Science Fair Project Encyclopedia
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.
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(n−k) so that
This shows that Vn−1,n is diffeomorphic to SO(n).
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