Singularities of 3d vector fields preserving the form of Martinet

TL;DR

This paper classifies the singularities and bifurcations of 3D vector fields that preserve a specific Martinet 1-form, providing a comprehensive understanding of their local structure and unfoldings.

Contribution

It offers the first complete classification of singularities and bifurcations for vector fields preserving the Martinet form in three dimensions.

Findings

Classification of singularities up to diffeomorphisms

Description of transversal unfoldings

Complete list of bifurcations

Abstract

We study the local structure of vector fields on $\mathbb{R}^3$ which preserve the Martinet $1$-form $\alpha=(1+x)dy\pm zdz$. We present the classification of their singularities, up to diffeomorphisms preserving the form $\alpha$, as well as their transversal unfoldings. We are thus able to provide a fairly complete list of the bifurcations such vector fields undergo.