with a large symmetry group.” Many classical manifolds arise as homogeneous spaces, and questions about invariants on $M$ can often be translated into representation-theoretic questions about $H$.