Author Description

The authors present the rolling (or development) of a smooth connected complete Riemannian manifold over another manifold with the same dimension. The rolling is assumed to be without spinning or slipping. Relying on geometric control theory, the authors provide an intrinsic description of the two constraints ``without spinning'' and ``without slipping'' in terms of the Levi-Civita connections, by defining corresponding vector fields distributions in the appropriate state space. The authors then address the issue of complete controllability and establish basic global properties for the reachable set and investigate the associated Lie bracket structure. In particular, they point out the role played by a curvature tensor defined on the state space that they call the rolling curvature. When the two manifolds are three dimensional, the authors give a complete local characterization of the reachable sets and, in particular, they identify necessary and sufficient conditions for the existence of a non-open orbit. In addition to the trivial case where the manifolds are (locally) isometric the authors show that (local) noncontrollability occurs if and only if the manifolds are either warped products or contact manifolds with additional restrictions that the authors precisely describe.