Local and global properties of Jacobi related geometries

Abstract

In this thesis local and global properties of Jacobi and related geometries are discussed, which means for us that so-called Dirac-Jacobi bundles are considered. The whole work is roughly divided in three parts, which are independent of each other up to preliminaries. In the rst part local and semi-local properties of Dirac-Jacobi bundles are considered, in particular it is proven that a Dirac-Jacobi bundle is always of a certain form close to suitable transversal manifolds. These semi-local structure theorems are usually refered to as normal form theorems. Using the normal form theorems, we prove local splitting theorems of Jacobi brackets, generalized contact bundles and homogeneous Poisson manifolds. The second part is dedicated to the study of weak dual pairs in Dirac-Jacobi geometry. It is proven that weak dual pairs give rise to an equivalence relation in the category of Dirac-Jacobi bundles. After that, the similarities of equivalent Dirac-Jacobi bundles are discussed in detail. The goal of the last part is to nd global obstructions for existence of generalized contact structures. With the main result of this chapter it is easy to nd nontrivial examples of theses structures and two classes are discussed in detail. [edited by Author]