# Extremal curves on Stiefel and Grassmann manifolds

arXiv: https://arxiv.org/abs/1809.07013 · HTML: https://mlsanity.com/paper/1809.07013/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Sep 2018.

## Built on by (most cited first, 1 of 1)

- [Parallel transport on matrix manifolds and Exponential Action](https://mlsanity.com/paper/2408.06054/) (2024)

## Abstract (arXiv)

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 Stkn(V). This result is valid not only for the real Stiefel manifolds in V=mathbb Rn, but also for the Stiefels in the Hermitian space V=mathbb Cn and the quaternion space V=mathbb Hn.