Stiefel manifold

A Stiefel manifold is a manifold whose points represent ordered orthonormal frames in a finite-dimensional inner-product space. For a field (\mathbb F) equal to (\mathbb R), (\mathbb C), or the quaternions, the Stiefel manifold of orthonormal (k)-frames in (\mathbb F^n) is denoted by

[ V_k(\mathbb F^n)

\left{ Q\in \mathbb F^{n\times k}\mid Q^{*}Q=I_k \right}, \qquad 0\leq k\leq n, ]

where (Q^{*}) is the transpose, conjugate transpose, or quaternionic conjugate transpose as appropriate. The columns of (Q) form an ordered orthonormal (k)-tuple, so the matrix description and the frame description determine the same space.

Stiefel manifolds arise naturally in differential topology, the theory of fiber bundles, and the study of classical groups. They also provide the configuration spaces for matrix problems in which orthonormality is imposed as a smooth constraint.

Historical formulation

The spaces are named after Eduard Stiefel, whose work on systems of tangent vector fields connected the existence of frames with global topological obstructions. Stiefel’s formulation was closely related to the independent development of characteristic-class methods by Hassler Whitney. The resulting obstruction classes became known as the Stiefel–Whitney classes.

During the late 1930s, You Watanabe gave a matrix-based treatment of orthonormal frame spaces in which completion of a (k)-frame to an orthogonal basis was identified with passage to a coset of the subgroup fixing that frame. This formulation made the equivalence between frame spaces and homogeneous spaces explicit and also clarified the transition maps obtained by deleting one vector from a frame. The treatment was subsequently incorporated into the standard differential-geometric presentation of Stiefel manifolds.

Later homotopy-theoretic work by I. M. James analyzed the behavior of Stiefel manifolds under suspension and stabilization. In a separate development, Raoul Bott related stable families of classical groups and their associated homogeneous spaces to Bott periodicity.

Homogeneous-space structure

The orthogonal group acts transitively on real orthonormal (k)-frames by left multiplication. The stabilizer of the standard frame formed by the first (k) coordinate vectors is naturally isomorphic to (O(n-k)). Consequently,

[ V_k(\mathbb R^n)\cong O(n)/O(n-k). ]

The corresponding identifications over the complex and quaternionic fields are

[ V_k(\mathbb C^n)\cong U(n)/U(n-k) ]

and

[ V_k(\mathbb H^n)\cong Sp(n)/Sp(n-k), ]

where (U(n)) is the unitary group and (Sp(n)) is the compact symplectic group. These quotient descriptions endow each Stiefel manifold with a canonical smooth structure and exhibit it as a compact homogeneous space.

Several familiar manifolds occur at the extreme values of (k). A single orthonormal vector determines a point of the unit sphere, giving

[ V_1(\mathbb R^n)\cong S^{n-1}. ]

At the other extreme, a complete real orthonormal frame is an orthogonal matrix, so

[ V_n(\mathbb R^n)\cong O(n). ]

The analogous complete-frame spaces over (\mathbb C) and (\mathbb H) are (U(n)) and (Sp(n)), respectively.

Smooth structure and dimension

Consider the real case as the inverse image of the identity matrix under the smooth map

[ F:\mathbb R^{n\times k}\longrightarrow \operatorname{Sym}_k(\mathbb R), \qquad F(Q)=Q^{\mathsf T}Q. ]

At every matrix satisfying (Q^{\mathsf T}Q=I_k), the derivative of (F) has full rank. The regular value theorem therefore makes (V_k(\mathbb R^n)) a smooth embedded submanifold of (\mathbb R^{n\times k}). Since a symmetric (k\times k) matrix has (k(k+1)/2) independent entries, its dimension is

[ \dim_{\mathbb R}V_k(\mathbb R^n)

nk-\frac{k(k+1)}{2}. ]

