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 (M,g) onto a space form (M ˆ,g ˆ) of the same dimension n2. This amounts to study an n-dimensional distribution 𝒟 R , that we call the rolling distribution, and which is defined in terms of the Levi-Civita connections g and g ˆ . We then address the issue of the complete controllability of the control system associated to 𝒟 R . The key remark is that the state space Q carries the structure of a principal bundle compatible with 𝒟 R . It implies that the orbits obtained by rolling along loops of (M,g) become Lie subgroups of the structure group of π Q,M . Moreover, these orbits can be realized as holonomy groups of either certain vector bundle connections 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 (M,g) is equal to SO (n). Moreover, when (M ˆ,g ˆ) has positive (constant) curvature we prove that, if the action of the holonomy group of 𝖱𝗈𝗅 is not transitive, then (M,g) admits (M ˆ,g ˆ) as its universal covering. In addition, we show that, for n even and n16, the rolling problem (R) of (M,g) against the space form (M ˆ,g ˆ) of positive curvature c>0, is completely controllable if and only if (M,g) 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://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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