1999/11/14 by William Pasillas-Lépine, William Pasillas-Lepine, Pasillas-Lepine, William +2 · 2 citations
Engineering · Mathematics · #58A30 (Primary) 53C15 93B29 58A17(Secondary) #Advanced Differential Equations and Dynamical Systems #Control and Dynamics of Mobile Robots #Differential Geometry (math.DG) #FOS: Mathematics #Mathematical Dynamics and Fractals #Optimization and Control (math.OC) #math.DG #math.OC #msc:53C15 #msc:58A30 #msc:93B29
paper · pdf · doi:10.48550/arxiv.math/9911101
LaTeX2e, 90 pages, 9 figures, submitted to ESAIM COCV
arxiv created 1999/11/14 · openalex publication_date 1999/11/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A Goursat structure on a manifold of dimension n is a rank two distribution D such that dim D(i)=i+2, for i=0,...,n-2, where D(i) denotes the derived flag of D, which is defined by D(0)=D and D(i+1)=D(i)+[D(i),D(i)]. Goursat structures appeared first in the work of E. von Weber and E. Cartan, who have shown that on an open and dense subset they can be converted into the so-called Goursat normal form. Later, Goursat structures have been studied by Kumpera and Ruiz. Contact structures on three manifolds and Engel structures on four manifolds are examples of Goursat structures. In the paper, we introduce a new invariant for Goursat structures, called the singularity type, and prove that the growth vector and the abnormal curves of all elements of the derived flag are determined by this invariant. Then we show, using a generalized version of Backlund's theorem, that abnormal curves of all elements of the derived flag do not determine the local equivalence class of a Goursat structure if n>8. We also propose a new proof of a classical theorem of Kumpera and Ruiz. All results are illustrated by the n-trailer system, which, as we show, turns out to be a universal model for all local Goursat structures.