Rolling Against a Sphere The Non-transitive Case

Yacine Chitour (1) , Mauricio Godoy Molina (2, 3) , Petri Kokkonen , Irina Markina (2)

1 L2S - Laboratoire des signaux et systèmes
2 UiB - Department of Mathematics [Bergen]
3 Universidad de la frontera [Chile]

We study the control system of a Riemannian manifold M of dimension n rolling on the sphere . 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.

Keywords: Rolling system; Sasakian manifold; 3-Sasakian manifold; Nonholonomic mechanics; Holonomy of connections

Domains: Automatic

Yacine Chitour, Mauricio Godoy Molina, Petri Kokkonen, Irina Markina. Rolling Against a Sphere The Non-transitive Case. The Journal of Geometric Analysis, 2016, 26 (4), pp.2542-2562. ⟨10.1007/s12220-015-9638-y⟩. ⟨hal-02307514⟩