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Extremal Curves on Stiefel and Grassmann Manifolds

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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 \(V={{\mathbb {H}}}^n\).

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, ON, M5S 3G3, Canada

    V. Jurdjevic

  2. Department of Mathematics, University of Bergen, P.O. Box 7803, 5020, Bergen, Norway

    I. Markina

  3. Institute of Systems and Robotics, University of Coimbra, Pólo II, 3030-290, Coimbra, Portugal

    F. Silva Leite

  4. Department of Mathematics, University of Coimbra, Apartado 3008, 3001-501, Coimbra, Portugal

    F. Silva Leite

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  1. V. Jurdjevic
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  2. I. Markina
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  3. F. Silva Leite
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Correspondence to F. Silva Leite.

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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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Jurdjevic, V., Markina, I. & Silva Leite, F. Extremal Curves on Stiefel and Grassmann Manifolds. J Geom Anal 30, 3948–3978 (2020). https://doi.org/10.1007/s12220-019-00223-1

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