Rolling manifolds on space forms

Chitour, Yacine ; Kokkonen, Petri

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012), p. 927-954 / Harvested from Numdam

Résumé

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

Détail

Publié le : 2012-01-01

MR 2995101 | Zbl 1321.53021

DOI : https://doi.org/10.1016/j.anihpc.2012.05.005

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