Rolling manifolds on space forms

Chitour, Yacine; Kokkonen, Petri

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 6, pp. 927-954.

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

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