# 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

## Abstract

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 (M\limits^{ˆ},g\limits^{ˆ}) of the same dimension $n⩾2$. This amounts to study an *n*-dimensional distribution $\mathscr{D}_{\mathrm{R}}$, that we call the rolling distribution, and which is defined in terms of the Levi-Civita connections $\mathrm{∇}^{g}$ and \mathrm{∇}^{g\limits^{ˆ}}. We then address the issue of the complete controllability of the control system associated to $\mathscr{D}_{\mathrm{R}}$. The key remark is that the state space *Q* carries the structure of a principal bundle compatible with $\mathscr{D}_{\mathrm{R}}$. 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 $\mathrm{∇}^{\mathsf{Rol}}$, 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 (M\limits^{ˆ},g\limits^{ˆ}) has positive (constant) curvature we prove that, if the action of the holonomy group of $\mathrm{∇}^{\mathsf{Rol}}$ is not transitive, then $(M,g)$ admits (M\limits^{ˆ},g\limits^{ˆ}) 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 (M\limits^{ˆ},g\limits^{ˆ}) of positive curvature $c > 0$, is completely controllable if and only if $(M,g)$ 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