Coset space

The quotient space $G/H$ of left cosets, a smooth manifold when $H$ is a closed Lie subgroup.

Coset space

Let $G$ be a Lie group and let $H\le G$ be a subgroup.

Definition

The left coset space $G/H$ is the set of left cosets

G/H := \{gH : g\in G\},

equipped with the quotient topology for the projection $\pi:G\to G/H$ , $\pi(g)=gH$ .

If $H$ is a closed Lie subgroup (equivalently, a closed subgroup that is automatically an embedded Lie subgroup by the Closed Subgroup Theorem ), then $G/H$ carries a canonical smooth manifold structure characterized by:

$\pi:G\to G/H$ is a smooth surjective submersion;
the left action of $G$ on $G/H$ given by $g'\cdot (gH)=(g'g)H$ is a smooth Lie group action .

In this case, $G/H$ is a basic example of a homogeneous space : the action is transitive and the stabilizer of the basepoint $eH$ is exactly $H$ (compare stabilizer and transitive action ).

Tangent space identification

Let $\mathfrak g=\mathrm{Lie}(G)$ and $\mathfrak h=\mathrm{Lie}(H)$ (see Lie algebra of a Lie group ). Then

T_{eH}(G/H)\;\cong\;\mathfrak g/\mathfrak h,

canonically, via the differential $d\pi_e:\mathfrak g\to T_{eH}(G/H)$ whose kernel is $\mathfrak h$ .

Context. Many geometric objects (spheres, Grassmannians, flag varieties) arise naturally as $G/H$ ; understanding $G/H$ is often the first step in studying invariant tensors such as left-invariant forms or bi-invariant metrics on $G$ and their descendants on $G/H$ .