Stiefel manifold

In mathematics, the Stiefel manifold, denoted V_k(Rⁿ) or V_k,n, is the set of all orthonormal k-frames in Rⁿ. That is, it is the set of ordered k-tuples of orthonormal vectors in Rⁿ.

The Stiefel manifold V_k,n can be thought of as living inside the product of k copies on Sⁿ⁻¹. With the induced topology V_k,n becomes a manifold of dimension

$\dim V_{k,n} = \sum_{i=1}^{k}(n-i) = nk - \frac{1}{2}k(k+1)$

When k = 1, the manifold V_1,n is just the set of unit vectors in Rⁿ; that is, V_1,n is diffeomorphic to the n − 1 sphere, Sⁿ⁻¹. At the other extreme, when k = n, the Stiefel manifold V_n,n is the set of all ordered orthonormal bases for Rⁿ. The orthogonal group O(n) acts simply transitively on this space, so that V_n,n is a principal homogeneous space for O(n) and therefore diffeomorphic to it.

In general, the orthogonal group O(n) acts transitively on V_k,n with stabilizer subgroup isomorphic to O(n−k). Therefore V_k,n can be viewed as the homogeneous space

$V_{k,n} \cong \mbox{O}(n)/\mbox{O}(n-k)$

If k is strictly less than n then the special orthogonal group SO(n) also acts transitively on V_k,n with stabilizer subgroup isomorphic to SO(n−k) so that

$V_{k,n} \cong \mbox{SO}(n)/\mbox{SO}(n-k)\qquad\mbox{for } k < n$

This shows that V_n−1,n is diffeomorphic to SO(n).

Stiefel manifold

See also

See also: Stiefel manifold, Diffeomorphic, Flag manifold, Grassmannian, Group action, Homogeneous space, Manifold, Mathematics, Orthogonal group