Geometric structures on Weil bundles: Canonical differential-geometric constructions

A. Ndiaye; C. Khoule; R. A. M. Mohameden; S. Tchuiaga

REVIEW 4 major objections 5 minor 13 references

Geometric structures on Weil bundles: Canonical differential-geometric constructions

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The Weil-bundle functor canonically lifts eleven classical geometric structures — lcs, lcc, cosymplectic, contact, Jacobi, Sasakian, Walker, sub-Riemannian, orientation, Riemannian, and Kähler — preserving the equations that define them.

desk verdict A survey of standard Weil lifts with new lift claims that are assumed rather than proved; the averaged-section mechanism is not well-defined globally. read the letter →

arxiv 2411.17212 v3 pith:ZCJMTQAV submitted 2024-11-26 math.DG

S. Tchuiaga , A. Ndiaye , C. Khoule , R. A. M. Mohameden This is my paper · ORCID

classification math.DG MSC 58A3253D0553D1053C1553C25

keywords WeilbundlecanonicalliftprolongationcosymplecticstructureJacobiSasakiancontactlocallyconformalsymplectic

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that a single functorial construction, the Weil bundle (or manifold of infinitely near points) $M^{\mathbf A}$ of a smooth manifold $M$, canonically inherits a wide family of geometric structures from $M$. The list includes locally conformal symplectic and cosymplectic structures, contact and Jacobi structures, Sasakian, Walker, sub-Riemannian, Riemannian, oriented, and Kähler structures. If the claims are right, tools from one manifold transfer to its 'thickened' bundle while preserving the defining equations, so symplectic, contact, and complex geometry can be studied on Weil bundles without ad hoc constructions. The paper also produces an explicit cosymplectic Weil bundle that is not a trivial suspension of a symplectic manifold, showing the lifted geometry is genuinely new.

What carries the argument

The load-bearing object is the Weil functor $T^{\mathbf A}:M\mapsto M^{\mathbf A}$, which sends a manifold to the bundle of algebra homomorphisms $C^\infty(M)\to\mathbf A$ over its points, together with the canonical lift $T\mapsto T^{\mathbf A}$ of functions, vector fields, forms, tensor fields, and connections. The lift is functorial: it commutes with $d$, $L_X$, $\iota_X$, brackets, and algebraic operations, and real-valued results are obtained by applying a normalized trace $\operatorname{Tr}:\mathbf A\to\mathbb R$. For characteristic vector fields the paper uses a second mechanism, the averaged lift $\frac1l\sum_{j=1}^l (S_j)_*$ built from sections $S_j$ associated to a basis of $\mathbf A$, which projects to the base vector field but is not a Lie-algebra homomorphism. These two lift types, canonical and averaged, are what carry every theorem in the paper.

What would settle it

Compute, in local coordinates on $(\mathbb R^3)^{\mathbf A}$ for a 3-dimensional Weil algebra such as $\mathbb R[u]/(u^3)$, the averaged Reeb field $\Xi=\frac13\sum_j(S_j)_*\partial_z$ for the standard cosymplectic structure $\omega=dx\wedge dy$, $\eta=dz$, then check whether $\eta^{\mathbf A}(\Xi)=1$ and $\iota_\Xi\omega^{\mathbf A}=0$ for the lifted forms used in the proof. A single choice of sections or trace where either identity fails would refute the Reeb-field construction in Theorem A(3).

Watch

Extended reading notes

Core claim

The central discovery is that canonical lifts—denoted $\omega^{\mathbf A}$, $g^{\mathbf A}$, $\beta^{\mathbf A}$, $J^{\mathbf A}$—preserve the algebraic identities characterizing each structure, provided the Weil algebra has odd dimension for contact, cosymplectic, lcc, and Sasakian cases. For example, $d(\omega^{\mathbf A})=(d\omega)^{\mathbf A}$ turns $d\omega=-\theta\wedge\omega$ into $d\omega^{\mathbf A}=-\theta^{\mathbf A}\wedge\omega^{\mathbf A}$; the lift of a Killing field is Killing; the lift of the Levi-Civita connection is the Levi-Civita connection of the lifted metric; and the lifted Nijenhuis tensor vanishes when the base one does. For Reeb fields, the paper distinguishes two lift types: the flow prolongation $X^{\mathbf A}$ for contact structures, and the averaged section lift $\frac1l\sum_j (S_j)_*X$ for cosymplectic and Jacobi structures. A further result shows that a Lagrangian submanifold lifts to a Lagrangian submanifold of the symplectic Weil bundle.

