Stiefel manifold

In mathematics, the Stiefel manifold formula_1 is the set of all orthonormal "k"-frames in formula_2 That is, it is the set of ordered orthonormal "k"-tuples of vectors in formula_2 It is named after Swiss mathematician Eduard Stiefel. Likewise one can define the complex Stiefel manifold formula_4 of orthonormal "k"-frames in formula_5 and the quaternionic Stiefel manifold formula_6 of orthonormal "k"-frames in formula_7. 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 formula_8 or formula_9 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 formula_10 stand for formula_11 or formula_12 The Stiefel manifold formula_13 can be thought of as a set of "n" × "k" matrices by writing a "k"-frame as a matrix of "k" column vectors in formula_14 The orthonormality condition is expressed by "A"*"A" = formula_15 where "A"* denotes the conjugate transpose of "A" and formula_16 denotes the "k" × "k" identity matrix. We then have
The topology on formula_13 is the subspace topology inherited from formula_19 With this topology formula_13 is a compact manifold whose dimension is given by
As a homogeneous space.
Each of the Stiefel manifolds formula_13 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 formula_23 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 formula_24 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 formula_4 with stabilizer subgroup U("n"−"k") and the symplectic group Sp("n") acts transitively on formula_6 with stabilizer subgroup Sp("n"−"k").
In each case formula_13 can be viewed as a homogeneous space:
When "k" = "n", the corresponding action is free so that the Stiefel manifold formula_29 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 formula_1 with stabilizer subgroup isomorphic to SO("n"−"k") so that
The same holds for the action of the special unitary group on formula_4
Thus for "k" = "n" − 1, the Stiefel manifold is a principal homogeneous space for the corresponding "special" classical group.
Uniform measure.
The Stiefel manifold can be equipped with a uniform measure, i.e. a Borel measure that is invariant under the action of the groups noted above. For example, formula_34 which is isomorphic to the unit circle in the Euclidean plane, has as its uniform measure the obvious uniform measure (arc length) on the circle. It is straightforward to sample this measure on formula_13 using Gaussian random matrices: if formula_36 is a random matrix with independent entries identically distributed according to the standard normal distribution on formula_10 and "A" = "QR" is the QR factorization of "A", then the matrices, formula_38 are independent random variables and "Q" is distributed according to the uniform measure on formula_39 This result is a consequence of the Bartlett decomposition theorem.
Special cases.
A 1-frame in formula_40 is nothing but a unit vector, so the Stiefel manifold formula_41 is just the unit sphere in formula_42 Therefore:
Given a 2-frame in formula_44 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 formula_45 may be identified with the unit tangent bundle 
When "k" = "n" or "n"−1 we saw in the previous section that formula_46 is a principal homogeneous space, and therefore diffeomorphic to the corresponding classical group:
Functoriality.
Given an orthogonal inclusion between vector spaces formula_49 the image of a set of "k" orthonormal vectors is orthonormal, so there is an induced closed inclusion of Stiefel manifolds, formula_50 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 formula_51 which is continuous, and thus yields a homeomorphism of top Stiefel manifolds formula_52 This is also functorial for isomorphisms of vector spaces.
As a principal bundle.
There is a natural projection
from the Stiefel manifold formula_13 to the Grassmannian of "k"-planes in formula_40 which sends a "k"-frame to the subspace spanned by that frame. The fiber over a given point "P" in formula_56 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 formula_1 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 formula_59 are just the tautological bundles over the Grassmannians. In other words, the Stiefel manifold formula_13 is the orthogonal, unitary, or symplectic frame bundle associated to the tautological bundle on a Grassmannian.
When one passes to the formula_61 limit, these bundles become the universal bundles for the classical groups.
Homotopy.
The Stiefel manifolds fit into a family of fibrations: 
thus the first non-trivial homotopy group of the space formula_1 is in dimension "n" − "k". Moreover, 
This result is used in the obstruction-theoretic definition of Stiefel–Whitney classes.