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

The Stiefel manifold V k , n can be thought of as living inside the product (topology) of k copies on S n −1. With the subspace (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 R n ; that is, V 1, n is diffeomorphic to the n − 1 sphere, S n −1. At the other extreme, when k = n , the Stiefel manifold V n , n is the set of all ordered orthonormal bases for R n . 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 ) group action 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)qquadmbox{for } k < n This shows that V n −1, n is diffeomorphic to SO ( n ).

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