Stiefel manifold

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". 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".

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:$V\_k(mathbb\; F^n)\; =\; left\{A\; in\; mathbb\; F^\{n\; imes\; k\}\; :\; A^\{ast\}A\; =\; 1\; ight\}.$

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:

:

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:The same holds for the action of the special unitary group on "V"_"k"(C^"n"):

pecial cases

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 at the right.

As a principal bundle

There is a natural projection:$p:\; V\_k(mathbb\; F^n)\; o\; G\_k(mathbb\; F^n)$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::

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.

Homotopy

The Stiefel manifolds fit into a family of fibrations $V\_\{k-1\}(Bbb\; R^\{n-1\})\; o\; V\_k(Bbb\; R^n)\; o\; S^\{n-1\}$, thus the first non-trivial homotopy group of the space $V\_k(Bbb\; R^n)$ is in dimension $n-k$. Moreover, $pi\_\{n-k\}\; V\_k(Bbb\; R^n)\; simeq\; Bbb\; Z$ if $n-k\; in\; 2\; Bbb\; Z$ 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.

ee also

*Flag manifold

References

*cite book | first = Allen | last = Hatcher | authorlink = Allen Hatcher | year = 2002 | title = Algebraic Topology | publisher = Cambridge University Press | id = ISBN 0-521-79540-0 | url = http://www.math.cornell.edu/~hatcher/AT/ATpage.html
*cite book | first = Dale | last = Husemoller | year = 1994 | title = Fibre Bundles | edition = (3rd ed.) | publisher = Springer-Verlag | location = New York | id = ISBN 0-387-94087-1