That is, it is the set of ordered orthonormal k-tuples of vectors in
It is named after Swiss mathematician Eduard Stiefel.
Likewise one can define the complex Stiefel manifold
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
this is homotopy equivalent to the more restrictive definition, as the compact Stiefel manifold is a deformation retract of the non-compact one, by employing the Gram–Schmidt process.
can be thought of as a set of n × k matrices by writing a k-frame as a matrix of k column vectors in
can be viewed as a homogeneous space for the action of a classical group in a natural manner.
In other words, the orthogonal group O(n) acts transitively on
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
with stabilizer subgroup U(n−k) and the symplectic group Sp(n) acts transitively on
can be viewed as a homogeneous space: When k = n, the corresponding action is free so that the Stiefel manifold
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
with stabilizer subgroup isomorphic to SO(n−k) so that The same holds for the action of the special unitary group on
Thus for k = n − 1, the Stiefel manifold is a principal homogeneous space for the corresponding special classical group.
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.
which is isomorphic to the unit circle in the Euclidean plane, has as its uniform measure the natural uniform measure (arc length) on the circle.
are independent random variables and Q is distributed according to the uniform measure on
This result is a consequence of the Bartlett decomposition theorem.
is a principal homogeneous space, and therefore diffeomorphic to the corresponding classical group: 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,
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
which is continuous, and thus yields a homeomorphism of top Stiefel manifolds
which sends a k-frame to the subspace spanned by that frame.
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.
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.
The Stiefel manifolds fit into a family of fibrations: thus the first non-trivial homotopy group of the space
is in dimension n − k. Moreover, This result is used in the obstruction-theoretic definition of Stiefel–Whitney classes.