Symmetries of the Rolling Model

In the present paper, we study the infinitesimal symmetries of the model of two Riemannian manifolds $(M,g)$ and $(\widehat{M},\widehat{g})$ 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 $\widehat{M}$ has constant sectional curvature. Additionally, we prove that when $M$ and $\widehat{M}$ 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 $1\text{\hspace{0.17em}⁣}:3$, which is a well-known system satisfying 'E. Cartan’s flatness condition.