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Stiefel manifold

In mathematics, the Stiefel manifold, denoted Vk(Rn) or Vk,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 Vk,n can be thought of as living inside the product of k copies on Sn−1. With the induced topology Vk,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 V1,n is just the set of unit vectors in Rn; that is, V1,n is diffeomorphic to the n − 1 sphere, Sn−1. At the other extreme, when k = n, the Stiefel manifold Vn,n is the set of all ordered orthonormal bases for Rn. The orthogonal group O(n) acts simply transitively on this space, so that Vn,n is a principal homogeneous space for O(n) and therefore diffeomorphic to it.

In general, the orthogonal group O(n) acts transitively on Vk,n with stabilizer subgroup isomorphic to O(nk). Therefore Vk,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 Vk,n with stabilizer subgroup isomorphic to SO(nk) so that

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

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