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.