Rolling of manifolds without spinning

Abstract

The control model of rolling of a Riemannian manifold (M; g) onto another one $\left( {\hat{M},\hat{g}} \right)$ consists of a state space Q of relative orientations (isometric linear maps) between their tangent spaces equipped with a so-called rolling distribution ${\mathcal D}$ _R, which models the natural constraints of no-spinning and no-slipping of the rolling motion. It turns out that the distribution ${\mathcal D}$ _R can be built as a sub-distribution of a so-called no-spinning distribution ${{\mathcal{D}}_{\overline{\nabla}}}$ on Q that models only the no-spinning constraint of the rolling motion. One is thus motivated to study the control problem associated to ${{\mathcal{D}}_{\overline{\nabla}}}$ and, in particular, the geometry of ${{\mathcal{D}}_{\overline{\nabla}}}$ -orbits. Moreover, the definition of ${{\mathcal{D}}_{\overline{\nabla}}}$ (contrary to the definition of ${\mathcal D}$ _R) makes sense in the general context of vector bundles equipped with linear connections.

The purpose of this paper is to study the distribution ${{\mathcal{D}}_{\overline{\nabla}}}$ determined by the product connection $\nabla \times \hat{\nabla}$ on a tensor bundle ${E^{*}}\otimes \hat{E}\to M\times \hat{M}$ induced by linear connections ∇, $\hat{\nabla}$ on vector bundles $E\to M,\,\,\,\hat{E}\to \hat{M}$. We describe completely the orbit structure of ${{\mathcal{D}}_{\overline{\nabla}}}$ in terms of the holonomy groups of ∇, $\hat{\nabla}$ and characterize the integral manifolds of it. Moreover, we describe the general formulas for the Lie brackets of vector elds in ${E^{*}}\otimes \hat{E}$ in terms of ${{\mathcal{D}}_{\overline{\nabla}}}$ and the vertical tangent distribution of ${E^{*}}\otimes \hat{E}\to M\times \hat{M}$.

In the particular case of tangent bundles $TM\to M,\,\,\,T\hat{M}\to \hat{M}$ and Levi-Civita connections, we describe in more detail how ${{\mathcal{D}}_{\overline{\nabla}}}$ is related to the above mentioned rolling model, where these Lie brackets formulas provide an important tool for the study of controllability of the related control system.