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# Stiefel manifold

In some contexts, a non-compact Stiefel manifold is defined as the set of all linearly independent k-frames in Rn, Cn, or Hn; this is homotopy equivalent, as the compact Stiefel manifold is a deformation retract of the non-compact one, by Gram–Schmidt. Statements about the non-compact form correspond to those for the compact form, replacing the orthogonal group (or unitary or symplectic group) with the general linear group.

In mathematics, the **Stiefel manifold** *V*_{k}(**R**^{n}) is the set of all orthonormal *k*-frames in **R**^{n}. That is, it is the set of ordered *k*-tuples of orthonormal vectors in **R**^{n}. It is named after Swiss mathematician Eduard Stiefel. Likewise one can define the complex Stiefel manifold *V*_{k}(**C**^{n}) of orthonormal *k*-frames in **C**^{n} and the quaternionic Stiefel manifold *V*_{k}(**H**^{n}) of orthonormal *k*-frames in **H**^{n}. 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 **R**^{n}, **C**^{n}, or **H**^{n}; this is homotopy equivalent, as the compact Stiefel manifold is a deformation retract of the non-compact one, by Gram–Schmidt. Statements about the non-compact form correspond to those for the compact form, replacing the orthogonal group (or unitary or symplectic group) with the general linear group.

Topology

Let **F** stand for **R**, **C**, or **H**. The Stiefel manifold *V*_{k}(**F**^{n}) can be thought of as a set of *n* × *k* matrices by writing a *k*-frame as a matrix of *k* column vectors in **F**^{n}. 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 *V*_{k}(**F**^{n}) is the subspace topology inherited from **F**^{n×k}. With this topology *V*_{k}(**F**^{n}) is a compact manifold whose dimension is given by

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

\( \dim V_k(\mathbb C^n) = 2nk - k^2 \)

\( \dim V_k(\mathbb H^n) = 4nk - k(2k-1). \)

As a homogeneous space

Each of the Stiefel manifolds *V*_{k}(**F**^{n}) 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 **R**^{n} 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 *V*_{k}(**R**^{n}). 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 *V*_{k}(**C**^{n}) with stabilizer subgroup U(*n*−*k*) and the symplectic group Sp(*n*) acts transitively on *V*_{k}(**H**^{n}) with stabilizer subgroup Sp(*n*−*k*).

In each case *V*_{k}(**F**^{n}) 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 *V*_{n}(**F**^{n}) 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 *V*_{k}(**R**^{n}) 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)

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

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

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 |
\(\begin{align} V_{n}(\mathbb R^n) &\cong \mathrm O(n)\\ V_{n}(\mathbb C^n) &\cong \mathrm U(n)\\ V_{n}(\mathbb H^n) &\cong \mathrm{Sp}(n) \end{align} \) |

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.

A 1-frame in **F**^{n} is nothing but a unit vector, so the Stiefel manifold *V*_{1}(**F**^{n}) is just the unit sphere in **F**^{n}.

Given a 2-frame in **R**^{n}, let the first vector define a point in *S*^{n−1} and the second a unit tangent vector to the sphere at that point. In this way, the Stiefel manifold *V*_{2}(**R**^{n}) may be identified with the unit tangent bundle to *S*^{n−1}.

When *k* = *n* or *n*−1 we saw in the previous section that *V*_{k}(**F**^{n}) is a principal homogeneous space, and therefore diffeomorphic to the corresponding classical group. These are listed in the table above.

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 *V*_{k}(**F**^{n}) to the Grassmannian of *k*-planes in **F**^{n} which sends a *k*-frame to the subspace spanned by that frame. The fiber over a given point *P* in *G*_{k}(**F**^{n}) 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 *V*_{k}(**R**^{n}) 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 **F**^{k} are just the tautological bundles over the Grassmannians. In other words, the Stiefel manifold *V*_{k}(**F**^{n}) 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.

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Homotopy

The Stiefel manifolds fit into a family of fibrations \( V_{k-1}(\Bbb R^{n-1}) \to V_k(\Bbb R^n) \to S^{n-1} \) , thus the first non-trivial homotopy group of the space Vk(Rn) is in dimension n - k. Moreover, \( \pi_{n-k} V_k(\Bbb R^n) \simeq \Bbb Z \) if n - k ∈ 2Z or if k = 1. \( \pi_{n-k} V_k(\Bbb R^n) \simeq \Bbb Z_2 \) if 'n - k is odd and k > 1. This result is used in the obstruction-theoretic definition of Stiefel-Whitney classes.

See also

Flag manifold

References

Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0.

Husemoller, Dale (1994). Fibre Bundles ((3rd ed.) ed.). New York: Springer-Verlag. ISBN 0-387-94087-1.

James, Ioan Mackenzie (1976). The topology of Stiefel manifolds. CUP Archive. ISBN 978-0-521-21334-9.

External links

Encyclopaedia of Mathematics » Stiefel manifold, Springer

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