Rolling Manifolds of Different Dimensions

Abstract

If $(M,g)$ and $(\hM,\hg)$ are two smooth connected complete oriented Riemannian manifolds of dimensions $n$ and $\hn$ respectively, we model the rolling of $(M,g)$ onto $(\hM,\hg)$ as a driftless control affine systems describing two possible constraints of motion: the first rolling motion $\Sigma_{NS}$ captures the no-spinning condition only and the second rolling motion $\Sigma_{R}$ corresponds to rolling without spinning nor slipping. Two distributions of dimensions $(n + \hn)$ and $n$, respectively, are then associated to the rolling motions $\Sigma_{NS}$ and $\Sigma_{R}$ respectively. This generalizes the rolling problems considered in \cite{ChitourKokkonen1} where both manifolds had the same dimension. The controllability issue is then addressed for both $\Sigma_{NS}$ and $\Sigma_{R}$ and completely solved for $\Sigma_{NS}$. As regards to $\Sigma_{R}$, basic properties for the reachable sets are provided as well as the complete study of the case $(n,\hn)=(3,2)$ and some sufficient conditions for non-controllability.