pith. sign in

arxiv: 2606.20030 · v1 · pith:CNII34ULnew · submitted 2026-06-18 · 🧮 math.DG · math-ph· math.MP

Poisson and Jacobi structures from 2-covariant tensors

Manuel de Le\'on , Xavier Gr\`acia , Rub\'en Izquierdo-L\'opez , \'Angel Mart\'inez-Mu\~noz , Xavier Rivas This is my paper

Pith reviewed 2026-06-26 16:11 UTC · model grok-4.3

classification 🧮 math.DG math-phmath.MP
keywords Poisson structuresJacobi structuresSchouten-Nijenhuis bracket2-covariant tensorssymplectic geometrycontact geometrycosymplectic structuresfat bundles
0
0 comments X

The pith

A formula expresses the Schouten-Nijenhuis bracket of the bivector induced by a 2-covariant tensor as curvature of a distribution plus the exterior derivative of a form, giving the obstruction to Poisson or Jacobi structures.

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

The paper constructs Poisson and Jacobi structures from 2-covariant tensors under suitable assumptions on the tensor. It derives an explicit formula for the Schouten-Nijenhuis bracket of the induced bivector field that involves the curvature of an associated distribution together with the exterior derivative of a differential form. This expression directly measures the obstruction to the bivector satisfying the Jacobi identity required for Poisson or Jacobi structures. The same formula recovers the standard brackets arising in symplectic, locally conformally symplectic, cosymplectic, and contact geometries. The framework further supplies necessary and sufficient conditions for fat bundles and almost cosymplectic structures of order p to induce Jacobi brackets.

Core claim

Under suitable assumptions on the tensor, the Schouten-Nijenhuis bracket of the associated bivector field is given by an expression involving the curvature of a certain distribution and the exterior derivative of a differential form; this expression is the obstruction to the existence of a Poisson or Jacobi structure.

What carries the argument

The formula computing the Schouten-Nijenhuis bracket of the bivector associated to the 2-covariant tensor, written in terms of distribution curvature and the exterior derivative of a form.

If this is right

  • The induced bivector on a symplectic manifold has vanishing Schouten-Nijenhuis bracket precisely when the curvature term and exterior derivative term both vanish.
  • The same obstruction formula specializes to the known Jacobi bracket on a contact manifold.
  • Fat bundles induce Jacobi structures exactly when the curvature and exterior derivative terms satisfy the stated algebraic relations.
  • Almost cosymplectic structures of order p yield Jacobi brackets under the explicit conditions given by the formula.

Where Pith is reading between the lines

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

  • The same curvature-plus-derivative expression may serve as a test for Poisson or Jacobi properties in other classes of 2-tensors not treated explicitly in the paper.
  • The distribution whose curvature appears in the formula could be compared with the characteristic distribution already studied in contact and cosymplectic geometry.

Load-bearing premise

The 2-covariant tensor satisfies suitable assumptions that permit construction of an associated bivector field together with a distribution whose curvature enters the obstruction formula.

What would settle it

Direct evaluation of the derived formula on the standard symplectic 2-form on Euclidean space, checking whether the right-hand side vanishes identically.

read the original abstract

Poisson and Jacobi structures play a fundamental role in the geometric description of many systems arising in classical mechanics. In most cases, the corresponding bivector field is induced by a non-degenerate 2-covariant tensor. In this paper, we present a unified framework for constructing the associated brackets by studying the Poisson and Jacobi structures induced by these tensors. More specifically, under suitable assumptions on the tensor, we derive a formula for computing the Schouten-Nijenhuis bracket of the associated bivector field in terms of the curvature of a certain distribution and the exterior derivative of a differential form. This formula provides the obstruction to the existence of a Poisson or Jacobi structure. To illustrate the theory, we recover the classical brackets associated with symplectic, locally conformally symplectic, cosymplectic, and contact geometries. Finally, we characterize the conditions under which fat bundles and almost cosymplectic structures of order $p$ determine a Jacobi bracket.

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.

Referee Report

1 major / 3 minor

Summary. The paper develops a unified framework for inducing Poisson and Jacobi structures from non-degenerate 2-covariant tensors. Under suitable assumptions on the tensor, it derives an explicit formula for the Schouten-Nijenhuis bracket of the associated bivector field expressed in terms of the curvature of an induced distribution and the exterior derivative of an auxiliary differential form; this formula is presented as the obstruction to the structure. The framework is tested by recovering the standard brackets in the symplectic, locally conformally symplectic, cosymplectic, and contact cases, and it characterizes the conditions under which fat bundles and almost cosymplectic structures of order p yield Jacobi brackets.

