Local moduli in the special 2-flags of length 5

TL;DR

This paper investigates the local classification of special 2-flags of length 5, identifying the emergence of numerical moduli in certain singularity classes, extending understanding beyond previously classified lengths.

Contribution

It demonstrates the first appearance of numerical moduli in the local classification of special 2-flags at length 5, expanding the classification framework for these geometric structures.

Findings

Numerical moduli appear in 3 out of 41 classes at length 5.

Classification is finite up to length 4, with moduli emerging at length 5.

The study extends the understanding of local geometries of special 2-flags.

Abstract

A number of key issues concerning distributions generating 1-flags(most often called Goursat flags) has been settled over the past 30 years. Presently similar questions are being discussed as regards distributions generating multi-flags. (More precisely, only so-called special multi-flags,to avoid functional moduli in local classifications.) In particular, special 2-flags of small lengths are a natural ground for the search of generalizations of theorems established earlier for Goursat structures. This includes the search for the first appearing modulus (or moduli) in the classification up to local diffeomorphisms of special 2-flags. (For Goursat flags the first modulus of the local classification appears in length 8.) It has been known in this respect that up to length 4 that classification is finite, and that in length 7 at least one numerical modulus exists. In the last fully classified length 4 possible are precisely 34 local geometries (local models) of special 2-flags. We now demonstrate that in the length 5 single numerical moduli show up in exactly three out of altogether 41 singularity classes existing in that length.