Homogeneous space

Let $G$ be a Lie group acting smoothly on a manifold $M$ (see smooth action ).

Definition (Homogeneous space).
$M$ is a homogeneous space for $G$ if the action is transitive , i.e. for any $p,q\in M$ there exists $g\in G$ with $g\cdot p=q$.

Fix $p\in M$ and let $H=G_p$ be its stabilizer . Then the orbit map $G\to M$, $g\mapsto g\cdot p$, induces a bijection

where $G/H$ is the coset space . If $H$ is a closed subgroup, then $G/H$ carries a unique smooth manifold structure making the induced map a $G$-equivariant diffeomorphism (compare the closed subgroup theorem ).

Special case.
If the action is also free , then $H$ is trivial and $M$ is (noncanonically) identified with $G$; in the free-and-transitive case, $M$ is a principal homogeneous space .

Context.
Homogeneous spaces provide the basic examples of manifolds built from Lie groups (spheres, Grassmannians, flag varieties), and their geometry is controlled by the subgroup $H$ and the induced action on tangent spaces.