Load-bearing premise

The load-bearing premise is that averaging vector fields by pushing them forward along sections interacts with lifted differential forms exactly as the canonical lift does; the paper asserts this compatibility rather than proving it, and the cosymplectic and Jacobi theorems collapse if it fails.

Editorial extensions

If this is right

A symplectic manifold lifts to a symplectic Weil bundle, and a Kähler manifold lifts to a Kähler Weil bundle with integrable complex structure; Lagrangian submanifolds lift to Lagrangian submanifolds.
For odd-dimensional Weil algebras, contact, cosymplectic, lcc, and Sasakian structures lift with explicit Reeb fields: flow prolongation for contact, section-averaged sums for cosymplectic.
The canonical connection on $M^{\mathbf A}$ is the Levi-Civita connection of the lifted metric, geodesics lift through canonical sections, and Killing fields lift to Killing fields.
The example on $(\mathbb R^{2n+1})^{\mathbf A}$ with $l>1$ is cosymplectic but not a product $P\times\mathbb R$, so the lifted structure is not a trivial suspension.
Bracket-generating sub-Riemannian distributions lift to bracket-generating distributions, and parallel null distributions lift to parallel null distributions, so Walker and sub-Riemannian geometries pass to Weil bundles.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the same averaged-lift recipe could be iterated, lifting a Jacobi structure on $M^{\mathbf A}$ to $M^{\mathbf A\otimes\mathbf B}$, which would give a hierarchy of nilpotent thickenings of Jacobi geometry.
Since Einstein and Ricci-flat conditions are not preserved, a natural next check is whether curvature invariants of $g^{\mathbf A}$ decompose into base curvature plus $\mathbf A$-algebraic terms; computing scalar curvature for the flat-base examples in Section 5 would be a short test.
The non-suspension cosymplectic example suggests that the cosymplectic topology of Weil bundles differs from the base, so the lifted Reeb flows are worth studying even when the base flow is trivial.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague, here's my read of arXiv:2411.17212. The paper is best treated as a survey of known canonical lifts to Weil bundles, plus a set of new claims that are not proved. The standard material—Riemannian, symplectic, contact, Kähler, orientation lifts—is correctly assembled from Kolář–Michor–Slovák and Okayama, and the functorial proofs for Kähler integrability and the Lagrangian submanifold lift are clean. The lcs and lcc lift statements also follow from functoriality and are likely fine.

The trouble is the new mechanism. The averaged lifts in §2.3 define (S_j)_*X only along the image of each section, not on all of M^A, and no extension rule is given. Even if that were fixed, the proofs of Theorem A(3) and Theorem F assume compatibility rules that are exactly the defining identities of the claimed structures: η^A(Ξ)=1, ι_Ξ ω^A=0, and [Λ^A,Λ^A]=2Ξ^A∧Λ^A. The authors admit as much in the text ("requires specific interaction rules," "Assumes compatibility rules [6]"). That is circular, not a proof. The Schouten bracket of averaged bivectors also has cross terms and a 1/l^2 prefactor; the paper gives no reason they collapse to the claimed (1/l)Σ(S_j)_*[Λ,Λ].

There's also a clear false statement: Lemma 2.12 says any smooth function on M^A is constant along fibers. This contradicts the local coordinates the paper itself uses (x_{i,k}, k≥2, are nonconstant along fibers). So the lemma is simply wrong.

One place where the reader's critique misses: Example 5.2. The construction gives M^A ≅ (R^{2n})^A × R^l with l>1. That is correctly not a suspension in the standard sense (P × R). So don't repeat that as a flaw.

Who gets value from this? Someone looking for a single place that lists which structures survive Weil prolongation will find the survey useful. Researchers should not rely on the new lift theorems as stated.

For peer review: I'd send it out, because the standard part is a legitimate survey and a referee can pin down the exact gaps in the new claims. But my own verdict is that the new results are unsupported as written, and the paper would need heavy revision to become a valid research contribution.

