Normalization of singular contact forms and primitive 1-forms

TL;DR

This paper studies the local normal forms of singular contact 1-forms on odd-dimensional manifolds, extending previous results by employing classical and toric normalization techniques.

Contribution

It provides new normal form classifications for singular contact and primitive 1-forms, improving upon earlier work by Lychagin, Webster, and Zhitomirskii.

Findings

Classifies local normal forms of singular contact forms.

Extends previous results with improved normalization techniques.

Connects singular contact forms to primitive 1-forms and conformal vector fields.

Abstract

A differential 1-form $\alpha$ on a manifold of odd dimension $2n+1$, which satisfies the contact condition $\alpha \wedge (d\alpha)^n \neq 0$ almost everywhere, but which vanishes at a point $O$, i.e. $\alpha (O) = 0$, is called a \textit{singular contact form} at $O$. The aim of this paper is to study local normal forms (formal, analytic and smooth) of such singular contact forms. Our study leads naturally to the study of normal forms of singular primitive 1-forms of a symplectic form $\omega$ in dimension $2n$, i.e. differential 1-forms $\gamma$ which vanish at a point and such that $d\gamma = \omega$, and their corresponding conformal vector fields. Our results are an extension and improvement of previous results obtained by other authors, in particular Lychagin \cite{Lychagin-1stOrder1975}, Webster \cite{Webster-1stOrder1987} and Zhitomirskii \cite{Zhito-1Form1986,Zhito-1Form1992}. We make use of both the classical normalization techniques and the toric approach to the normalization problem for dynamical systems \cite{Zung_Birkhoff2005, Zung_Integrable2016, Zung_AA2018}.