Rolling manifolds on space forms

Abstract

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, ĝ) 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 ∇ g and ∇ ĝ . 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, ĝ) has positive (constant) curvature we prove that, if the action of the holonomy group of ∇ Rol is not transitive, then (M, g) admits ( M, ĝ) 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, ĝ) of positive curvature c > 0, is completely controllable if and only if (M, g) is not of constant curvature c.

Horizontal Holonomy for Affine Manifolds

“…Using the previous proposition, one can extend immediately the simple but crucial Proposition 4.1 in [11] to derive the next result.…”

“…In particular, the affine group of R n is denoted by Aff(n). One can extend readily Proposition 3.10 of [11] to get the following result.…”

Rolling Against a Sphere: The Non-transitive Case

“…Rolling against a space form. In [6,8], the authors study the rolling system when ( M, g) is a space form, that is, a complete and simply connected Riemannian manifold of constant sectional curvature c ∈ R. In this case, the natural projection π Q,M : Q → M is a principal bundle of a special form, which we proceed to explain.…”

“…In the spherical case, that is c = 1, the situation is more complicated due to the nontrivial topology of spheres, see [6]. In fact, there is an almost complete answer in the case of even dimensional spheres.…”

Properties of an affine transport equation and its holonomy

“…The vector ∆ξ a also appears as the development of a curve on a manifold into the affine tangent space at the curve's starting point. This is sometimes equivalently described as rolling the manifold along the initial tangent space without slipping or twisting [8,9,10]. More specifically, a vector ∆ξ a at x is equivalent to the displacement vector in the initial affine tangent space that points between the initial and final values of the rolling (or developing) curve, and ∆ξ a = Λ a a ∆ξ a is its parallel transport along the curve in the manifold from x to x.…”