Rolling of manifolds and controllability in dimension three (Q2829378)

The present booklet concerns the rolling or development of one smooth connected Riemannian manifold \((M, g)\) on another \((\hat{M}, \hat{g})\) of equal dimension \(n\geq 2\) when there is no relative spin or slip of one manifold with respect to the other one. The state space of the rolling problem \((R)\) is \(Q=\{A:T_xM\rightarrow T_{\hat{x}}\hat{M}; A=\text{o-isometry}, x\in M, \hat{x}\in \hat{M}\}\) where ``o-isometry'' means positively oriented isometry.NEWLINENEWLINEThis study consists of 5 sections and three appendices. In Section 2, the notations used throughout the work are gathered. The control system associated to the problem \((R)\) is presented in Section 3 by given a precise definition of the set of admissible controls, which is equal to the set of locally absolutely continuous curves on \(M\) only. Then the main tools are some driftless control systems affine in the control and the first appendix provides expressions in local coordinates for these control systems.NEWLINENEWLINEThe Agrachev-Sachkov's approach based on an \(n\)-dimensional distribution on \(Q\) is represented in detail in Section 4 while the last section deals with the 3-dimensional case. The second appendix studies the problem \((R)\) embedded in \(\mathbb{R}^N\) while the last appendix is devoted to special manifolds (like warped products) in 3D Riemannian geometry.