Homogeneous Space
A homogeneous space M 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, M is isomorphic to the quotient space G/H where H is the isotropy group G_x. The choice of x in M does not affect the isomorphism type of G/G_x because all of the isotropy groups are conjugate.
Many common spaces are homogeneous spaces, such as the hypersphere,
| S^n∼O(n+1)/O(n), |
(1)
|
and the complex projective space
| CP^n∼U(n+1)/U(n)×U(1). |
(2)
|
The real Grassmannian of k-dimensional subspaces in R^(n+k) is
| O(n+k)/O(n)×O(k). |
(3)
|
The projection pi:G->G/H makes G a principal bundle on G/H with fiber H. For example, pi:SO(3)->SO(3)/SO(2)∼S^2 is a SO(2) bundle, i.e., a circle bundle, on the sphere. The subgroup
![画像: SO(2)=[1 0 0; 0 cost -sint; 0 sint cost]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/HomogeneousSpace/NumberedEquation4.svg){kind=link}
acts on the right, and does not affect the first column so pi(v_1v_2v_3)=v_1 in S^2 is well-defined.
See also
Effective Action, Free Action, Group, Group Orbit, Group Representation, Isotropy Group, Lie Group Quotient Space, Matrix Group, Topological Group, TransitiveThis entry contributed by Todd Rowland
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Rowland, Todd. "Homogeneous Space." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HomogeneousSpace.html