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Pairs of differential forms: a framework for precontact geometry

Xavier Gràcia, Àngel Martínez-Muñoz, Xavier Rivas

TL;DR

The paper develops a unified framework for precontact geometry by studying pairs $(\tau,\omega)$ of a 1-form and a 2-form under a constant-class regularity assumption. It introduces the class of a pair, the characteristic and extended characteristic distributions, and the characteristic tensor $B=\tau\otimes\tau+\omega$, linking these to Reeb and Liouville vector fields. The framework recovers and interrelates a range of classical geometries—contact, cosymplectic, symplectic, and locally conformally symplectic—and extends Hamiltonian dynamics to precontact settings, including cases without distinguished vector fields. It further develops precontact forms as special cases $(\eta, d\eta)$ and analyzes how conformal changes can flip or preserve the parity of the class, with precise criteria in terms of Lie derivatives along Liouville or Reeb fields. The results provide a robust, general toolkit for analyzing singular Lagrangian systems and constrained dynamics within a single geometric language, with broad potential applications in geometric mechanics and beyond.

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.

Pairs of differential forms: a framework for precontact geometry

TL;DR

The paper develops a unified framework for precontact geometry by studying pairs of a 1-form and a 2-form under a constant-class regularity assumption. It introduces the class of a pair, the characteristic and extended characteristic distributions, and the characteristic tensor , linking these to Reeb and Liouville vector fields. The framework recovers and interrelates a range of classical geometries—contact, cosymplectic, symplectic, and locally conformally symplectic—and extends Hamiltonian dynamics to precontact settings, including cases without distinguished vector fields. It further develops precontact forms as special cases and analyzes how conformal changes can flip or preserve the parity of the class, with precise criteria in terms of Lie derivatives along Liouville or Reeb fields. The results provide a robust, general toolkit for analyzing singular Lagrangian systems and constrained dynamics within a single geometric language, with broad potential applications in geometric mechanics and beyond.

Abstract

Precontact manifolds extend contact geometry by weakening the maximal non-integrability condition of the defining -form. We clarify the geometric foundations of this structure by studying general pairs of a -form and a -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 -form and its exterior derivative. We also define Hamiltonian dynamics on precontact manifolds. Several examples are presented to illustrate the theory.
Paper Structure (23 sections, 37 theorems, 124 equations)

This paper contains 23 sections, 37 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Doublets of a one-form and a two-form
  3. Rank of a differential form
  4. Pairs of differential forms
  5. Doublets of a 1-form and a 2-form
  6. Reeb and Liouville vector fields
  7. Other characterizations of the class
  8. Geometrical structures associated with a doublet
  9. The characteristic tensor field
  10. The extended characteristic distribution
  11. The extended 3-form
  12. Some important examples
  13. Contact structures
  14. Cosymplectic and precosymplectic structures
  15. Symplectic and presymplectic structures
  16. ...and 8 more sections

Key Result

Theorem 2.2

Given a $p$-form $\alpha \in \Omega^{p} (M)$ on a manifold $M$, the image of the multilinear map is $(\mathrm{Ker \,} \alpha_x)^{\circ}$.

Theorems & Definitions (84)

  • Definition 2.1
  • Theorem 2.2: god_1969
  • Theorem 2.3: god_1969LM_1987
  • Proposition 2.4: god_1969
  • Definition 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Example 2.8
  • Example 2.9
  • ...and 74 more