Stiefel manifold

In mathematics, the Stiefel manifold ${\displaystyle V_{k}(\mathbb {R} ^{n})}$ is the set of all orthonormal k-frames in ${\displaystyle \mathbb {R} ^{n}.}$ That is, it is the set of ordered orthonormal k-tuples of vectors in ${\displaystyle \mathbb {R} ^{n}.}$ It is named after Swiss mathematician Eduard Stiefel. Likewise one can define the complex Stiefel manifold ${\displaystyle V_{k}(\mathbb {C} ^{n})}$ of orthonormal k-frames in ${\displaystyle \mathbb {C} ^{n}}$ and the quaternionic Stiefel manifold ${\displaystyle V_{k}(\mathbb {H} ^{n})}$ of orthonormal k-frames in ${\displaystyle \mathbb {H} ^{n}}$. 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 ${\displaystyle \mathbb {R} ^{n},\mathbb {C} ^{n},}$ or ${\displaystyle \mathbb {H} ^{n};}$ 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. 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 ${\displaystyle \mathbb {F} }$ stand for ${\displaystyle \mathbb {R} ,\mathbb {C} ,}$ or ${\displaystyle \mathbb {H} .}$ The Stiefel manifold ${\displaystyle V_{k}(\mathbb {F} ^{n})}$ can be thought of as a set of n × k matrices by writing a k-frame as a matrix of k column vectors in ${\displaystyle \mathbb {F} ^{n}.}$ The orthonormality condition is expressed by A*A = ${\displaystyle I_{k}}$ where A* denotes the conjugate transpose of A and ${\displaystyle I_{k}}$ denotes the k × k identity matrix. We then have

The topology on ${\displaystyle V_{k}(\mathbb {F} ^{n})}$ is the subspace topology inherited from ${\displaystyle \mathbb {F} ^{n\times k}.}$ With this topology ${\displaystyle V_{k}(\mathbb {F} ^{n})}$ is a compact manifold whose dimension is given by

Each of the Stiefel manifolds ${\displaystyle V_{k}(\mathbb {F} ^{n})}$ 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 ${\displaystyle \mathbb {R} ^{n}}$ 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 ${\displaystyle V_{k}(\mathbb {R} ^{n}).}$ 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 ${\displaystyle V_{k}(\mathbb {C} ^{n})}$ with stabilizer subgroup U(n−k) and the symplectic group Sp(n) acts transitively on ${\displaystyle V_{k}(\mathbb {H} ^{n})}$ with stabilizer subgroup Sp(n−k).

In each case ${\displaystyle V_{k}(\mathbb {F} ^{n})}$ can be viewed as a homogeneous space: