Definition
A homogeneous space of a Lie group is a smooth manifold with a differentiable transitive left action of on .
Relation to stabilizer groups
Let be a Lie group, and a -homogeneous space.
Then the map
is well-defined and a diffeomorphism for the isotropy subgroup for .
Proof
Well-defined: Let , that is for .
Then
Diffeomorphism:
First, we show that is injective.
Let such that , that is .
This means
Therefore, , so , and .
Next, since the action is transitive, then the action map
is surjective, and this bijective.
Additionally, it is smooth, so is also smooth.
Finally, since is equivariant, then it has constant rank, so by the constant rank theorem it is a diffeomorphism.
Examples
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