Godbillon-Vey classes of regular Jacobi manifolds
Shuhei Yonehara
TL;DR
This work extends the classical Godbillon–Vey framework from regular Poisson (symplectic) foliations to regular Jacobi manifolds by giving explicit GV-class formulas in terms of Jacobi data $(\pi,E)$, distinguishing LCS-type and contact-type leaves. Using a volume-form based isomorphism $\varphi$ and the operator $\psi$, the authors derive concrete expressions for $GV(\mathcal{F}_{(\pi,E)})$ involving $(\pi,E)$, with separate treatments for the LCS-type (even leaves) and the contact-type (odd leaves) cases, and provide a Poissonization-based alternative proof for the contact-type scenario. The paper also analyzes the codimension-one situation, discusses naturality and unimodularity relations, and offers codimension-one examples, thereby linking foliation invariants to Jacobi-structure data. The results advance understanding of characteristic classes of foliations arising from Jacobi geometry and connect Jacobi theory with established Poisson-foliation methods, potentially impacting the study of regular foliations in contact/LCS settings.
Abstract
The notion of a Jacobi manifold is a natural generalization of that of a Poisson manifold. A Jacobi manifold has a natural foliation in which each leaf has either a contact structure or a locally conformal symplectic structure. In this paper, we study a characteristic class called the Godbillon-Vey class for Jacobi manifolds with regular foliation and express it explicitly in terms of Jacobi structures.