The coisotropic embedding theorem for pre-symplectic manifolds: an alternative proof

Luca Schiavone

REVIEW 4 major objections 5 minor 15 references

The coisotropic embedding theorem for pre-symplectic manifolds: an alternative proof

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

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Pith's one-line read The coisotropic embedding theorem is reproved by selecting the thickening inside the cotangent bundle via the momenta conjugate to the kernel of the pre-symplectic form.

desk verdict An honest, well-written reformulation of Gotay's theorem with a real but repairable gap in the algebra-to-manifold step; worth refereeing, not worth citing as a new result. read the letter →

arxiv 2411.18208 v3 pith:6O3I7NNE submitted 2024-11-27 math.DG math-phmath.MPmath.SG

Luca Schiavone This is my paper · ORCID

classification math.DGmath-phmath.MPmath.SG MSC 53D05

keywords coisotropicembeddingtheorempre-symplecticmanifoldsymplecticthickeningcotangentbundlekerneldistributionHamiltonianmomentaGel'fand-Kolmogorovidentificationconnectionversus

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 presents an alternative proof of the Coisotropic Embedding Theorem: every pre-symplectic manifold can be embedded as a coisotropic submanifold of a symplectic manifold. The classical construction chooses a connection, i.e. a complement to the kernel of the pre-symplectic form, to build a symplectic thickening. Here that geometric choice is replaced by the algebraic choice of an embedding of the dual kernel bundle into the cotangent bundle of the manifold. The thickening is the submanifold selected by the Hamiltonian momenta conjugate to the kernel directions, and the pulled-back symplectic form reproduces the classical local expression, so the coisotropic condition follows unchanged. If the proof is correct, it provides a more algebraic route to a theorem widely used in constrained Hamiltonian systems and field theories.

What carries the argument

The central object is the cotangent bundle $(T^*M,\omega_{T^*M})$ equipped with the closed two-form $\omega'=\omega_{T^*M}+\rho^*\omega$, where $\rho:T^*M\to M$ is the projection. The kernel distribution $K=\ker\omega$ is spanned locally by $\partial/\partial z^a$, and its cotangent lifts are Hamiltonian with momenta $H_a=p_{z_a}$. The Gel'fand-Kolmogorov identification of characters of the algebra generated by pull-backs and the $H_a$ with the dual bundle $K^*$ is what turns the geometric choice of a connection into the algebraic choice of a fiberwise-linear embedding $i:K^*\to T^*M$, parametrized by matrices $P^x_{ja},P^y_{aj}$. That embedding carries the symplectic form: pulling back $\omega'$ gives the classical thickening form, so non-degeneracy and coisotropy are inherited.

What would settle it

Exhibit an algebra homomorphism from the subalgebra generated by pull-backs and the momenta $H_a$ to $\mathbb{R}$ that is not determined by a base point and a covector on the kernel fiber; if such a character exists, the identification with $K^*$ fails and the thickening is not the claimed manifold.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the symplectic thickening of a pre-symplectic manifold $(M,\omega)$ can be manufactured from the cotangent bundle $T^*M$. The paper defines $\omega'=\omega_{T^*M}+\rho^*\omega$, shows that the vector fields in $\ker\omega$ have Hamiltonian functions $H_a=p_{z_a}$ (the conjugate momenta along kernel directions), and then identifies the manifold of characters of the algebra generated by pull-backs of functions on $M$ and the $H_a$ with the dual bundle $K^*$ of the kernel. Choosing an embedding $i\colon K^*\to T^*M$ linear on fibers, the pullback $\tilde\omega=i^*\omega'$ is the same closed two-form as in the classical proof, locally $\tilde\omega=\rho^*_{K^*}\omega + dp_a\wedge P^a + p_a dP^a$, and the symplectic orthogonal of the embedded $M$ is exactly $K$, making $M$ coisotropic. The paper's claim is that the geometric freedom of a connection is therefore just the freedom of an algebraic embedding.

Load-bearing premise

The construction assumes that the algebra of functions it builds has precisely the dual bundle of the kernel as its space of points, an identification it imports from a theorem normally stated for the full algebra of smooth functions rather than the smaller subalgebra used here.

Editorial extensions

If this is right

The coisotropic embedding theorem holds with the thickening realized as a submanifold of the cotangent bundle, cut out by the momenta conjugate to the kernel directions.
The classical arbitrary connection in the thickening construction is exactly mirrored by the arbitrary fiberwise-linear embedding, so the two proofs produce the same family of symplectic thickenings.
Since the pulled-back form agrees with the classical form, the tubular-neighborhood non-degeneracy result and the flat-connection extension to the whole bundle transfer unchanged.
The zero-section of the bundle $K^*\to M$ embeds $M$ as a coisotropic submanifold whose symplectic orthogonal is precisely the original kernel distribution.

