Extremal curves on Stiefel and Grassmann manifolds

TL;DR

This paper explores the geometric structure of extremal curves on Stiefel and Grassmann manifolds, revealing explicit solutions for certain sub-Riemannian systems and linking quasi-geodesics to projections of sub-Riemannian geodesics on Lie groups.

Contribution

It introduces a class of explicit solutions for left-invariant sub-Riemannian systems and establishes geometric origins for quasi-geodesics on Stiefel and Grassmann manifolds across real, complex, and quaternionic spaces.

Findings

Quasi-geodesics are projections of sub-Riemannian geodesics on Lie groups.

Results apply to real, complex, and quaternionic Stiefel manifolds.

Explicit solutions for a large class of sub-Riemannian systems are provided.

Abstract

This paper uncovers a large class of left-invariant sub-Rie\-mannian systems on Lie groups that admit explicit solutions with certain properties, and provides geometric origins for a class of important curves on Stiefel manifolds, called quasi-geodesics, that project on Grassmann manifolds as Riemannian geodesics. We show that quasi-geodesics are the projections of sub-Riemannian geodesics generated by certain left-invariant distributions on Lie groups that act transitively on each Stiefel manifold $St_k^n(V)$. This result is valid not only for the real Stiefel manifolds in $V=\mathbb R^n$, but also for the Stiefels in the Hermitian space $V=\mathbb C^n$ and the quaternion space $V=\mathbb H^n$.