A Jacobi Coupling Construction on Associated Bundles

Aissa Wade; Djideme Franck Houenou; Emmanuel Davakan

arxiv: 2606.30614 · v1 · pith:57XTJRLKnew · submitted 2026-06-29 · 🧮 math.DG

A Jacobi Coupling Construction on Associated Bundles

Emmanuel Davakan , Djideme Franck Houenou , Aissa Wade This is my paper

Pith reviewed 2026-06-30 03:28 UTC · model grok-4.3

classification 🧮 math.DG

keywords Jacobi structuresassociated bundlescoupling constructionprincipal bundlesconnectionscurvatureReeb vector fieldHamiltonian G-spaces

0 comments

The pith

The associated bundle carries a Jacobi structure induced from a Jacobi Hamiltonian G-space and a principal bundle with nondegenerate curvature connection.

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

The paper extends the Sternberg-Weinstein coupling construction to Jacobi geometry. It shows that given a Jacobi Hamiltonian G-space and a principal bundle with a connection satisfying a nondegeneracy condition on its curvature, the associated bundle obtains a Jacobi structure. This structure is compatible with the canonical Jacobi structures on the fibers. The construction unifies the symplectic, locally conformal symplectic, and contact cases into one framework. It also brings to light new coupling effects involving the Reeb vector field.

Core claim

Starting from a Jacobi Hamiltonian G-space and a principal bundle equipped with a connection whose curvature satisfies some nondegeneracy condition, the associated bundle naturally carries a Jacobi structure compatible with the canonical ones on the fibers. This construction provides a unified framework encompassing the symplectic, locally conformal symplectic, and contact cases. It reveals new coupling phenomena related to the presence of the Reeb vector field.

What carries the argument

The Jacobi coupling construction on associated bundles using a connection with nondegenerate curvature to induce a compatible Jacobi structure.

If this is right

The construction encompasses the symplectic, locally conformal symplectic, and contact cases.
New coupling phenomena related to the Reeb vector field are revealed.
The induced Jacobi structure is compatible with the canonical ones on the fibers.

Where Pith is reading between the lines

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

This framework may enable the construction of additional examples of Jacobi manifolds by varying the principal bundle and connection.
Potential connections to reduction procedures in Jacobi geometry could be investigated further.
The emphasis on the Reeb vector field suggests new ways to study contact-type structures within the Jacobi setting.

Load-bearing premise

The curvature of the connection on the principal bundle must satisfy a nondegeneracy condition.

What would settle it

An explicit example of a principal bundle and Jacobi Hamiltonian G-space where the curvature is degenerate but a compatible Jacobi structure still exists on the associated bundle.

read the original abstract

We extend the Sternberg--Weinstein coupling construction to the Jacobi geometry setting. Starting from a Jacobi Hamiltonian $G$-space and a principal bundle equipped with a connection whose curvature satisfies some nondegeneracy condition, we show that the associated bundle naturally carries a Jacobi structure compatible with the canonical ones on the fibers. This construction provides a unified framework encompassing the symplectic, locally conformal symplectic, and contact cases. It reveals new coupling phenomena related to the presence of the Reeb vector field.

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 an explicit Jacobi version of the Sternberg-Weinstein coupling on associated bundles, but the nondegeneracy condition on curvature is the load-bearing step.

read the letter

The core result is a construction that starts with a Jacobi Hamiltonian G-space and a principal bundle with connection, then equips the associated bundle with a Jacobi structure compatible with the fiberwise ones, as long as the curvature meets a nondegeneracy condition. This single setup recovers the symplectic, locally conformal symplectic, and contact cases and flags some new coupling behavior tied to the Reeb field.

The extension itself is the main contribution. It is not just a routine rewrite; the authors carry the known symplectic argument over and adjust for the extra 1-form and Reeb direction that appear in the Jacobi setting. That unification is useful for anyone who already works with these geometries side by side.

The soft spot is the nondegeneracy condition on the curvature. It is required for the induced bivector and 1-form to satisfy the Jacobi identity and to stay compatible with the fibers. The paper verifies the identity under the assumption but does not show necessity or supply examples where the condition fails, so the range of the construction stays narrower than the abstract suggests. No counter-examples or boundary cases are given.

The work is aimed at people already inside Jacobi geometry or geometric mechanics. A reader who knows the symplectic Sternberg-Weinstein construction will see the extension clearly and can judge how much the extra condition restricts applications.

I would send it to referees. The central claim is new and formally stated; the limitation around the curvature condition is real but not fatal, and a referee can ask for more on its scope.

Referee Report

2 major / 2 minor

Summary. The manuscript extends the Sternberg-Weinstein coupling construction to the Jacobi setting. Starting from a Jacobi Hamiltonian G-space and a principal G-bundle equipped with a connection whose curvature satisfies a nondegeneracy condition, the associated bundle is shown to carry a Jacobi structure compatible with the canonical Jacobi structures on the fibers. The construction unifies the symplectic, locally conformal symplectic, and contact cases and identifies new coupling phenomena involving the Reeb vector field.

