The structural invariants of Goursat distributions

TL;DR

This paper investigates the local structural invariants of Goursat distributions, establishing a foundation for understanding their geometric properties and their relation to singularities of curves on surfaces.

Contribution

It introduces the concept of structural invariants of Goursat distributions and connects them to the geometry of the monster tower and curve singularities.

Findings

Defined the structural invariants of Goursat distributions

Linked these invariants to the geometry of the monster tower

Set the stage for relating invariants to small growth invariants in the sequel

Abstract

This is the first of a pair of papers devoted to the local invariants of Goursat distributions. The study of these distributions naturally leads to a tower of spaces over an arbitrary surface, called the monster tower, and thence to connections with the topic of singularities of curves on surfaces. Here we study those invariants of Goursat distributions akin to those of curves on surfaces, which we call structural invariants. In the subsequent paper we will relate these structural invariants to the small growth invariants.