Remarks on Contact and Jacobi Geometry

TL;DR

This paper simplifies Jacobi and contact geometry by using Kirillov manifolds and linear Kirillov structures, relating homogeneity to principal bundle structures, thus providing clearer insights and extending the theory to nontrivial line bundles.

Contribution

It introduces a natural approach to Jacobi and contact geometry via principal bundle structures, simplifying existing complex proofs and extending the framework to nontrivial line bundles.

Findings

Simplifies Jacobi and contact geometry using Kirillov manifolds.

Relates homogeneity to principal ${ m GL}(1,{ m R})$-bundles, not just vector fields.

Provides a unified framework that extends to nontrivial line bundles.

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 ${\rm GL}(1,{\mathbb R})$-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.