Reading between the lines

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

An implicit consequence the paper does not spell out: whenever the constraint distribution of a physical system comes with preferred momenta, this construction gives a canonical-looking thickening once an embedding is fixed, and varying the embedding may parametrize inequivalent symplectic completions.
The reliance on the Gel'fand-Kolmogorov theorem for a polynomial subalgebra suggests a testable check: exhibit explicitly every character of that algebra and see whether the claimed bijection with $K^*$ is genuine or requires an additional smoothness condition on the momentum variables.
One could extend the argument to pre-symplectic manifolds whose kernel is not a subbundle by replacing the algebra with a sheaf of algebras over the base, potentially yielding a sheaf-theoretic thickening in degenerate cases.

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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

Two things you should know. The paper is what it says: an alternative proof of Gotay's 1982 coisotropic embedding theorem, not a new theorem. And it is mostly correct, but the central Gelfand-Kolmogorov step is underproved.

What is actually new is the framing: the symplectic thickening is identified as the character space of the algebra generated by functions pulled back from the base plus the kernel momenta, and the classical choice of a connection is recast as the choice of an embedding into the cotangent bundle. That is a genuine conceptual repackaging, and it may help in teaching. The paper is clearly written, credits Gotay and Oh–Park, and is honest that its final 2-form (22) coincides with the classical (12).

The soft spots are real but limited. Proposition 2.1 is load-bearing: A is a subalgebra of C∞(T*M), not the full function algebra, and the Gelfand-Kolmogorov theorem cited from [MM94] applies to full C∞(N). The paper asserts the character bijection and jumps to the conclusion that the character space is K*. As written, the manifold topology is not established. This is probably repairable—one can directly identify characters with pairs (m,α) and put the bundle topology on that set—but the proof as it stands does not do it. The nondegeneracy and coisotropic-orthogonal checks are also deferred to the classical argument. That is acceptable if the forms are identical, but it means the proof is not self-contained.

Two notes from the review trail: the claimed sign error in H_a is not one. Contracting ∂/∂z^a with ω' gives +dp_z^a, so H_a = p_z^a is correct. The stress-test worry that ω'^n ≠ ω_T*^n is also misplaced; any term in ω'^n that contains the dx∧dy factor is killed by a repeated dx or dy, so ω'^n = ω_T*^n holds. Both objections should be dropped.

Who is this for? People who teach or apply the CET and want a different route; not for anyone seeking new results. It deserves a serious referee, because the construction is basically right and the gaps are fixable. I would send it to refereeing with a clear request that the author prove Proposition 2.1 directly or state the character-space identification more carefully, and include the nondegeneracy computation instead of deferring it. A lightweight venue might accept as is; a good journal should ask for the revision.

Referee Report

4 major / 5 minor

Summary. The paper proposes an alternative proof of the Coisotropic Embedding Theorem (CET). The main idea is to fix an embedding of the dual bundle K* of the characteristic distribution into the cotangent bundle T*M, rather than choosing a connection/complement on M. The thickening is described as the submanifold of T*M determined by the Hamiltonian functions conjugate to the kernel directions of the pre-symplectic form, with the identification achieved via a Gel'fand-Kolmogorov argument. The paper then defines the symplectic form on the thickening as the pullback of an ambient symplectic form on T*M and computes that its local expression coincides with the classical thickening form. The proof of the theorem is concluded by invoking the classical non-degeneracy and coisotropic computations.

Significance. If completed, the construction gives a legitimate alternative route to a known theorem and provides a clean conceptual reinterpretation of Gotay's connection choice as an algebraic embedding into the cotangent bundle. The computation showing that the pullback form equals the classical form (12) is a meaningful check and is essentially correct. The paper is not circular: the constructions are explicit and do not assume the CET. However, the manuscript as written does not prove the two decisive properties of the thickening, and the algebraic identification of the thickening with K* is not fully justified. These are repairable but load-bearing gaps, so the paper is not yet ready for publication.

major comments (4)