Referee Report

4 major / 5 minor

Summary. The paper claims that a wide range of geometric structures on a smooth manifold M, including lcs, lcc, cosymplectic, contact, Jacobi, Sasakian, Walker, sub-Riemannian, orientation, Riemannian, and Kähler structures, admit canonical lifts to the Weil bundle M^A. The proofs rely on the Weil functor formalism, on known canonical lifts from the literature, and, for the Reeb field of a cosymplectic structure and for Jacobi structures, on averaged lifts defined by (1/l)Σ_j (S_j)_* over sections S_j. The paper also gives coordinate examples, discusses non-preserved Einstein and Calabi-Yau conditions, and includes a short appendix of definitions.

Significance. Several of the claimed lifts are already standard: contact, symplectic, Riemannian, and Kähler lifts appear in the cited literature by Kolář–Michor–Slovák, Morimoto, and Okayama. The genuinely new mechanism in this manuscript is the averaged section lift in §2.3, used for the Reeb field in Theorem A(3) and for the Jacobi structure in Theorem F. That mechanism is not rigorously defined, and the proofs explicitly assume the very identities that must be proved. The paper also contains a plainly false lemma about smooth functions on Weil bundles. If the averaging construction were replaced by a well-defined global construction with genuine proofs of the bracket and contraction identities, the cosymplectic and Jacobi results would be a useful contribution; as written, the central claims are not established. The paper does have useful expository parts: Section 2.2's distinction between pullbacks and canonical lifts is clear, and the coordinate examples in Section 5 are concrete.

major comments (4)

[§2.3 / Theorem A(3)] The averaged vector field X~ = (1/l)Σ_j (S_j)_*X is not shown to be a well-defined global vector field on M^A. Each S_j:M→M^A is an embedding whose image has positive codimension when l>1, so (S_j)_*X is defined only along S_j(M); no extension to all of M^A is given. The proof of Theorem A(3) states only that Ξ 'matches' ξ_{M^A} under an assumption of compatibility with the averaged construction, and that verifying η^A(Ξ)=1 and ι_Ξω^A=0 'requires specific interaction rules.' These two equations are exactly the defining Reeb identities of the asserted cosymplectic structure, so the theorem is not proved.
[Theorem F / §3] The proof of Theorem F assumes [Λ^A,Λ^A]_{SN} = (1/l)Σ_j (S_j)_*[Λ,Λ]_{SN} and [Ξ^A,Λ^A]_{SN}=0. The Schouten bracket is bilinear, so the left-hand side of the first identity contains l^2 cross terms with a 1/l^2 prefactor; no rule is given that eliminates or combines those cross terms. The second identity is also assumed. These are precisely the Jacobi identities that characterize a Jacobi structure, so the argument is circular rather than a proof.
[Lemma 2.12 / Remark 2.13] Lemma 2.12 asserts that every smooth real-valued function on M^A is constant along the fibers of π~_M. This is false: for A = R[ε]/(ε^2), M^A is the tangent bundle TM, and fiber coordinates such as the components of a tangent vector are smooth, nonconstant functions on the fibers. Consequently the identification C^∞(M^A) ≅ C^∞(M) in Remark 2.13 is incorrect. This lemma is not directly used in the main proofs, but it is a stated foundational result and should either be corrected or removed.
[Example 5.2 / Theorem A(3)] Example 5.2 defines a cosymplectic lift using η^A = dz_1, whose Reeb field is ∂/∂z_1, a canonical section-type lift of ∂/∂z. This is not the averaged field (1/l)Σ_j (S_j)_*(∂/∂z) featured in Theorem A(3). The relation between the two constructions is never explained, so the example does not illustrate the theorem's Reeb-field construction and leaves the status of the averaged Reeb field unclear.

minor comments (5)

[Introduction] In the paragraph after the pullback discussion, the phrase 'lifts the full algebraic structure of A and the functorial nature of T^A' is grammatically incomplete; 'using' or 'exploiting' appears to be missing.
[Proposition H] The notation D^A and (g_D)^A for the lifted distribution and metric is not defined; the proof should specify how the Weil functor is applied to a subbundle of TM and why the lifted distribution is a smooth subbundle of TM^A.
[Corollary 3.1] Corollary 3.1 states that the map [θ]↦[θ^A] is an isomorphism on H^1 because M^A deformation retracts onto M, but it does not prove that the canonical lift of an exact form is exact; this naturality should be justified explicitly.
[Theorem E proof] The displayed comparison of transition Jacobians, det(J(ψ^A_{αβ})) ≈ (det J(ψ_{αβ}))^l, is only heuristic; the orientation form used and the convention for the lifted volume element should be specified.
[References] References [7] and [12] do not appear to be cited in the text, while several citations to [6] and [8] would benefit from precise proposition or chapter numbers.

