An intrinsic formulation of the rolling manifolds problem

We present an intrinsic formulation of the kinematic problem of two $n-$dimensional manifolds rolling one on another without twisting or slipping. We determine the configuration space of the system, which is an $\frac{n(n+3)}2-$dimensional manifold. The conditions of no-twisting and no-slipping are decoded by means of a distribution of rank $n$. We compare the intrinsic point of view versus the extrinsic one. We also show that the kinematic system of rolling the $n$-dimensional sphere over $\mathbb R^n$ is controllable. In contrast with this, we show that in the case of $SE(3)$ rolling over $\mathfrak{se}(3)$ the system is not controllable, since the configuration space of dimension 27 is foliated by submanifolds of dimension 12.