Significance. If the derivation is correct, the obstruction formula supplies a concrete, computable criterion that unifies several classical geometries and may facilitate the discovery of new Poisson or Jacobi structures. The explicit recovery of the vanishing bracket in the four standard cases constitutes a non-trivial consistency check. The final characterization of fat bundles and almost cosymplectic structures adds concrete applications. The manuscript does not claim parameter-free derivations or machine-checked proofs, but the geometric test against known examples strengthens its value.

major comments (1)
  1. [Main derivation / statement of the obstruction formula] The central obstruction formula (the expression for [π,π] in terms of curvature and d of the auxiliary form) is stated under “suitable assumptions on the tensor” (abstract and the paragraph beginning “More specifically”). These assumptions—rank conditions, compatibility between the tensor and the induced distribution, and the precise relation that defines the auxiliary form—are never collected in a single numbered list before the main theorem. Because the curvature term and the exterior-derivative term are constructed from these assumptions, their explicit enumeration is load-bearing for any reader who wishes to apply or verify the formula.
minor comments (3)
  1. [Throughout, especially §§2–3] Notation for the 2-covariant tensor, the induced bivector π, and the auxiliary 1-form should be introduced once in a dedicated “Notation and conventions” paragraph and then used consistently; several passages re-introduce symbols without cross-reference.
  2. [Introduction and §3] The Schouten-Nijenhuis bracket is used without a reference to a standard text (e.g., Cannas da Silva–Weinstein or Vaisman). Adding one citation would help readers unfamiliar with the sign conventions employed.
  3. [§4 (illustrative examples)] In the recovery of the contact and cosymplectic cases, the precise identification of the auxiliary form with the contact or cosymplectic form is stated only in prose; a short displayed equation would make the specialization of the general formula immediate.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive comment on the presentation of the main result. We address the point below.

read point-by-point responses

  1. Referee: [Main derivation / statement of the obstruction formula] The central obstruction formula (the expression for [π,π] in terms of curvature and d of the auxiliary form) is stated under “suitable assumptions on the tensor” (abstract and the paragraph beginning “More specifically”). These assumptions—rank conditions, compatibility between the tensor and the induced distribution, and the precise relation that defines the auxiliary form—are never collected in a single numbered list before the main theorem. Because the curvature term and the exterior-derivative term are constructed from these assumptions, their explicit enumeration is load-bearing for any reader who wishes to apply or verify the formula.

    Authors: We agree that an explicit, numbered enumeration of the assumptions would improve readability and facilitate verification of the obstruction formula. In the revised version we will insert, immediately before the statement of the main theorem, a clearly labeled list (or a short dedicated paragraph) that collects: (i) the rank conditions on the 2-covariant tensor, (ii) the compatibility requirements between the tensor and the induced distribution, and (iii) the precise relation that defines the auxiliary differential form. This change does not alter any mathematical content but addresses the referee’s concern directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives an explicit formula for the Schouten-Nijenhuis bracket of the bivector induced by a 2-covariant tensor, expressed via distribution curvature and exterior derivative, under stated rank/compatibility assumptions. This is then verified by recovering the known vanishing conditions in the symplectic, cosymplectic, contact, and locally conformally symplectic cases. No quoted step reduces a claimed result to a fitted parameter, self-citation loop, or definitional tautology; the central obstruction formula is an independent identity tested against external geometries. The construction therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard differential geometry; no free parameters or invented entities are mentioned.

axioms (1)
  • standard math Standard properties of the Schouten-Nijenhuis bracket, exterior derivative, and curvature of distributions on smooth manifolds
    Invoked to obtain the obstruction formula from the 2-covariant tensor.

pith-pipeline@v0.9.1-grok · 5714 in / 1212 out tokens · 38560 ms · 2026-06-26T16:11:34.178589+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

33 extracted references · 26 canonical work pages

  1. [1]

    Abraham and J

    R. Abraham and J. E. Marsden.Foundations of mechanics, volume 364 ofAMS Chelsea publishing. Benjamin/Cummings Pub. Co., New York, 2nd edition, 1978. 10.1090/chel/364

    work page doi:10.1090/chel/364 1978

  2. [2]

