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Symmetries of the rolling model

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Abstract

In the present paper, we study the infinitesimal symmetries of the model of two Riemannian manifolds (M, g) and \(({\hat{M}},\hat{g})\) rolling without twisting or slipping. We show that, under certain genericity hypotheses, the natural bundle projection from the state space Q of the rolling model onto M is a principal bundle if and only if \({\hat{M}}\) has constant sectional curvature. Additionally, we prove that when M and \({\hat{M}}\) have different constant sectional curvatures and dimension \(n\ge 3\), the rolling distribution is never flat, contrary to the two dimensional situation of rolling two spheres of radii in the proportion \(1{:}3\), which is a well-known system satisfying É. Cartan’s flatness condition.

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Acknowledgments

We would like to thank the anonymous reviewer for the insightful comments communicated to us, and docent J. Tervo for reading a preliminary version of this paper and suggesting some important improvements.

Author information

Authors and Affiliations

  1. L2S, Université Paris-Sud XI, CNRS and Supélec, 91192, Gif-sur-Yvette, France

    Yacine Chitour

  2. Department of Mathematics, University of Bergen, Bergen, Norway

    Mauricio Godoy Molina

  3. Departamento de Matemática y Estadística, Universidad de la Frontera, Temuco, Chile

    Mauricio Godoy Molina

  4. Varian Medical Systems, Helsinki, Finland

    Petri Kokkonen

Authors
  1. Yacine Chitour
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  2. Mauricio Godoy Molina
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  3. Petri Kokkonen
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Corresponding author

Correspondence to Mauricio Godoy Molina.

Additional information

This work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissement d’Avenir” program, through the “iCODE Institute project” funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-0. The work of the second author is partially supported by the grants of the Norwegian Research Council NFR-FRINAT 213440/BG and NFR ISP 809290. The work of the third author is supported by Finnish Academy of Science and Letters, KAUTE Foundation and l’Institut français de Finlande.

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Chitour, Y., Godoy Molina, M. & Kokkonen, P. Symmetries of the rolling model. Math. Z. 281, 783–805 (2015). https://doi.org/10.1007/s00209-015-1508-6

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  • DOI: https://doi.org/10.1007/s00209-015-1508-6

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