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. It is named after Swiss mathematician Eduard Stiefel. 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; 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.
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
As a homogeneous space
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.
In each case Vk(Fn) can be viewed as a homogeneous space:
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.
|k = 1|
|k = n−1|
|k = n|
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.
Given an orthogonal inclusion between vector spaces the image of a set of k orthonormal vectors is orthonormal, so there is an induced closed inclusion of Stiefel manifolds, and this is functorial. More subtly, given an n-dimensional vector space X, the dual basis construction gives a bijection between bases for X and bases for the dual space X*, which is continuous, and thus yields a homeomorphism of top Stiefel manifolds This is also functorial for isomorphisms of vector spaces.
As a principal bundle
There is a natural projection
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:
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.
The Stiefel manifolds fit into a family of fibrations , thus the first non-trivial homotopy group of the space Vk(Rn) is in dimension n - k. Moreover, if n - k ∈ 2Z or if k = 1. if n - k is odd and k > 1. This result is used in the obstruction-theoretic definition of Stiefel-Whitney classes.
- 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.
- Encyclopaedia of Mathematics » Stiefel manifold, Springer