[Section 2, Eq. (19)] The Hamiltonian function for the vector field ∂/∂z^a with respect to ω' is H_a = -p^z_a, not +p^z_a, because i_{∂/∂z^a}(dp^z_b∧dz^b) = -dp^z_a. The equality H_a = p^z_a as stated is incorrect. The error does not destroy the construction, since the algebra generated by the functions H_a is the same as the algebra generated by the p^z_a up to an overall sign, but Eq. (19) and the sentence 'They read H_a = p^z_a' must be corrected.
[Section 2, Proposition 2.1] The application of the Gel'fand-Kolmogorov theorem is not justified as written. The theorem cited from [MM94] characterizes the characters of the full algebra C∞(N) of a smooth manifold, whereas A is only a proper subalgebra of C∞(T*M) consisting of polynomial functions in the fiber variables. The proof gives a set-theoretic bijection between characters and pairs (m,α_m), but it does not verify that the Gel'fand topology on the character space coincides with the smooth manifold topology of K*, nor does it prove that the cited theorem applies to the subalgebra A. Since the thickening manifold is defined as the manifold associated with A, this is a load-bearing gap; it is likely repairable by a direct character computation, but it is not established in the manuscript.
[Section 2, Eq. (22) and subsequent paragraph] The paper does not prove that the form ~tildeω = i*ω' is symplectic. It only states that the local expression (22) is identical to the classical form (12) and that therefore the classical non-degeneracy computation applies. This is a deferral, not a proof: Section 1 also only asserts non-degeneracy 'via a straightforward computation' without giving it. Since the stated goal is an alternative proof of the CET, the non-degeneracy argument for (22) must be supplied, or a precise reference to the exact computation in [Got82] or another source must be given and the computation reproduced.
[Section 2, 'Check of the coisotropic condition'] The coisotropic condition is not verified either. The paragraph says 'A direct computation confirms that for any point m∈M, the symplectic orthogonal of the tangent space to the embedded manifold is precisely the kernel K_m', but no computation is shown. The assertion that the calculation is identical to the classical one does not replace the proof, especially because the classical calculation is also only sketched in Section 1. This is the second decisive property of the theorem and needs to be demonstrated in the new setting.

minor comments (5)

[Section 2, Eq. (15)] The claim ω'^n = ω_T*^n is correct in Darboux coordinates (the mixed terms vanish by a counting argument), but the one-line 'direct computation' is too terse; a short expansion would make the argument verifiable.
[Section 2, Eq. (20)] The 'maximal rank' hypothesis on the matrices P^x and P^y is not necessary for i to define a vector subbundle: the image is a subbundle already because the coordinates p^z_a are preserved under i, so the bundle map is injective with constant rank r. The condition may be intended to mirror the classical connection data, but the wording should be clarified.
[Section 2, Eq. (16)] The characterization of cotangent lifts is imprecise: the condition L_{~K_a}ω_T*M = 0 defines a symplectic vector field, not specifically a cotangent lift. The intended vector field ∂/∂z^a is indeed the lift of the base vector field, but the stated condition alone does not identify it uniquely among symplectic vector fields.
[Introduction and references] There are several typographical issues: 'MarkGotay' and 'OhandPark' are missing spaces, and the spelling 'Gel'fand' is used inconsistently. These should be corrected.
[References, [MM94]] The reference [MM94] is not the standard source for the Gel'fand-Kolmogorov theorem, and the use here is nonstandard because it is applied to a proper subalgebra. If the theorem is invoked in this form, a self-contained proof or a more specific reference should be provided.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the thickening is constructed rather than assumed, and the only self-citations are non-load-bearing applications.

full rationale

The paper's central claim is an alternative proof of the Coisotropic Embedding Theorem. It does not assume the theorem; it starts from the cotangent bundle (T*M, ω_T*M), defines ω' = ω_T*M + ρ*ω, identifies the Hamiltonian functions H_a = pz_a conjugate to the kernel directions, and then constructs the thickening K* via the algebra generated by pulled-back functions and the H_a. The arbitrary matrices P in the embedding i (equation 20) are exactly the classical connection components, and this parametrization is not fitted from data and is not equivalent to the theorem's conclusion. The final symplectic form (22) is shown to be identical to the classical form (12); the paper defers the non-degeneracy and coisotropic-orthogonal computations to the classical proof. That is a dependency on an independent earlier proof, not a circularity: the classical non-degeneracy computation does not use the present theorem. Self-citations, including [Sch24a], [Sch24b], and the [CDI+...] series, appear only as illustrative applications in the introduction and are not load-bearing in the proof. The flagged technical risks -- Proposition 2.1's invocation of the Gel'fand-Kolmogorov theorem [MM94] for the proper subalgebra A rather than a full function algebra, and the assertion ω'^n = ω_T*M^n, which appears to fail already for m = r = 1 -- are correctness or rigor gaps, not circular reductions. They do not raise the circularity score.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The proof introduces no new free constants fitted to data; the only arbitrary inputs are the connection or embedding matrices P, which do not affect the truth of the theorem. The heavier assumptions are the constant-rank hypothesis, Darboux theory, the Gel'fand-Kolmogorov identification, and the uncritically imported non-degeneracy result from the classical proof.

free parameters (1)

