Symmetries of the rolling model

International audienceIn the present paper, we study the infinitesimal symmetries of the model of two Riemannian manifolds (M, g) and ({\hat{M}},\hat{g}) 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 if {\hat{M}} has constant sectional curvature. Additionally, we prove that when M and {\hat{M}} have different constant sectional curvatures and dimension n\ge 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