Regular Jacobi Structures and Generalized Contact Bundles

TL;DR

This paper explores the conditions under which regular Jacobi structures and transverse complex structures induce generalized contact structures, providing new examples and counterexamples in the field of geometric structures on manifolds.

Contribution

It establishes criteria for constructing generalized contact bundles from regular Jacobi structures and transverse complex structures, and demonstrates their existence on nilmanifolds and contact bundles.

Findings

Every 5-dimensional nilmanifold admits an invariant generalized contact structure.

All contact bundles over complex manifolds have compatible generalized contact structures.

Counterexample of a conformal symplectic bundle lacking a compatible generalized contact structure.

Abstract

A Jacobi structure $J$ on a line bundle $L\to M$ is weakly regular if the sharp map $J^\sharp : J^1 L \to DL$ has constant rank. A generalized contact bundle with regular Jacobi structure possess a transverse complex structure. Paralleling the work of Bailey in generalized complex geometry, we find condition on a pair consisting of a regular Jacobi structure and an transverse complex structure to come from a generalized contact structure. In this way we are able to construct interesting examples of generalized contact bundles. As applications: 1) we prove that every 5-dimensional nilmanifold is equipped with an invariant generalized contact structure, 2) we show that, unlike the generalized complex case, all contact bundles over a complex manifold possess a compatible generalized contact structure. Finally we provide a counterexample presenting a locally conformal symplectic bundle over a generalized contact manifold of complex type that do not possess a compatible generalized contact structure.