Extremal curves on Stiefel and Grassmann manifolds

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.