On Poisson structures on R4

TL;DR

This paper investigates Poisson structures on four-dimensional Euclidean space, providing explicit formulas for their geometric objects, analyzing unimodularity, and exploring conditions for characteristic foliations.

Contribution

It introduces explicit formulas for Poisson geometric objects in R4 and discusses unimodular structures and characteristic foliation realizability.

Findings

Explicit formulas for Poisson structures in R4

Characterization of unimodular Poisson structures

Existence results for characteristic foliations

Abstract

This paper is devoted to the study of Poisson structures on the Euclidean four dimensional space R4. By using the properties of the trace operator associated to a volumen form and the elementary vector calculus operations in R3, we give explicit formulas for the main geometric objects associated to the Poisson structures in R4, including its characteristic foliation, the Hamiltonian and Poisson vector fields, normal forms and some useful decomposition formulae for Poisson tensors. We also discuss the class of unimodular Poisson structures and give two results about the existence of Poisson structures having as its characteristic foliation a given arbitrary regular foliation.