Rolling manifolds on space forms

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 $(\hat{M},\hat{g})$ 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 ${\nabla }^g$ and ${\nabla }^{\hat{g}}$. 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 ${\nabla }^{\mathsf{Rol}}$, 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 $(\hat{M},\hat{g})$ has positive (constant) curvature we prove that, if the action of the holonomy group of ${\nabla }^{\mathsf{Rol}}$ 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 $c$.