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On symmetries of a sub-Riemannian structure with growth vector (4, 7)

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Abstract

We study symmetries of specific left-invariant sub-Riemannian structure with filtration (4, 7) and their impact on sub-Riemannian geodesics of corresponding control problem. We show that there are two very different types of geodesics, they either do not intersect the fixed point set of symmetries or are contained in this set for all times. We use the symmetry reduction to study properties of geodesics.

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Authors and Affiliations

  1. Institute of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology, Technická 2896/2, 616 69, Brno, Czech Republic

    Jaroslav Hrdina & Aleš Návrat

  2. Institute of Mathematics, Faculty of Science, University of South Bohemia, Branišovská 1760, 370 05, České Budějovice, Czech Republic

    Lenka Zalabová

  3. Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic

    Lenka Zalabová

Authors
  1. Jaroslav Hrdina
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  2. Aleš Návrat
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  3. Lenka Zalabová
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Correspondence to Lenka Zalabová.

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The first and second authors were supported by the grant no. FSI-S-20-6187. Third author is supported by grant no. 20-11473S Symmetry and invariance in analysis, geometric modeling and control theory from the Czech Science Foundation. We thank to Luca Rizzi for useful discussions during Winter School Geometry and Physics, Srní, 2020. Finally, we thank the referee for valuable comments.

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Hrdina, J., Návrat, A. & Zalabová, L. On symmetries of a sub-Riemannian structure with growth vector (4, 7). Annali di Matematica 202, 293–306 (2023). https://doi.org/10.1007/s10231-022-01242-6

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  • DOI: https://doi.org/10.1007/s10231-022-01242-6

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