Description

We present the rolling (or development) of one smooth connected complete Riemannian manifold (M, g) onto another one (...) of equal dimension n (...) 2 where there is no relative spin or slip of one manifold with respect to the other one. Relying on geometric control theory, we provide an intrinsic description of the two constraints « without spinning » and « without slipping » in terms of the Levi-Civita connections (...) and (...) by defining corresponding vector fields distributions in the appropriate state space. We then address the issue of complete controllability for that rolling problem. We first establish basic global properties for the reachable set and investigate the associated Lie bracket structure. In particular, we point out the role played by a curvature tensor defined on the state space, that we call the rolling curvature. When the two manifolds are three-dimensional, we give a complete local characterization of the reachable sets and, in particular, we identify necessary and sufficient conditions for the existence of a non open orbit. In addition to the trivial case where the manifolds (M, g) and (...) are (locally) isometric, we show that (local) non controllability occurs if and only if (M, g) and (...) are either warped products or contact manifolds with additional restrictions that we precisely describe.