Rolling manifolds on space forms
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 6, p. 927-954
In this paper, we consider the rolling problem (R) without spinning nor slipping of a smooth connected oriented complete Riemannian manifold $\left(M,g\right)$ onto a space form $\left(\stackrel{ˆ}{M},\stackrel{ˆ}{g}\right)$ of the same dimension $n⩾2$. This amounts to study an n-dimensional distribution ${𝒟}_{\mathrm{R}}$, that we call the rolling distribution, and which is defined in terms of the Levi-Civita connections ${\nabla }^{g}$ and ${\nabla }^{\stackrel{ˆ}{g}}$. We then address the issue of the complete controllability of the control system associated to ${𝒟}_{\mathrm{R}}$. The key remark is that the state space Q carries the structure of a principal bundle compatible with ${𝒟}_{\mathrm{R}}$. It implies that the orbits obtained by rolling along loops of $\left(M,g\right)$ become Lie subgroups of the structure group of ${\pi }_{Q,M}$. Moreover, these orbits can be realized as holonomy groups of either certain vector bundle connections ${\nabla }^{\mathrm{𝖱𝗈𝗅}}$, called the rolling connections, when the curvature of the space form is non-zero, or of an affine connection (in the sense of Kobayashi and Nomizu, 1996 [14]) in the zero curvature case. As a consequence, we prove that the rolling (R) onto an Euclidean space is completely controllable if and only if the holonomy group of $\left(M,g\right)$ is equal to $\mathrm{SO}\left(n\right)$. Moreover, when $\left(\stackrel{ˆ}{M},\stackrel{ˆ}{g}\right)$ has positive (constant) curvature we prove that, if the action of the holonomy group of ${\nabla }^{\mathrm{𝖱𝗈𝗅}}$ is not transitive, then $\left(M,g\right)$ admits $\left(\stackrel{ˆ}{M},\stackrel{ˆ}{g}\right)$ as its universal covering. In addition, we show that, for n even and $n⩾16$, the rolling problem (R) of $\left(M,g\right)$ against the space form $\left(\stackrel{ˆ}{M},\stackrel{ˆ}{g}\right)$ of positive curvature $c>0$, is completely controllable if and only if $\left(M,g\right)$ is not of constant curvature c.
@article{AIHPC_2012__29_6_927_0,
author = {Chitour, Yacine and Kokkonen, Petri},
title = {Rolling manifolds on space forms},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {29},
number = {6},
year = {2012},
pages = {927-954},
doi = {10.1016/j.anihpc.2012.05.005},
zbl = {1321.53021},
mrnumber = {2995101},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2012__29_6_927_0}
}

Chitour, Yacine; Kokkonen, Petri. Rolling manifolds on space forms. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 6, pp. 927-954. doi : 10.1016/j.anihpc.2012.05.005. http://www.numdam.org/item/AIHPC_2012__29_6_927_0/

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