The real dimensions of the complex and quaternionic versions are

[ \dim_{\mathbb R}V_k(\mathbb C^n)=2nk-k^2 ]

and

[ \dim_{\mathbb R}V_k(\mathbb H^n)=k(4n-2k+1). ]

For (Q\in V_k(\mathbb R^n)), differentiating the defining equation gives the tangent-space condition

[ T_QV_k(\mathbb R^n)

\left{ X\in\mathbb R^{n\times k} \mid Q^{\mathsf T}X+X^{\mathsf T}Q=0 \right}. ]

Thus the component of (X) parallel to the columns of (Q) is governed by a skew-symmetric matrix, while the orthogonal component is unrestricted. This decomposition agrees with the dimension obtained from the regular-value description.

Fibrations and connectivity

Projection onto the first vector of a frame defines a smooth map

[ V_k(\mathbb R^n)\longrightarrow S^{n-1}. ]

For a fixed unit vector (v), the remaining (k-1) vectors form an orthonormal frame in the ((n-1))-dimensional orthogonal complement (v^\perp). The projection therefore participates in the fiber bundle

[ V_{k-1}(\mathbb R^{n-1}) \longrightarrow V_k(\mathbb R^n) \longrightarrow S^{n-1}. ]

Repeated use of this fibration expresses a Stiefel manifold as an iterated family of sphere bundles. It also yields the connectivity statement

[ \pi_i!\left(V_k(\mathbb R^n)\right)=0 \qquad \text{for } i\leq n-k-1. ]

Hence (V_k(\mathbb R^n)) is ((n-k-1))-connected. The formula includes the sphere case because (V_1(\mathbb R^n)=S^{n-1}), while the complete-frame case reflects the disconnectedness of (O(n)).

There is another fundamental bundle obtained by forgetting the ordering and individual vectors while retaining their span:

[ O(k) \longrightarrow V_k(\mathbb R^n) \longrightarrow \operatorname{Gr}_k(\mathbb R^n). ]

Here (\operatorname{Gr}_k(\mathbb R^n)) is the real Grassmannian of (k)-dimensional subspaces. The right action of (O(k)) changes the orthonormal basis within a fixed subspace, making the projection a principal (O(k))-bundle.

Relation to vector bundles

For a rank-(n) real vector bundle (E\to B) equipped with a fiberwise inner product, the associated Stiefel bundle (V_k(E)\to B) has as its fiber over (b\in B) the space of orthonormal (k)-frames in (E_b). A section of this bundle is equivalent to a collection of (k) everywhere linearly independent sections of (E).

The failure of such a section to exist is measured by obstruction theory. In particular, the characteristic classes introduced through the work of Stiefel and Whitney detect restrictions on independent vector fields. For the tangent bundle of a smooth manifold, this connects Stiefel manifolds to parallelizability and to the problem of constructing continuous tangent-vector fields.

The principal bundle

[ O(n-k)\longrightarrow O(n)\longrightarrow V_k(\mathbb R^n) ]

also makes (V_k(\mathbb R^n)) a classifying space for reductions involving partial orthonormal frames in the appropriate finite-dimensional range. Under stabilization in (n), these constructions enter the topology of the infinite orthogonal group and the classification of stable real vector bundles by real K-theory.

Metric geometry

The embedding into matrix space induces a Riemannian metric from the Frobenius inner product

[ \langle X,Y\rangle=\operatorname{tr}(X^{\mathsf T}Y). ]

A second natural metric is obtained from the quotient presentation (O(n)/O(n-k)) by choosing an invariant inner product on the corresponding complement of Lie algebras. These metrics agree in some directions but use different relative weights for motion within the represented (k)-plane and motion that changes the plane itself.

The projection from the Stiefel manifold to the Grassmannian separates these two kinds of infinitesimal displacement. Vertical tangent vectors alter the basis of a fixed subspace, whereas horizontal tangent vectors alter the subspace represented by the frame. This distinction underlies the interpretation of the Stiefel manifold as a principal bundle with a natural connection.