Normal Forms for Dirac-Jacobi bundles and Splitting Theorems for Jacobi Structures

TL;DR

This paper establishes a normal form theorem for Dirac-Jacobi bundles, leading to splitting theorems for Jacobi structures and an alternative proof for homogeneous Poisson structures, advancing the understanding of geometric structures in differential geometry.

Contribution

It introduces a normal form theorem for Dirac-Jacobi bundles and applies it to prove splitting theorems for Jacobi pairs and homogeneous Poisson structures.

Findings

Proved a normal form theorem for Dirac-Jacobi bundles.

Established splitting theorems for Jacobi pairs.

Provided an alternative proof for the splitting theorem of homogeneous Poisson structures.

Abstract

The aim of this paper is to prove a normal form Theorem for Dirac-Jacobi bundles using the recent techniques from Bursztyn, Lima and Meinrenken. As the most important consequence, we can prove the splitting theorems of Jacobi pairs which was proposed by Dazord, Lichnerowicz and Marle. As an application we provide a alternative proof of the splitting theorem of homogeneous Poisson structures.