Rolling manifolds on space forms
Résumé
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) over cap, (g) over cap) of the same dimension n >= 2. 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 del(g) and del((g) over cap). 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 pi(Q,M). Moreover, these orbits can be realized as holonomy groups of either certain vector bundle connections del(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) over cap, (g) over cap) has positive (constant) curvature we prove that, if the action of the holonomy group of del(Rol) is not transitive. then (M g) admits ((M) over cap, (g) over cap) as its universal covering. In addition, we show that, for n even and n >= 16, the rolling problem (R) of (M, g) against the space form ((M) over cap, (g) over cap) of positive curvature c > 0, is completely controllable if and only if (M, g) is not of constant curvature c. (C) 2012 Elsevier Masson SAS. All rights reserved.