Symmetries of the Rolling Model

Yacine Chitour; Mauricio Godoy Molina; Petri Kokkonen

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arXiv · 1301.2579

Symmetries of the Rolling Model

Yacine Chitour·Mauricio Godoy Molina·Petri Kokkonen

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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 $n\geq3$, the rolling distribution is never flat, contrary to the two dimensional situation of rolling two spheres of radii in the proportion $1\colon3$, which is a well-known system satisfying É. Cartan's flatness condition.

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BibTeX RIS

Yacine Chitour, Mauricio Godoy Molina, Petri Kokkonen. 2013-01-11. Symmetries of the Rolling Model. https://arxiv.org/abs/1301.2579

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