滚动模型的对称性
Symmetries of the Rolling Model
摘要：
本文研究两个黎曼流形$(M,g)$和$(\hat M,\hat g)$无扭无滑滚动模型的无穷小对称性。我们证明了，在一定的一般性假设下，从滚动模型的状态空间$Q$到$M$的自然丛投影是主丛当且仅当$\hat M$具有恒定的截面曲率。此外，我们证明了当$M$和$\hat M$具有不同的常截面曲率和尺寸$n\geq3$时，滚动分布永远不是平坦的，这与两个半径球按$1\colon3$的比例滚动的二维情形相反，这是一个众所周知的系统，满足π. Cartan平坦性条件
英文摘要：
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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