Extremal Curves on Stiefel and Grassmann Manifolds

Abstract

This paper uncovers a large class of left-invariant sub-Riemannian 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 $$\mathrm{St}_k^n(V)$$ . 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 $$V={{\mathbb {H}}}^n$$ .