This paper is the first of two parts which considers the rolling (or development) of two Riemannian connected manifolds (M,g) and \(\left (\hat {M},\hat {g}\right )\) of dimensions 2 and 3 respectively, with the constraints of no-spinning and no-slipping. The present work is a continuation of Mortada et al. (Acta Appl Math 139:105–31, 2015), which modeled the general setting of the rolling of two Riemannian connected manifolds with different dimensions as a driftless control affine system on a fibered space Q of dimension eighth, with an emphasis on understanding the local structure of the rolling orbits, i.e., the reachable sets in Q. We show that the possible dimensions of non open rolling orbits belong to the set {2, 5, 6, 7}. In this first part, we describe the structures of orbits of dimension 2, one of the two possible local structure of rolling orbits of dimension 5 and special cases of dimension 7.

本文是考虑两个黎曼连通流形 ( M , g ) 和\(\left (\hat {M},\hat {g}\right )\)维度的滚动（或展开）的两部分中的第一部分分别为 2 和 3，具有无旋转和无滑移的约束。目前的工作是 Mortada 等人的延续。( Acta Appl Math 139:105–31, 2015)，将两个不同维数的黎曼连接流形的滚动的一般设置建模为一个无漂移的控制仿射系统，在第 8 维的纤维空间Q上，重点是理解滚动轨道的局部结构，即Q 中的可达集. 我们表明非开放滚动轨道的可能维度属于集合 {2, 5, 6, 7}。在第一部分中，我们描述了 2 维轨道的结构，这是 5 维滚动轨道的两种可能局部结构之一和 7 维特殊情况。