Homogeneous Spaces As Coset Spaces
In general, if X is a homogeneous space, and Ho is the stabilizer of some marked point o in X (a choice of origin), the points of X correspond to the left cosets G/Ho, 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 Ho′ which is related to Ho 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 Ho.
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.
Read more about this topic: Homogeneous Space
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