Significance. If the central construction holds under the stated hypotheses, the result supplies a unified framework that recovers and generalizes several known coupling constructions in Jacobi geometry. The explicit treatment of the Reeb vector field in the coupling is a potentially useful new feature.

major comments (2)

[Main theorem (likely §3)] The nondegeneracy condition on the curvature is the load-bearing hypothesis for both the Jacobi identity and fiberwise compatibility. The precise statement of this condition (and its interaction with the Jacobi Hamiltonian G-action) must be given explicitly in the main theorem, together with a verification that the induced bivector and 1-form satisfy the Jacobi identity under exactly this hypothesis.
[Main theorem and §4 (compatibility)] It is not shown whether the nondegeneracy condition is necessary. Either a necessity argument or a concrete counter-example (a connection whose curvature violates the condition and for which the induced structure fails to be Jacobi) should be supplied to clarify the scope of the construction.

minor comments (2)

The abstract refers to 'some nondegeneracy condition' without a forward reference; the introduction should state the condition at the outset.
Notation for the Jacobi structure (bivector plus 1-form) should be fixed consistently from the preliminaries onward.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the constructive major comments. We address each point below and commit to revisions that will strengthen the presentation of the main theorem.

read point-by-point responses

Referee: [Main theorem (likely §3)] The nondegeneracy condition on the curvature is the load-bearing hypothesis for both the Jacobi identity and fiberwise compatibility. The precise statement of this condition (and its interaction with the Jacobi Hamiltonian G-action) must be given explicitly in the main theorem, together with a verification that the induced bivector and 1-form satisfy the Jacobi identity under exactly this hypothesis.

Authors: We agree that the nondegeneracy condition must be stated with full precision in the main theorem. In the revised manuscript we will restate Theorem 3.1 so that the nondegeneracy hypothesis on the curvature appears explicitly, together with its precise interaction with the Jacobi Hamiltonian G-action. We will also expand the proof to contain a self-contained verification that the induced bivector and 1-form satisfy the Jacobi identity if and only if this condition holds. revision: yes
Referee: [Main theorem and §4 (compatibility)] It is not shown whether the nondegeneracy condition is necessary. Either a necessity argument or a concrete counter-example (a connection whose curvature violates the condition and for which the induced structure fails to be Jacobi) should be supplied to clarify the scope of the construction.

Authors: The present version establishes sufficiency of the nondegeneracy condition but does not treat necessity. In the revision we will add to §4 an explicit counter-example: a principal G-bundle with connection whose curvature fails the nondegeneracy condition, together with a direct computation showing that the induced bivector and 1-form on the associated bundle do not satisfy the Jacobi identity. This will make the sharpness of the hypothesis transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity: construction from independent external inputs

full rationale

The paper describes a direct geometric construction: inputs are a Jacobi Hamiltonian G-space and a principal bundle with connection whose curvature meets a stated nondegeneracy condition; the output is an induced Jacobi structure on the associated bundle that is compatible with the fiberwise structures. This matches the Sternberg-Weinstein pattern but in the Jacobi setting and does not reduce any derived object to a fitted parameter, self-definition, or self-citation chain. The nondegeneracy condition is an explicit hypothesis required for the Jacobi identity to hold, not a quantity obtained by fitting or renaming within the paper. No load-bearing step quotes a prior result by the same authors as the sole justification for a uniqueness or ansatz claim. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review is based solely on the abstract; the ledger is therefore minimal and provisional. The construction rests on the existence of a Jacobi Hamiltonian G-space and a principal bundle whose connection curvature meets an unspecified nondegeneracy condition.

axioms (2)

domain assumption Existence of a Jacobi Hamiltonian G-space
The construction begins from this object as input.
domain assumption Principal bundle with connection whose curvature satisfies a nondegeneracy condition
This condition is required for the associated bundle to carry the Jacobi structure.

pith-pipeline@v0.9.1-grok · 5599 in / 1232 out tokens · 53341 ms · 2026-06-30T03:28:34.644755+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references

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Brahic, R.-L

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2008

[2] [2]

Dazord, A

P. Dazord, A. Lichnerowicz, C.-M. Marle,Structure locale des variétés de Jacobi, J. Math. Pures Appl.70(1991), 101–152

1991

[3] [3]

Gotay, R

M. Gotay, R. Lashof, J.`Sniatycki, A. Weinstein,Closed forms on symplectic bundles, Comment. Math. Helv.58(1983), 617–621

1983

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Guillemin, S

V. Guillemin, S. Sternberg,Symplectic Techniques in Physics, Cambridge University Press, 1984

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Houenou, A

F. Houenou, A. Wade, E. Davakan,Coupling 2-forms and Locally Conformal Symplectic Fibrations, Preprint 2025

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E. Lerman,Contact fiber bundles, J. Geom. Phys.49(2004), 52–66

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1978

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1985

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2004

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A. Wade,Poisson fiber bundles and coupling Dirac structures, Ann. Global Anal. Geom. 33(2008), no. 3, 207–217

2008

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Wade,Integration of conformal Jacobi fibrations and prequantization of Poisson fibrations, Afr

A. Wade,Integration of conformal Jacobi fibrations and prequantization of Poisson fibrations, Afr. Diaspora J. Math.14(2013), no. 2, 187–196

2013

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Weinstein,Fat bundles and symplectic manifolds, Adv

A. Weinstein,Fat bundles and symplectic manifolds, Adv. Math.37(1980), 239–250. 19

1980