Homogeneous Space

A homogeneous space in the smooth setting is a smooth manifold with a smooth transitive group action by a Lie group . Because a transitive group action implies that there is only one group orbit, is diffeomorphic to the quotient space , where is the isotropy group of a point and is a closed subgroup of . The choice of does not affect the diffeomorphism type of because all the isotropy groups are conjugate.

Many common spaces are homogeneous spaces, such as the hypersphere,

(1)

and complex projective space,

(2)

The real Grassmannian of -dimensional subspaces in is

(3)

The projection makes a principal bundle on with fiber and structure group . For example, is a bundle, i.e., a circle bundle, on the sphere. The subgroup

(4)

acts on the right and does not affect the first column vector, so is well-defined.