Pairs of differential forms: a framework for precontact geometry

TL;DR

This paper develops a comprehensive geometric framework for precontact manifolds by analyzing pairs of differential forms, characterizing their properties, and establishing Hamiltonian dynamics, thereby extending contact geometry.

Contribution

It introduces a unified approach to precontact structures via pairs of forms, clarifies their geometric properties, and defines Hamiltonian dynamics on these manifolds.

Findings

Characterization of pairs of 1-forms and 2-forms under regularity conditions

Analysis of Reeb and Liouville vector fields in precontact geometry

Development of Hamiltonian dynamics on precontact manifolds

Abstract

Precontact manifolds extend contact geometry by weakening the maximal non-integrability condition of the defining $1$-form. We clarify the geometric foundations of this structure by studying general pairs of a $1$-form and a $2$-form under mild regularity conditions. We characterize them through their class, analyse the role of distinguished vector fields, such as Reeb or Liouville fields, and study other associated geometrical objects. Precontact structures are then treated as the special case of pairs formed by a nowhere-vanishing $1$-form and its exterior derivative. We also define Hamiltonian dynamics on precontact manifolds. Several examples are presented to illustrate the theory.