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.