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Abstract. 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 .

Abstract. In the present paper, we study the infinitesimal symmetries of the model of two Riemannian manifolds (M, g) and (M ,ĝ) rolling without twisting or slipping. We show that, under certain genericity hypotheses, the natural bundle projection from the state space Q of the rolling model onto M is a principal bundle if and only ifM has constant sectional curvature. Additionally, we prove that when M andM have different constant sectional curvatures and dimension n ≥ 3, the rolling distribution is never flat, contrary to the two dimensional situation of rolling two spheres of radii in the proportion 1 : 3, which is a well-known system satisfying É. Cartan's flatness condition.