Godbillon-Vey classes of regular Jacobi manifolds

Shuhei Yonehara

arxiv: 2412.05257 · v3 · submitted 2024-12-06 · 🧮 math.DG · math.SG

Godbillon-Vey classes of regular Jacobi manifolds

Shuhei Yonehara This is my paper

Pith reviewed 2026-05-23 08:11 UTC · model grok-4.3

classification 🧮 math.DG math.SG

keywords Godbillon-Vey classJacobi manifoldregular foliationcharacteristic classJacobi structurebivectorcontact structurelocally conformal symplectic

0 comments

The pith

The Godbillon-Vey class for Jacobi manifolds with regular foliation reduces to an explicit expression in the Jacobi bivector and vector field.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the Godbillon-Vey class for Jacobi manifolds whose induced foliation is regular. It derives an explicit formula for this class directly from the bivector and vector field that define the Jacobi structure. Jacobi manifolds generalize Poisson manifolds, and their leaves carry contact structures or locally conformal symplectic structures. The explicit expression lets the class be computed from the structure tensors alone, without separate singularity handling. This supplies a uniform way to obtain the class across the contact and conformal symplectic cases that arise on the leaves.

Core claim

For a Jacobi manifold whose foliation is regular, the Godbillon-Vey class equals an explicit expression constructed from the Jacobi bivector and the vector field that together define the Jacobi structure.

What carries the argument

The Godbillon-Vey class of the regular foliation induced by a Jacobi structure, written directly in terms of the bivector and vector field.

If this is right

The class becomes computable directly from the defining tensors on both contact and locally conformal symplectic leaves.
Previous formulas for the class in the Poisson and contact settings appear as special cases of the single expression.
The class functions as a topological invariant that can be read off the Jacobi data without additional foliation analysis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The formula may simplify explicit calculations of the class on concrete families of Jacobi manifolds such as Lie groups or homogeneous spaces.
When the vector field vanishes identically the expression should recover the known Godbillon-Vey formula for Poisson manifolds.
The same reduction technique might apply to other characteristic classes of the foliation once regularity is assumed.

Load-bearing premise

The foliation induced by the Jacobi structure is regular.

What would settle it

A regular Jacobi manifold whose Godbillon-Vey class computed from the foliation differs from the explicit expression in its bivector and vector field would falsify the claim.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a direct formula for the Godbillon-Vey class on regular Jacobi manifolds pulled from the bivector and vector field.

read the letter

The main takeaway is that this paper writes down an explicit expression for the Godbillon-Vey class on a Jacobi manifold whose characteristic foliation is regular, and it does so straight from the Jacobi bivector and vector field. The regularity assumption lets them skip any singularity handling and just reduce the standard foliation construction to the given data. That reduction appears to be the new concrete piece not already in the Poisson or contact references they cite. The abstract is clear that this is the goal, and the stress-test note confirms the hypothesis is stated up front and does exactly the work needed. If the derivation holds, it is a modest but usable reference for people who want to compute the class on concrete Jacobi examples. The paper stays focused and does not overclaim. It treats the Jacobi setting as a natural extension and simply carries the class over under the regularity condition. That is honest work. The only real limitation visible from the abstract is that we do not see the actual steps or any verification examples, so the soundness cannot be checked in detail yet. The citation pattern is standard and appropriate for the area. No circularity or invented objects show up. This is the sort of short, targeted note that belongs in a differential geometry journal. Readers already working with Jacobi structures or with characteristic classes on foliations will get immediate value from the formula. It will not reorganize the field, but it fills a small, well-defined gap. I would send it to referees. A serious editor should give it a review rather than desk-reject it; the central claim is narrow enough that referees can check it quickly and the regularity condition is handled cleanly.

Referee Report

0 major / 2 minor

Summary. The manuscript studies the Godbillon-Vey class of Jacobi manifolds whose characteristic foliation is regular. Under this hypothesis it derives an explicit formula expressing the class directly in terms of the Jacobi bivector and the associated vector field.

