Stiefel manifold

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In mathematics, the Stiefel manifold Vk(Rn) is the set of all orthonormal k-frames in Rn. That is, it is the set of ordered k-tuples of orthonormal vectors in Rn. Likewise one can define the complex Stiefel manifold Vk(Cn) of orthonormal k-frames in Cn and the quaternionic Stiefel manifold Vk(Hn) of orthonormal k-frames in Hn. More generally, the construction applies to any real, complex, or quaternionic inner product space.

In some contexts, a non-compact Stiefel manifold is defined as the set of all linearly independent k-frames in Rn, Cn, or Hn.

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[edit] Topology

Let F stand for R, C, or H. The Stiefel manifold Vk(Fn) can be thought of as a set of n × k matrices by writing a k-frame as a matrix of k column vectors in Fn. The orthonormality condition is expressed by A*A = 1 where A* denotes the conjugate transpose of A and 1 denotes the k × k identity matrix. We then have

V_k(\mathbb F^n) = \left\{A \in \mathbb F^{n\times k} : A^{\ast}A = 1\right\}.

The topology on Vk(Fn) is the subspace topology inherited from Fn×k. With this topology Vk(Fn) is a compact manifold whose dimension is given by

\dim V_k(\mathbb R^n) = nk - \frac{1}{2}k(k+1)
\dim V_k(\mathbb C^n) = 2nk - k^2
\dim V_k(\mathbb H^n) = 4nk - k(2k-1)

[edit] 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

V_k(\mathbb R^n) \cong \mbox{SO}(n)/\mbox{SO}(n-k)\qquad\mbox{for } k < n.

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

V_k(\mathbb C^n) \cong \mbox{SU}(n)/\mbox{SU}(n-k)\qquad\mbox{for } k < n.

[edit] 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.

[edit] As a principal bundle

There is a natural projection

p: V_k(\mathbb F^n) \to G_k(\mathbb F^n)

from the Stiefel manifold Vk(Fn) to the Grassmannian of k-planes in Fn which sends a k-frame to the subspace spanned by that frame. The fiber over a given point P in Gk(Fn) is the set of all orthonormal k-frames contained in the space P.

This projection has the structure of a principal G-bundle where G is the associated classical group of degree k. Take the real case for concreteness. There is a natural right action of O(k) on Vk(Rn) which rotates a k-frame in the space it spans. This action is free but not transitive. The orbits of this action are precisely the orthonormal k-frames spanning a given k-dimensional subspace; that is, they are the fibers of the map p. Similar arguments hold in the complex and quaternionic cases.

We then have a sequence of principal bundles:

\begin{align} \mathrm O(k) &\to V_k(\mathbb R^n) \to G_k(\mathbb R^n)\\ \mathrm U(k) &\to V_k(\mathbb C^n) \to G_k(\mathbb C^n)\\ \mathrm{Sp}(k) &\to V_k(\mathbb H^n) \to G_k(\mathbb H^n) \end{align}

The vector bundles associated to these principal bundles via the natural action of G on Fk are just the tautological bundles over the Grassmannians. In other words, the Stiefel manifold Vk(Fn) is the orthogonal, unitary, or symplectic frame bundle associated to the tautological bundle on a Grassmannian.

When one passes to the n → ∞ limit, these bundles become the universal bundles for the classical groups.

[edit] See also

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