Circularity Check

2 steps flagged · score 6.0 of 10

The averaged-section constructions in Theorems A(3) and F assume the very Reeb and Jacobi identities they are meant to establish, while the canonical-lift theorems rest on the independent references [6,8].

self definitional [Section 3, Proof of Theorem A(3)]
"The Reeb field construction Ξ := 1/l∑(S_j)_*ξ_M matches ξ_{M^A} under the assumption that the canonical lifts (ω^A,η^A) used are compatible with this averaging construction (e.g., if they are themselves defined via averaging, potentially requiring specific choices of trace/sections). Verifying η^A(Ξ)=1 and ι_Ξω^A=0 requires specific interaction rules between the averaged forms and averaged vector field."

In the appendix, a Reeb field for a cosymplectic structure is defined by η(ξ)=1 and ι_ξω=0. The proof does not derive these equations for the averaged field Ξ; it states only that verifying them requires unspecified interaction rules. Those two equations are exactly the defining property of the claimed Reeb field ξ_{M^A}. Thus the theorem's assertion that ξ_{M^A} can be constructed as Ξ is conditional on an unstated rule whose content is the conclusion itself, so the cosymplectic structure is assumed rather than established from the definitions.
self definitional [Section 3, Proof of Theorem F]
"Let Λ^A = 1/l∑(S_j)_*Λ, Ξ^A=1/l∑(S_j)_*Ξ. Assumes compatibility rules [6] such that [Λ^A,Λ^A]_{SN}=1/l∑(S_j)_*([Λ,Λ]_{SN}) = 1/l∑(S_j)_*(2Ξ∧Λ) equals 2Ξ^A∧Λ^A, and [Ξ^A,Λ^A]_{SN}=1/l∑(S_j)_*([Ξ,Λ]_{SN})=0."

A Jacobi structure is defined in Appendix item 5 by [Λ,Λ]_{SN}=2Ξ∧Λ and [Ξ,Λ]_{SN}=0. The proof explicitly assumes compatibility rules that assert exactly these two lifted identities. Moreover, the intermediate identity [Λ^A,Λ^A]=(1/l)∑(S_j)_*[Λ,Λ] does not follow from bilinearity of the Schouten bracket, which would produce cross terms and a 1/l^2 prefactor. The claimed Jacobi structure on M^A is therefore true only by inserting the target identities as axioms; the central new claim of Theorem F is not derived.

full rationale

The canonical-lift results are largely self-contained against the external references [6,8]: Theorems A(1)-(2), B, C, D, E, and G verify structure preservation using functorial identities such as d(ω^A)=(dω)^A, [X^A,Y^A]=[X,Y]^A, and ∇^A_{X^A}Y^A=(∇_XY)^A, so those derivations are not circular. The self-citations [11,12] are not load-bearing. The circularity is concentrated in the new averaged-section mechanism. In Theorem A(3), the proof sets Ξ=(1/l)∑(S_j)_*ξ_M and then says that verifying η^A(Ξ)=1 and ι_Ξω^A=0 requires specific interaction rules; these are exactly the defining Reeb equations, so the conclusion is assumed. In Theorem F, the proof explicitly adopts compatibility rules asserting the Jacobi identities [Λ^A,Λ^A]=2Ξ^A∧Λ^A and [Ξ^A,Λ^A]=0, which is the target statement itself. Independently, the averaged construction is not well-defined globally, since (S_j)_*X is defined only along the submanifold S_j(M) and no extension rule to all of M^A is given. These gaps affect the paper's novel claims rather than the standard canonical-lift results, so a partial circularity score of 6 is appropriate; the independent content of the rest of the paper prevents a higher score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central results depend on the standard Weil functor machinery (cited) plus unproved compatibility rules for averaged lifts and a false lemma about functions on fibers. No numerical free parameters are fitted; the main uncharged assumptions are algebraic compatibility assumptions that carry the load for the new claims.