    C. Albert. Le th´ eor` eme de r´ eduction de Marsden–Weinstein en g´ eom´ etrie cosymplectique et de contact.J. Geom. Phys.,6(4):627–649, 1989. 10.1016/0393-0440(89)90029-6

    work page doi:10.1016/0393-0440(89)90029-6 1989

  3. [3]

    V. I. Arnold.Mathematical Methods of Classical Mechanics, volume 60 ofGraduate Texts in Mathematics. Springer, New York, 2nd edition, 1989. 10.1007/978-1-4757-1693-1

    work page doi:10.1007/978-1-4757-1693-1 1989

  4. [4]

    Bhaskara and K

    K. Bhaskara and K. Viswanath.Poisson Algebras and Poisson Manifolds. Number 174 in Pitman research notes in mathematics series. Longman Scientific & Technical, 1988

    1988

  5. [5]

    Bravetti

    A. Bravetti. Contact geometry and thermodynamics.Int. J. Geom. Methods Mod. Phys., 16(supp01):1940003, 2018. 10.1142/S0219887819400036

    work page doi:10.1142/s0219887819400036 2018

  6. [6]

    J. F. Cari˜ nena and P. Guha. Lichnerowicz–Witten differential, symmetries and locally conformal symplectic structures.J. Geom. Phys.,210:105418, 2025. 10.1016/j.geomphys.2025.105418

    work page doi:10.1016/j.geomphys.2025.105418 2025

  7. [7]

    Corbas and G

    B. Corbas and G. D. Williams. Bilinear forms over an algebraically closed field.J. Pure Appl. Algebra, 165(3):255–266, 2001. 10.1016/S0022-4049(01)00123-2

    work page doi:10.1016/s0022-4049(01)00123-2 2001

  8. [8]

    de Le´ on and J

    M. de Le´ on and J. Bajo. A geometric description of some thermodynamical systems.J. Phys. A: Math. Theor.,58(17):175203, 2025. 10.1088/1751-8121/adcd14

    work page doi:10.1088/1751-8121/adcd14 2025

  9. [9]

    de Le´ on, J

    M. de Le´ on, J. Gaset, X. Gr` acia, M. C. Mu˜ noz-Lecanda, and X. Rivas. Time-dependent contact mechanics.Monatsh. Math.,201:1149–1183, 2023. 10.1007/s00605-022-01767-1

    work page doi:10.1007/s00605-022-01767-1 2023

  10. [10]

    de Le´ on and M

    M. de Le´ on and M. Lainz-Valc´ azar. Singular Lagrangians and precontact Hamiltonian systems.Int. J. Geom. Methods Mod. Phys.,16(10):1950158, 2019. 10.1142/S0219887819501585

    work page doi:10.1142/s0219887819501585 2019

  11. [11]

    de Le´ on and M

    M. de Le´ on and M. La´ ınz-Valc´ azar. Contact Hamiltonian systems.J. Math. Phys.,60(10):102902,

  12. [12]

    10.1063/1.5096475. 29 M. de Le´ on, X. Gr` acia, R. Izquierdo-L´ opez, ´A. Mart´ ınez-Mu˜ noz and X. Rivas

    work page doi:10.1063/1.5096475

  13. [13]

    de Le´ on and P

    M. de Le´ on and P. R. Rodrigues.Methods of differential geometry in analytical mechanics, volume 158 ofMathematics Studies. North-Holland, Amsterdam, 1989. 10.1016/s0304-0208(08)x7115-4

    work page doi:10.1016/s0304-0208(08)x7115-4 1989

  14. [14]

    An introduction to contact topology , url =

    H. Geiges.An introduction to contact topology, volume 109 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, New York, NY, 2008. 10.1017/CBO9780511611438

    work page doi:10.1017/cbo9780511611438 2008

  15. [15]

    Grabowska, J

    K. Grabowska, J. Grabowski, M. Ku´ s, and G. Marmo. Contactifications: a Lagrangian description of compact Hamiltonian systems.J. Phys. A: Math. Theor.,57(39):395204, 2024. 10.1088/1751- 8121/ad75d8

    work page doi:10.1088/1751- 2024

  16. [16]

    Greub.Multilinear Algebra

    W. Greub.Multilinear Algebra. Universitext. Springer, 1978. 10.1007/978-1-4613-9425-9

    work page doi:10.1007/978-1-4613-9425-9 1978

  17. [17]

