Extremal curves on Stiefel and Grassmann manifolds

2018/09/19 by Velimir, Jurdjevic, Irina, Markina, Fatima, Silva Leite · 3 citations
#30C80 (Primary) #49J15 #53B21 #53C17 #53C22 #53C25 #58E40 (Secondary) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1809.07013

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_kⁿ(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.

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• Sub-Riemannian geometry on infinite-dimensional manifolds
• The Riemannian geometry of orbit spaces. The metric, geodesics, and integrable systems
• Sub-semi-Riemannian geometry on H-type groups
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