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.

arXiv comment22 pages

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• Variational Approach to Differential Invariants of Rank 2 Vector Distributions
• 3-Sasakian Manifolds
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• Geometric conditions for the existence of an intrinsic rolling
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Cited by in corpus (2)

• The rolling problem: overview and challenges
• Rolling Manifolds of Different Dimensions