Path-connectedness of the intersection of translates of St(n,H)

TL;DR

This paper proves that the intersection of certain translated Stiefel manifolds in a Hilbert space is path-connected, under specific codimension conditions, advancing the topological understanding of these manifolds.

Contribution

It establishes a new path-connectedness result for intersections of translated Stiefel manifolds in Hilbert spaces, with a novel lemma supporting the proof.

Findings

Intersection of translates is path-connected under codimension conditions

Polygonal paths can connect points in the intersection

Provides a new topological insight into Stiefel manifolds

Abstract

If $H$ is a Hilbert space, the Stiefel manifold $St(n,H)$ is formed by all the independent $n$-tuples in $H$. In this article, we contribute to the topological study of Stiefel manifolds by proving a path-connectedness result. We prove that the intersection of translates of $St(n,H)$ is path-connected by polygonal paths under a condition on the codimension of the span of the components of the translating $n$-tuples. We rely on a lemma that we prove for the occasion.