Significance. If the derivation is correct, the result supplies a concrete computational expression that reduces the foliation-theoretic class to data on the Jacobi structure, removing the need for separate singularity analysis precisely when the regularity assumption holds. This is a natural and potentially useful extension of analogous explicit formulae known for Poisson manifolds.

minor comments (2)

[Abstract] Abstract: the phrase 'express it explicitly in terms of Jacobi structures' is accurate but gives no hint of the form of the expression; a single sentence indicating the main ingredients (e.g., 'in terms of the bivector, the vector field, and their Lie derivatives') would improve readability for specialists.
The regularity hypothesis is stated clearly, but a brief remark comparing the resulting formula with the classical Godbillon-Vey class on a contact manifold (when the Jacobi structure is contact) would help situate the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The referee's summary correctly identifies the main result: an explicit expression for the Godbillon-Vey class of regular Jacobi manifolds in terms of the Jacobi bivector and Reeb vector field.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claim is that, under the explicit regularity hypothesis on the characteristic foliation of a Jacobi manifold, the Godbillon-Vey class reduces directly to an explicit expression in the underlying Jacobi bivector and vector field. This is a standard foliation-to-structure translation enabled by the regularity assumption stated at the outset; no equations, parameters, or self-citations are shown to create a definitional loop, fitted-input prediction, or load-bearing self-reference. The derivation chain is therefore self-contained against the stated inputs and does not reduce to them by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard axioms of smooth manifold theory, Lie algebroid cohomology, and the definition of Jacobi structures; no free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Standard axioms of differential geometry: smooth manifolds, vector bundles, and Lie algebroid structures.
Invoked to define Jacobi manifolds and their induced foliations.
domain assumption Existence and basic properties of the Godbillon-Vey class for regular foliations.
Taken from foliation theory and applied to the Jacobi case.

pith-pipeline@v0.9.0 · 5571 in / 1140 out tokens · 19624 ms · 2026-05-23T08:11:02.988560+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Dazord, A

P. Dazord, A. Lichnerowicz, and C.M. Marle. Structure locale des vari \'e t \'e s de Jacobi . Journal de Math \'e matiques Pures et Appliqu \'e es , 70(1):101--152, 1991

work page 1991
[2]

Godbillon and J

C. Godbillon and J. Vey. Un invariant des feuilletages de codimension 1. Comptes rendus de l'Acad\' e mie des Sciences , 273:92--95, 1971

work page 1971
[3]

Lichnerowicz

A. Lichnerowicz. Les vari \'e t \'e s de Jacobi et leurs alg \`e bres de Lie associ \'e es. Journal de Math \'e matiques Pures et Appliqu \'e es , 57:453--488, 1978

work page 1978
[4]

K. Mikami. Godbillon - Vey classes of symplectic foliations. Pacific Journal of Mathematics , 194(1):165--174, 2000

work page 2000
[5]

Moerdijk and J

I. Moerdijk and J. Mr c un. Introduction to foliations and Lie groupoids , volume 91. Cambridge university press, 2003

work page 2003

[1] [1]

Dazord, A

P. Dazord, A. Lichnerowicz, and C.M. Marle. Structure locale des vari \'e t \'e s de Jacobi . Journal de Math \'e matiques Pures et Appliqu \'e es , 70(1):101--152, 1991

work page 1991

[2] [2]

Godbillon and J

C. Godbillon and J. Vey. Un invariant des feuilletages de codimension 1. Comptes rendus de l'Acad\' e mie des Sciences , 273:92--95, 1971

work page 1971

[3] [3]

Lichnerowicz

A. Lichnerowicz. Les vari \'e t \'e s de Jacobi et leurs alg \`e bres de Lie associ \'e es. Journal de Math \'e matiques Pures et Appliqu \'e es , 57:453--488, 1978

work page 1978

[4] [4]

K. Mikami. Godbillon - Vey classes of symplectic foliations. Pacific Journal of Mathematics , 194(1):165--174, 2000

work page 2000

[5] [5]

Moerdijk and J

I. Moerdijk and J. Mr c un. Introduction to foliations and Lie groupoids , volume 91. Cambridge university press, 2003

work page 2003