On generalized Stiefel manifolds (Q1085494)

A generalized Stiefel manifold M(m,n;k) is the space of \(m\times n\) matrices of a fixed rank k. An ''orthogonal version'' of M(m,n;k) is shown to fiber over the Grassmannian \(G_{m,k}\) as a (split for \(n\geq m)\) fibre bundle with typical fibre a Stiefel manifold \(V_{n,k}\). This is implicitly already contained in the book by \textit{U. Koschorke} [Vector fields and other vector bundle morphisms - a singularity approach (Lect. Notes Math. 847) (1981; Zbl 0459.57016)]. The associated Serre spectral sequence is shown to collapse which allows computations of cohomology groups mod 2. There are applications to subimmersions (mappings of a fixed rank) between manifolds using Gromov theory.