Editing Orthogonal group (section)

== Principal homogeneous space: Stiefel manifold ==
{{Main|Stiefel manifold}}
The [[principal homogeneous space]] for the orthogonal group {{math|O(''n'')}} is the [[Stiefel manifold]] {{math|''V<sub>n</sub>''('''R'''<sup>''n''</sup>)}} of [[Orthonormal basis|orthonormal bases]] (orthonormal [[k-frame|{{math|''n''}}-frames]]).

In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given an orthogonal space, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group. Concretely, a linear map is determined by where it sends a basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any ''orthogonal'' basis to any other ''orthogonal'' basis.

The other Stiefel manifolds {{math|''V<sub>k</sub>''('''R'''<sup>''n''</sup>)}} for {{math|''k'' < ''n''}} of ''incomplete'' orthonormal bases (orthonormal {{math|''k''}}-frames) are still homogeneous spaces for the orthogonal group, but not ''principal'' homogeneous spaces: any {{math|''k''}}-frame can be taken to any other {{math|''k''}}-frame by an orthogonal map, but this map is not uniquely determined.