Sub-Riemannian geometry of Stiefel manifolds

TL;DR

This paper investigates the geometric structure of Stiefel manifolds using sub-Riemannian geometry, providing a complete description of the cut locus for certain cases and conditions for the general case.

Contribution

It offers a detailed analysis of the cut locus on Stiefel manifolds, including explicit descriptions for specific cases and conditions for the general case.

Findings

Complete description of the cut locus on V_{n;1}

Sufficient conditions for the cut locus in the general case

Analysis of the complement to the cut locus of V_{2k;k}

Abstract

In the paper we consider the Stiefel manifold $V_{n;k}$ as a principal $U(k)$- bundle over the Grassmann manifold and study the cut locus from the unit element. We gave the complete description of this cut locus on $V_{n;1}$ and presented the sufficient condition on the general case. At the end, we study the complement to the cut locus of $V_{2k;k}$.