assumptions (4)

standard math The Weil functor T^A is a product-preserving bundle functor with the stated lifting properties for vector fields, forms, tensor fields, connections, and the Lie bracket.
Invoked throughout Section 3; taken from [6,8].
domain assumption There exists a normalized trace map Tr:A→R with Tr(1_A)=1 that converts A-valued lifts into real-valued geometric objects preserving nondegeneracy.
The real-valued canonical lifts g^A, ω^A, etc. are defined via Tr; not all choices of trace preserve the claimed nondegeneracy.
ad hoc to paper The averaged section lifts (1/l)∑(S_j)_* satisfy the same algebraic identities as the canonical lifts (Lie bracket, Schouten bracket, contractions with averaged forms).
This compatibility is assumed in the proofs of Theorem A(3) and Theorem F without proof; it is exactly the nontrivial content.
ad hoc to paper Smooth real-valued functions on M^A are constant along the fibers of π~M.
Lemma 2.12 states this, but it is false when dim A>1; the fibers R^{n(l-1)} admit nonconstant functions. If relied upon, it would break the proof.

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Pith. "Pith review of Geometric structures on Weil bundles: Canonical differential-geometric constructions." pith.science (2026). https://pith.science/paper/ZCJMTQAV

@misc{pith2026241117212,
  author       = {Pith},
  title        = {Pith review of: Geometric structures on Weil bundles: Canonical differential-geometric constructions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZCJMTQAV}},
  note         = {Machine review of arXiv:2411.17212}
}

abstract

This paper investigates the transfer of classical geometric structures from a smooth manifold $M$ to its Weil bundle $(M^\mathbf A, \tilde\pi_M, M)$ associated with a Weil algebra $\mathbf A$. We show that various structures including locally conformal symplectic (lcs), locally conformal cosymplectic (lcc), contact, Jacobi, Sasakian, Walker, sub Riemannian, orientation, Riemannian, and K\"ahlerian structures admit canonical lifts to $M^\mathbf A$. Our approach emphasizes the differential geometric properties of these canonical constructions, utilizing the Weil projection $\tilde{\pi}_M$ and related functorial tools. This provides a unified perspective on endowing Weil bundles with rich geometric structure inherited from the base manifold. Furthermore, we highlight a specific construction yielding a cosymplectic manifold on $M^\mathbf{A}$ (for suitable $M$ and $\mathbf{A}$) that is demonstrably not a trivial suspension of a symplectic manifold. We also explicitly show how integrability of almost complex structures is preserved and clarify the nature of lifted characteristic vector fields.

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [6]

Kol´ aˇ r, P

I. Kol´ aˇ r, P. W. Michor, J. Slov´ ak,Natural operations in diﬀerential geometry , Springer-Verlag, Berlin, 1993

work page 1993

[2] [1]

Bouesso et al, Symplectic structure and applications on Weil bundles, Afr

M. Bouesso et al, Symplectic structure and applications on Weil bundles, Afr. Mat. 35 (2024)

work page 2024

[3] [2]

B. G. R. Bossoto, E. Okassa, Champs de vecteurs et formes diﬀerentielles sur une vari´ et´ e des points proches. Arch. Math. (Brno) 44(2) (2008), 159–171

work page 2008

[4] [3]

D. R. Grigore, D. Krupka, Invariants of velocities and higher order Grassmann bundles . J. Geom. Phys. 24(3) (1998), 244–264

work page 1998

[5] [4]

Li, Topology of cosymplectic/cokaehler manifolds , Asian J

H. Li, Topology of cosymplectic/cokaehler manifolds , Asian J. Math. 12(4) (2008), 527–544. 11

work page 2008

[6] [5]

Kol´ aˇ r,Weil bundles as generalized jet spaces , In: Handbook of Global Analysis, Elsevier, 2008, pp

I. Kol´ aˇ r,Weil bundles as generalized jet spaces , In: Handbook of Global Analysis, Elsevier, 2008, pp. 625–664

work page 2008

[7] [7]

Libermann, Sur les automorphismes inﬁnit´ esimaux des structures symp lectiques et des struc- tures de contact , Colloque G´ eom

P. Libermann, Sur les automorphismes inﬁnit´ esimaux des structures symp lectiques et des struc- tures de contact , Colloque G´ eom. Diﬀ. Globale (Bruxelles, 1958), Centre Bel ge Rech. Math., Louvain, 1959, pp. 37–59

work page 1958

[8] [8]

Morimoto, Prolongations of connections to bundles of inﬁnitely near p oints, J

A. Morimoto, Prolongations of connections to bundles of inﬁnitely near p oints, J. Diﬀ. Geom. 11(4) (1976), 479–498

work page 1976