Remarks on Contact and Jacobi Geometry

Abstract

We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and linear Kirillov structures, i.e., homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal $\mathrm{GL}(1,\mathbb{R})$-bundle structure on the manifold and not just to a vector field. This allows for working with Jacobi bundle structures on nontrivial line bundles and drastically simplifies the picture of Jacobi and contact geometry. Our results easily reduce to various basic theorems of Jacobi and contact geometry when the principal bundle structure is trivial, while giving new insights into the theory.

References in corpus (9)

• Generalized Contact Bundles
• Higher order mechanics on graded bundles
• A note on actions of some monoids
• Graded Bundles in the Category of Lie Groupoids
• On Morphic Actions and Integrability of LA-Groupoids
• Polarisation of Graded Bundles
• On the integration of LA-groupoids and duality for Poisson groupoids
• Contact structures on Lie algebroids
• Locally conformal symplectic groupoids

Cited by in corpus (16)

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• A novel approach to contact Hamiltonians and contact Hamilton-Jacobi theory
• Generalized Contact Bundles
• Unified Lagrangian-Hamiltonian formalism for contact systems
• Infinitesimal Automorphisms of VB-groupoids and algebroids
• Lie groupoids in information geometry
• Reductions: precontact versus presymplectic
• Graded Bundles in the Category of Lie Groupoids
• Contact Dual Pairs
• Contactifications: a Lagrangian description of compact Hamiltonian systems
• Contact structures on Lie algebroids
• Contact Dynamics versus Legendrian and Lagrangian Submanifolds
• On the Linearization Covering Technique and its Application to Integrable Nonlinear Differential Systems
• Homogeneous G-structures
• Gauge transformations of Jacobi structures and contact groupoids
• Carrollian ${\mathbb{R}}^{\times }$-bundles: Connections and Beyond