Controllability Results for the Rolling of 2-Dimensional Against 3-Dimensional Riemannian Manifolds—Part 1

This paper is the first of two parts which considers the rolling (or development) of two Riemannian connected manifolds ( M , g ) and M ̂ , ĝ of dimensions 2 and 3 respectively, with the constraints of no-spinning and no-slipping. The present work is a continuation of Mortada et al. ( Acta Appl Math 139:105–31, 2015 ), which modeled the general setting of the rolling of two Riemannian connected manifolds with different dimensions as a driftless control affine system on a fibered space Q of dimension eighth, with an emphasis on understanding the local structure of the rolling orbits, i.e., the reachable sets in Q . We show that the possible dimensions of non open rolling orbits belong to the set {2, 5, 6, 7}. In this first part, we describe the structures of orbits of dimension 2, one of the two possible local structure of rolling orbits of dimension 5 and special cases of dimension 7.