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Foundations on k-contact geometry

Javier de Lucas, Xavier Rivas, Tomasz Sobczak

TL;DR

The paper develops foundations of $k$-contact geometry by introducing a distributional framework: $k$-contact distributions as kernels of $k$-valued 1-forms that are distributionally maximally non-integrable and admit locally $k$ commuting Lie symmetries, linking them to Engel, Goursat, and related distributions. It establishes a deep structural equivalence: a distribution is $k$-contact iff it is maximally non-integrable and locally spanned by $k$ Lie-symmetry fields supplementing the distribution, with Reeb fields generating the symmetry algebra and commuting. It then develops polarisation theory, Darboux-type theorems, and jet-bundle representations (e.g., $J^1(N,E)$) to provide canonical local models, and connects $k$-contact manifolds to presymplectic and $k$-symplectic covers, enabling Hamiltonian formalisms via HDW equations and Lie symmetries for first-order PDEs, including Hamilton–Jacobi and Dirac-type equations. The work also extends to compact examples, Weinstein-type conjectures in the $k$-contact setting, and practical applications to non-holonomic and control systems, while offering multiple routes for symplectification and submanifold theory within this richer geometric framework.

Abstract

k-Contact geometry is a generalisation of contact geometry to analyse field theories. We develop an approach to k-contact geometry based on distributions that are distributionally maximally non-integrable and admit, locally, k commuting supplementary Lie symmetries: the k-contact distributions. We related k-contact distributions with Engel, Goursat and other distributions, which have mathematical and physical interest. We give necessary topological conditions for the existence of globally defined Lie symmetries, k-contact Lie groups are defined and studied, and we study and propose a k-contact Weinstein conjecture for co-oriented k-contact manifolds. Polarisations for k-contact distributions are introduced and it is shown that a polarised k-contact distribution is locally diffeomorphic to the Cartan distribution of the first-order jet bundle over a fibre bundle of order k. We relate k-contact manifolds to presymplectic and k-symplectic manifolds on fibre bundles of larger dimension and define types of submanifolds in k-contact geometry. We study Hamilton-De Donder-Weyl equations in Lie groups for the first time. A theory of k-contact Hamiltonian vector fields is developed, and we describe characteristics of Lie symmetries for first-order partial differential equations in a k-contact Hamiltonian manner. We use our techniques to analyse Hamilton-Jacobi and Dirac equations. Other potential applications of k-contact distributions to non-holonomic and control systems are briefly described.

Foundations on k-contact geometry

TL;DR

The paper develops foundations of -contact geometry by introducing a distributional framework: -contact distributions as kernels of -valued 1-forms that are distributionally maximally non-integrable and admit locally commuting Lie symmetries, linking them to Engel, Goursat, and related distributions. It establishes a deep structural equivalence: a distribution is -contact iff it is maximally non-integrable and locally spanned by Lie-symmetry fields supplementing the distribution, with Reeb fields generating the symmetry algebra and commuting. It then develops polarisation theory, Darboux-type theorems, and jet-bundle representations (e.g., ) to provide canonical local models, and connects -contact manifolds to presymplectic and -symplectic covers, enabling Hamiltonian formalisms via HDW equations and Lie symmetries for first-order PDEs, including Hamilton–Jacobi and Dirac-type equations. The work also extends to compact examples, Weinstein-type conjectures in the -contact setting, and practical applications to non-holonomic and control systems, while offering multiple routes for symplectification and submanifold theory within this richer geometric framework.

Abstract

k-Contact geometry is a generalisation of contact geometry to analyse field theories. We develop an approach to k-contact geometry based on distributions that are distributionally maximally non-integrable and admit, locally, k commuting supplementary Lie symmetries: the k-contact distributions. We related k-contact distributions with Engel, Goursat and other distributions, which have mathematical and physical interest. We give necessary topological conditions for the existence of globally defined Lie symmetries, k-contact Lie groups are defined and studied, and we study and propose a k-contact Weinstein conjecture for co-oriented k-contact manifolds. Polarisations for k-contact distributions are introduced and it is shown that a polarised k-contact distribution is locally diffeomorphic to the Cartan distribution of the first-order jet bundle over a fibre bundle of order k. We relate k-contact manifolds to presymplectic and k-symplectic manifolds on fibre bundles of larger dimension and define types of submanifolds in k-contact geometry. We study Hamilton-De Donder-Weyl equations in Lie groups for the first time. A theory of k-contact Hamiltonian vector fields is developed, and we describe characteristics of Lie symmetries for first-order partial differential equations in a k-contact Hamiltonian manner. We use our techniques to analyse Hamilton-Jacobi and Dirac equations. Other potential applications of k-contact distributions to non-holonomic and control systems are briefly described.
Paper Structure (23 sections, 40 theorems, 240 equations, 1 figure, 2 tables)

This paper contains 23 sections, 40 theorems, 240 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Basic notions
  3. Generalised subbundles
  4. Contact manifolds
  5. --Vector fields and its integral sections
  6. -Symplectic manifolds
  7. A review on --contact geometry
  8. Field theories on co-oriented -contact manifolds
  9. A new approach: -contact distributions
  10. k-Contact distributions and their Lie symmetries
  11. Topological properties of co-oriented k-contact manifolds
  12. Representing -contact distributions
  13. Compact -contact manifolds
  14. Polarised -contact distributions
  15. -symplectization of -contact structures
  16. ...and 8 more sections

Key Result

Theorem 2.3

Given a co-oriented contact manifold $(M,\eta)$ such that $\dim M=2n+1$, there exist local coordinates $\{q^i, p_i, s\}$ with $i = 1,\dotsc,n$ around every point $x\in M$, called Darboux coordinates, such that In these coordinates, the Reeb vector field reads $R = \partial /\partial s$.

Figures (1)

  • Figure 1: Representation of the first three coordinates in \ref{['Eq:Dense']} in terms of $\theta_1\in[0,200\pi]$ and the last three coordinates of \ref{['Eq:Cyclic']} for every $\theta_2\in \mathbb{R}$.

Theorems & Definitions (130)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Contact Darboux theorem AM_78LM_87
  • Example 2.4: Canonical co-oriented contact manifold
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Example 2.9: Canonical $k$-symplectic manifold
  • Theorem 2.10: Darboux theorem for polarised $k$-symplectic manifolds Awa_92GLRR_24
  • ...and 120 more