Contact Isotropic Realisations of Jacobi Manifolds via Spencer Operators

TL;DR

This paper studies contact isotropic realisations of Jacobi manifolds, classifying them and exploring their relation to symplectic realisations, using Spencer operators to generalise and clarify existing results.

Contribution

It provides a classification of contact isotropic realisations of Jacobi manifolds and links their existence to symplectic realisations via Spencer operators.

Findings

Classification of contact isotropic realisations

Relation between symplectic and contact realisations

Use of Spencer operators for conceptual proofs

Abstract

Motivated by the importance of symplectic isotropic realisations in the study of Poisson manifolds, this paper investigates the local and global theory of contact isotropic realisations of Jacobi manifolds, which are those of minimal dimension. These arise naturally when considering multiplicity-free actions in contact geometry, as shown in this paper. The main results concern a classification of these realisations up to a suitable notion of isomorphism, as well as establishing a relation between the existence of symplectic and contact isotropic realisations for Poisson manifolds. The main tool is the classical Spencer operator which is related to Jacobi structures via their associated Lie algebroid, which allows to generalise previous results as well as providing more conceptual proofs for existing ones.