On the geometry of Goursat structures

TL;DR

This paper studies Goursat structures on manifolds, introducing a new invariant called the singularity type, and explores their properties, equivalence, and applications using the n-trailer system as a universal model.

Contribution

It introduces the singularity type invariant for Goursat structures and analyzes its implications for local equivalence and abnormal curves.

Findings

The singularity type determines growth vector and abnormal curves.

Abnormal curves do not fully determine local equivalence for dimensions greater than 8.

The n-trailer system serves as a universal model for Goursat structures.

Abstract

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.