Connectivity properties of the Schur–Horn map for real Grassmannians

To any V in the Grassmannian \(\textrm{Gr}_k({\mathbb R}^n)\) ${\text{Gr}}_k({\mathbb{R}}^n)$ of k-dimensional vector subspaces in \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ one can associate the diagonal entries of the ( \(n\times n\) $n\times n$ ) matrix corresponding to the orthogonal projection of \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ to V. One obtains a map \(\textrm{Gr}_k({\mathbb {R}}^n)\rightarrow {\mathbb {R}}^n\) ${\text{Gr}}_k({\mathbb{R}}^n)\to {\mathbb{R}}^n$ (the Schur–Horn map). The main result of this paper is a criterion for pre-images of vectors in \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ to be connected. This will allow us to deduce connectivity criteria for a certain class of subspaces of the real Stiefel manifold which arise naturally in frame theory. We extend in this way results of Cahill et al. (SIAM J Appl Algebra Geom 1:38–72, 2017).