Stiefel Manifold - Special Cases

Special Cases

k = 1 \begin{align} V_1(\mathbb R^n) &= S^{n-1}\\ V_1(\mathbb C^n) &= S^{2n-1}\\ V_1(\mathbb H^n) &= S^{4n-1} \end{align}
k = n−1 \begin{align} V_{n-1}(\mathbb R^n) &\cong \mathrm{SO}(n)\\ V_{n-1}(\mathbb C^n) &\cong \mathrm{SU}(n) \end{align}
k = n \begin{align} V_{n}(\mathbb R^n) &\cong \mathrm O(n)\\ V_{n}(\mathbb C^n) &\cong \mathrm U(n)\\ V_{n}(\mathbb H^n) &\cong \mathrm{Sp}(n) \end{align}

A 1-frame in Fn is nothing but a unit vector, so the Stiefel manifold V1(Fn) is just the unit sphere in Fn.

Given a 2-frame in Rn, let the first vector define a point in Sn−1 and the second a unit tangent vector to the sphere at that point. In this way, the Stiefel manifold V2(Rn) may be identified with the unit tangent bundle to Sn−1.

When k = n or n−1 we saw in the previous section that Vk(Fn) is a principal homogeneous space, and therefore diffeomorphic to the corresponding classical group. These are listed in the table at the right.

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