Extremal Curves on Stiefel and Grassmann Manifolds
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Extremal Curves on Stiefel and Grassmann Manifolds V. Jurdjevic1 · I. Markina2 · F. Silva Leite3,4 Received: 19 November 2018 © Mathematica Josephina, Inc. 2019
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 quasigeodesics are the projections of sub-Riemannian geodesics generated by certain leftinvariant distributions on Lie groups that act transitively on each Stiefel manifold St nk (V ). This result is valid not only for the real Stiefel manifolds in V = Rn , but also for the Stiefels in the Hermitian space V = Cn and the quaternion space V = Hn . Keywords Sub-Riemannian geometry · Quasi-geodesic curves · Horizontal distributions · Grassmann and Stiefel manifolds · Lie groups actions on manifolds · Pontryagin Maximum Principle Mathematics Subject Classification Primary: 53C17 · 53C22 · 53B21 · 53C25 · 30C80 · Secondary: 49J15 · 58E40
This work was partially supported by ISP Project 239033/F20 of Norwegian Research Council. The work of I. Markina was also partially supported by joint BFS-TSF mathematics program, and F. Silva Leite thanks Fundação para a Ciência e a Tecnologia (FCT-Portugal) and COMPETE 2020 Program for the partial financial support through Project UID-EEA-00048-2013.
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F. Silva Leite [email protected] V. Jurdjevic [email protected] I. Markina [email protected]
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Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, Canada
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Department of Mathematics, University of Bergen, P.O. Box 7803, 5020 Bergen, Norway
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Institute of Systems and Robotics, University of Coimbra, Pólo II, 3030-290 Coimbra, Portugal
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Department of Mathematics, University of Coimbra, Apartado 3008, 3001-501 Coimbra, Portugal
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V. Jurdjevic et al.
1 Introduction This paper provides geometric origins for a class of curves on Stiefel manifolds, called quasi-geodesic, that have proved to be particularly important in solving interpolation problems arising in real applications [13]. We show that quasi-geodesic curves are the projections of sub-Riemannian geodesics generated by certain left-invariant distributions on Lie groups G that act on Stiefel manifolds. This quest for the geometric characterization of quasi-geodesic curves uncovered a large class of left-invariant subRiemannian systems on Lie groups that admit explicit solutions, in the form that will be made clear below. As a result, the paper is as much about sub-Riemannian structures on Lie groups as it is about quasi-geodesic curves. The first part of the paper deals with sub-Riemannian structures associated with homogeneous spaces M = G/K induced by a transitive left action of a semi-simple Lie group G on a smooth manifold M, where K denotes the isotropy subgroup relative to a fixed point m 0 ∈ M. The sub-Riemannian structures will be defined by a le
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