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
![SO(2)=[1 0 0; 0 cost -sint; 0 sint cost]](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FHomogeneousSpace%2FNumberedEquation4.svg) |
(4) |
acts on the right, and does not affect the first column so 
is well-defined.
See also
Effective Action, Free Action, Group, Group Orbit, Group Representation, Isotropy Group, Lie Group Quotient Space, Matrix Group, Topological Group, Transitive
This entry contributed by Todd Rowland
Explore with Wolfram|Alpha
Cite this as:
Rowland, Todd. "Homogeneous Space." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HomogeneousSpace.html