Nambu structures and integrable 1-forms

TL;DR

This paper explores the application of Nambu structures to the study of singularities in integrable 1-forms, providing a classification and new insights inspired by Takhtajan's formalism.

Contribution

It demonstrates that certain integrable 1-forms can be viewed as pull-backs of 2D forms and classifies quadratic integrable 1-forms, offering a novel perspective on singularities.

Findings

Integrable 1-forms vanishing at a point are pull-backs of 2D forms.

Classification of quadratic integrable 1-forms.

New geometric insights from Nambu structures.

Abstract

Some years ago Mosh\'e Flato pointed up that it could be interesting to develop the Nambu's idea to generalize Hamiltonian mechanic. An interesting new formalism in that direction was proposed by T. Takhtajan. His theory gave new perspectives concerning deformation quantization, and many authors have developed its mathematical features. The purpose of this paper is to show that this theory, at first designated to physic, gives a new point of view for the study of singularities of integrable 1-forms. Namely, we will prove that any integrable 1-form which vanishes at a point and has a non-zero linear part at this point is, up to multiplication by a non-vanishing function, the formal pull-back of a two dimensional 1-form. We also obtain a classification of quadratic integrable 1-forms.