Homogeneous Space

A homogeneous space is a space with a transitive group action by a Lie group. Because a transitive group action implies that there is only one group orbit, is isomorphic to the quotient space where is the isotropy group . The choice of does not affect the isomorphism type of because all of the isotropy groups are conjugate.

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

(1)

and the complex projective space

(2)

The real Grassmannian of -dimensional subspaces in is

(3)

The projection makes a principal bundle on with fiber . 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 so is well-defined.