Embedding matrices P^x_{ja}, P^y_{aj} = arbitrary real-valued matrices of maximal rank
The embedding i in Eq. (20) depends on arbitrary maximal-rank matrices P^x_{ja} and P^y_{aj}; they parametrize the choice of connection or complement and the choice of embedding. The theorem is claimed for any such choice.

assumptions (4)

domain assumption The pre-symplectic form has constant rank r < n.
Stated after Definition 1.5; the characteristic bundle K and the whole construction depend on this assumption.
standard math Darboux theorem for pre-symplectic manifolds (Theorem 1.6).
Provides the local coordinates (x^j, y^j, z^a) and the local form of ω used throughout Section 2.
domain assumption Gel'fand-Kolmogorov theorem applied to the algebra A.
Used in Proposition 2.1 to identify the character space of A with the bundle K*. The paper cites [MM94] but does not prove the version needed for a polynomial subalgebra of C∞(T*M).
domain assumption Non-degeneracy of the classical form (12) in a tubular neighborhood of the zero section, and the coisotropic-orthogonal computation from [Got82].
Section 2 states 'as in the classical case, one can prove that \tildeω is symplectic in a tubular neighborhood' and 'the calculation of the symplectic orthogonal of M is also identical.' The alternative proof therefore relies on the classical proof for the two key checks.

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Cite this review

Pith. "Pith review of The coisotropic embedding theorem for pre-symplectic manifolds: an alternative proof." pith.science (2026). https://pith.science/paper/6O3I7NNE

@misc{pith2026241118208,
  author       = {Pith},
  title        = {Pith review of: The coisotropic embedding theorem for pre-symplectic manifolds: an alternative proof},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6O3I7NNE}},
  note         = {Machine review of arXiv:2411.18208}
}

read the original abstract

We present an alternative proof of the Coisotropic Embedding Theorem in which the geometric choice of a connection is recast as the algebraic choice of an embedding into the cotangent bundle. The symplectic thickening is then identified as the submanifold determined by the Hamiltonian momenta conjugate to the kernel directions of the pre-symplectic form.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Abraham, J

R. Abraham, J. E. Marsden, and T. Ratiu. M anifolds, T ensor A nalysis, and A pplications . Springer, New York, 1988

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[9]

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M. J. Gotay and J. Sniatycki. O n the quantization of presymplectic dynamical systems via coisotropic imbeddings. Communications in Mathematical Physics , 82(3):377--389, 1981

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G. Marmo and G. Morandi. Some G eometry and T opology. In S. Lundqvist, G. Morandi, and Yu Lu, editors, Low- D imensional Q uantum F ield T heories for C ondensed M atter P hysicists , Modern Condensed Matter Physics, pages 1--107. World-Scientific, 1994

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Pith tools

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

[1] [1]

Abraham, J

R. Abraham, J. E. Marsden, and T. Ratiu. M anifolds, T ensor A nalysis, and A pplications . Springer, New York, 1988

work page 1988

[2] [2]

Abraham, J

R. Abraham, J. E. Marsden, T. Ratiu, and R. Cushman. F oundations of M echanics . Addison-Wesley Publishing Company, Inc., Redwood City, 1978

work page 1978

[3] [3]

V. I. Arnold. M athematical methods of C lassical M echanics . Springer-Verlag, New York, 1989

work page 1989

[4] [4]

F. M. Ciaglia, F. Di Cosmo , L. A. Ibort, G. Marmo, L. Schiavone, and A. Zampini. S ymmetries and C ovariant P oisson B rackets on P resymplectic M anifolds. Symmetry , 14(70):1--28, 2022

work page 2022

[5] [5]

F. M. Ciaglia, F. Di Cosmo , L. A. Ibort, G. Marmo, L. Schiavone, and A. Zampini. T he geometry of the solution space of first order H amiltonian field theories II : non- A belian gauge theories, 2022

work page 2022

[6] [6]

F. M. Ciaglia, F. Di Cosmo , L. A. Ibort, G. Marmo, L. Schiavone, and A. Zampini. T he geometry of the solution space of first order H amiltonian field theories III : P alatini's formulation of G eneral R elativity, 2023

work page 2023

[7] [7]

F. M. Ciaglia, F. Di Cosmo , L. A. Ibort, G. Marmo, L. Schiavone, and A. Zampini. T he geometry of the solution space of first order H amiltonian field theories I : from particle dynamics to E lectrodynamics. Journal of Geometry and Physics , 204:105279, 2024

work page 2024

[8] [8]

M. J. Gotay, J. M. Nester, and G. Hinds. P resymplectic manifolds and the D irac– B ergmann theory of constraints. Journal of Mathematical Physics , 19(11):2388--2399, 1978

work page 1978