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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 andno-slipping are decoded by means of a distribution of rank $n$. We compare theintrinsic point of view versus the extrinsic one. We also show that thekinematic system of rolling the $n$-dimensional sphere over $\mathbb R^n$ iscontrollable. In contrast with this, we show that in the case of $SE(3)$rolling over $\mathfrak{se}(3)$ the system is not controllable, since theconfiguration space of dimension 27 is foliated by submanifolds of dimension12.

An intrinsic formulation of the rolling manifolds problem