    Gr` acia, A

    X. Gr` acia, A. Mart´ ınez-Mu˜ noz, and X. Rivas. Pairs of differential forms: a framework for precontact geometry. 2602.04882, 2026

    arXiv 2026

  18. [18]

    Ibort, M

    A. Ibort, M. de Le´ on, and G. Marmo. Reduction of Jacobi manifolds.J. Phys. A: Math. Theor., 30(8):2783, 1997. 10.1088/0305-4470/30/8/022

    work page doi:10.1088/0305-4470/30/8/022 1997

  19. [19]

    V. M. Jim´ enez and M. de Le´ on. The nonholonomic bracket on contact mechanical systems.J. Geom. Phys.,213:105484, 2025. 10.1016/j.geomphys.2025.105484

    work page doi:10.1016/j.geomphys.2025.105484 2025

  20. [20]

    A. A. Kirillov. Local Lie algebras.Russ. Math. Surv.,31(4):55, 1976. 10.1070/RM1976v031n04ABEH001556

    work page doi:10.1070/rm1976v031n04abeh001556 1976

  21. [21]

    Libermann and C.-M

    P. Libermann and C.-M. Marle.Symplectic geometry and analytical mechanics, volume 35 of Mathematics and Its Applications. Springer Dordrecht, 1987. 10.1007/978-94-009-3807-6

    work page doi:10.1007/978-94-009-3807-6 1987

  22. [22]

    Lichnerowicz

    A. Lichnerowicz. Les vari´ et´ es de Poisson et leurs alg` ebres de Lie associ´ ees.J. Differ. Geom., 12:253–300, 1977

    1977

  23. [23]

    Lichnerowicz

    A. Lichnerowicz. Les vari´ et´ es de Jacobi et leurs alg` ebres de Lie associ´ ees.J. Math. Pures Appl., 57:453–488, 1978

    1978

  24. [24]

    C.-M. Marle. The Schouten–Nijenhuis bracket and interior products.J. Geom. Phys.,23(3–4):350–359,

  25. [25]

    10.1016/S0393-0440(97)80009-5

    work page doi:10.1016/s0393-0440(97)80009-5

  26. [26]

    Fujikawa and H

    D. McDuff and D. Salamon.Introduction to Symplectic Topology. Oxford University Press, 03 2017. 10.1093/oso/9780198794899.001.0001

    work page doi:10.1093/oso/9780198794899.001.0001 2017

  27. [27]

    P. J. Morrison. A paradigm for joined Hamiltonian and dissipative systems.Phys. D: Nonlinear Phenom.,18(1):410–419, 1986. 10.1016/0167-2789(86)90209-5

    work page doi:10.1016/0167-2789(86)90209-5 1986

  28. [28]

    M. C. Mu˜ noz-Lecanda and N. Rom´ an-Roy.Geometry of Mechanics. World Scientific, 2025. 10.1142/q0490

    work page doi:10.1142/q0490 2025

  29. [29]

    C. Riehm. The equivalence of bilinear forms.J. Algebra,31(1):45–66, 1974. 10.1016/0021- 8693(74)90004-0

    work page doi:10.1016/0021- 1974

  30. [30]

    W. M. Tulczyjew. The Graded Lie Algebra of Multivector Fields and the Generalized Lie Derivative of Forms.Bull. Acad. Polon. Sci., Ser. Math. Astr. et Phys.,22(9):937–942, 1974

    1974

  31. [31]

    Vaisman.Lectures on the Geometry of Poisson Manifolds, volume 118 ofProgress in Mathematics

    I. Vaisman.Lectures on the Geometry of Poisson Manifolds, volume 118 ofProgress in Mathematics. Birkh¨ auser Basel, 1994. 10.1007/978-3-0348-8495-2

    work page doi:10.1007/978-3-0348-8495-2 1994

  32. [32]

    Vitagliano

    L. Vitagliano. Dirac–Jacobi bundles.J. Symplectic Geom.,16(2):485–561, 2018. 10.4310/JSG.2018.v16.n2.a4

    work page doi:10.4310/jsg.2018.v16.n2.a4 2018

  33. [33]

    Weinstein

    A. Weinstein. Fat bundles and symplectic manifolds.Adv. Math.,37(3):239–250, 1980. 10.1016/0001- 8708(80)90035-3

    work page doi:10.1016/0001- 1980