Stiefel Manifold - As A Homogeneous Space

As A Homogeneous Space

Each of the Stiefel manifolds Vk(Fn) can be viewed as a homogeneous space for the action of a classical group in a natural manner.

Every orthogonal transformation of a k-frame in Rn results in another k-frame, and any two k-frames are related by some orthogonal transformation. In other words, the orthogonal group O(n) acts transitively on Vk(Rn). The stabilizer subgroup of a given frame is the subgroup isomorphic to O(n−k) which acts nontrivially on the orthogonal complement of the space spanned by that frame.

Likewise the unitary group U(n) acts transitively on Vk(Cn) with stabilizer subgroup U(n−k) and the symplectic group Sp(n) acts transitively on Vk(Hn) with stabilizer subgroup Sp(n−k).

In each case Vk(Fn) can be viewed as a homogeneous space:

\begin{align} V_k(\mathbb R^n) &\cong \mbox{O}(n)/\mbox{O}(n-k)\\ V_k(\mathbb C^n) &\cong \mbox{U}(n)/\mbox{U}(n-k)\\ V_k(\mathbb H^n) &\cong \mbox{Sp}(n)/\mbox{Sp}(n-k). \end{align}

When k = n, the corresponding action is free so that the Stiefel manifold Vn(Fn) is a principal homogeneous space for the corresponding classical group.

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

The same holds for the action of the special unitary group on Vk(Cn)

Thus for k = n - 1, the Stiefel manifold is a principal homogeneous space for the corresponding special classical group.

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