JournalsaihpcVol. 29, No. 6pp. 927–954

Rolling manifolds on space forms

  • Yacine Chitour

    L2S, Université Paris-Sud XI, CNRS and Supélec, Gif-sur-Yvette 91192, France
  • Petri Kokkonen

    L2S, Université Paris-Sud XI, CNRS and Supélec, Gif-sur-Yvette 91192, France, University of Eastern Finland, Department of Applied Physics, 70211 Kuopio, Finland
Rolling manifolds on space forms cover
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Abstract

In this paper, we consider the rolling problem (R) without spinning nor slipping of a smooth connected oriented complete Riemannian manifold (M,g)(M,g) onto a space form (M\limits^{ˆ},g\limits^{ˆ}) of the same dimension n2n⩾2. This amounts to study an n-dimensional distribution DR\mathscr{D}_{\mathrm{R}}, that we call the rolling distribution, and which is defined in terms of the Levi-Civita connections g\mathrm{∇}^{g} and \mathrm{∇}^{g\limits^{ˆ}}. We then address the issue of the complete controllability of the control system associated to DR\mathscr{D}_{\mathrm{R}}. The key remark is that the state space Q carries the structure of a principal bundle compatible with DR\mathscr{D}_{\mathrm{R}}. It implies that the orbits obtained by rolling along loops of (M,g)(M,g) become Lie subgroups of the structure group of πQ,M\pi _{Q,M}. Moreover, these orbits can be realized as holonomy groups of either certain vector bundle connections Rol\mathrm{∇}^{\mathsf{Rol}}, called the rolling connections, when the curvature of the space form is non-zero, or of an affine connection (in the sense of Kobayashi and Nomizu, 1996 [14]) in the zero curvature case. As a consequence, we prove that the rolling (R) onto an Euclidean space is completely controllable if and only if the holonomy group of (M,g)(M,g) is equal to SO(n)\mathrm{SO}(n). Moreover, when (M\limits^{ˆ},g\limits^{ˆ}) has positive (constant) curvature we prove that, if the action of the holonomy group of Rol\mathrm{∇}^{\mathsf{Rol}} is not transitive, then (M,g)(M,g) admits (M\limits^{ˆ},g\limits^{ˆ}) as its universal covering. In addition, we show that, for n even and n16n⩾16, the rolling problem (R) of (M,g)(M,g) against the space form (M\limits^{ˆ},g\limits^{ˆ}) of positive curvature c>0c > 0, is completely controllable if and only if (M,g)(M,g) is not of constant curvature c.

Cite this article

Yacine Chitour, Petri Kokkonen, Rolling manifolds on space forms. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 6, pp. 927–954

DOI 10.1016/J.ANIHPC.2012.05.005