Homogeneous Space - Homogeneous Spaces As Coset Spaces

Homogeneous Spaces As Coset Spaces

In general, if X is a homogeneous space, and H_o is the stabilizer of some marked point o in X (a choice of origin), the points of X correspond to the left cosets G/H_o, and the marked point o corresponds to the coset of the identity. Conversely, given a coset space G/H, it is a homogeneous space for G with a distinguished point, namely the coset of the identity. Thus a homogeneous space can be thought of as a coset space without a choice of origin.

In general, a different choice of origin o will lead to a quotient of G by a different subgroup H_o′ which is related to H_o by an inner automorphism of G. Specifically,

(1)

where g is any element of G for which go = o′. Note that the inner automorphism (1) does not depend on which such g is selected; it depends only on g modulo H_o.

If the action of G on X is continuous, then H is a closed subgroup of G. In particular, if G is a Lie group, then H is a closed Lie subgroup by Cartan's theorem. Hence G/H is a smooth manifold and so X carries a unique smooth structure compatible with the group action.

If H is the identity subgroup {e}, then X is a principal homogeneous space.

One can go further to double coset spaces, notably Clifford–Klein forms Γ\G/H, where Γ is a discrete subgroup (of G) acting properly discontinuously.