Rolling Against a Sphere: The Non-transitive Case

We study the control system of a Riemannian manifold M of dimension n rolling on the sphere S n . The controllability of this system is described in terms of the holonomy of a vector bundle connection which, we prove, is isomorphic to the Riemannian holonomy group of the cone C ( M ) of M . Using Berger’s list, we reduce the possible holonomies to a few families. In particular, we focus on the cases where the holonomy is the unitary and the symplectic group. In the first case, using the rolling formalism, we construct explicitly a Sasakian structure on M ; and in the second case, we construct a 3-Sasakian structure on M .