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.

Read more about this topic:  Stiefel Manifold

Famous quotes containing the words special and/or cases:

    “Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
    —Walter Pater (1839–1894)

    “Medication alone is not to be relied on. In one half the cases medicine is not needed, or is worse than useless. Obedience to spiritual and physical laws—hygeine [sic] of the body, and hygeine of the spirit—is the surest warrant for health and happiness.”
    —Harriot K. Hunt (1805–1875)