Symmetries of the Rolling Model

Abstract

In 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.

22 pages

References in corpus (2)

• Rolling Manifolds: Intrinsic Formulation and Controllability
• An intrinsic formulation for rolling pseudo-Riemannian manifolds