Controllability of the rolling system of a Lorentzian manifold on \mathbb R^n,1

2024/08/11 by Abraham Bobadilla Osses, Osses, Abraham Bobadilla, Mauricio Godoy Molina +1
Mathematics · #53A17 #53C50 #93B05 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2408.05863

openalex publication_date 2024/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the mechanical system associated with rolling a Lorentzian manifold (M,g) of dimension n+1≥2 on flat Lorentzian space \widehatM=\mathbb R^n,1, without slipping or twisting. Using previous results, it is known that there exists a distribution D_R of rank (n+1) defined on the configuration space Q(M,\widehatM) of the rolling system, encoding the no-slip and no-twist conditions. Our objective is to study the problem of complete controllability of the control system associated with D_R. The key lies in examining the holonomy group of the distribution D_R and, following the approach of \citeChKok, establishing that the rolling problem is completely controllable if and only if the holonomy group of (M,g) equals SO₀(n,1).

Related

• Rolling Manifolds: Intrinsic Formulation and Controllability
• Rolling Manifolds of Different Dimensions
• Controllability results for the rolling of 2-dim. against 3-dim. Riemannian Manifolds
• Controllability of rolling without twisting or slipping in higher dimensions
• Rolling against a sphere: The non transitive case