The symplectic Hadamard question

Dan Cristofaro-Gardiner

A simply connected symplectic manifold with a complete nonpositively curved compatible metric is symplectomorphic to standard R^{2n}.

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2026-07-15 02:09 UTC pith:GJZDUY2F

load-bearing objection Abstract claims an affirmative answer to McDuff–Salamon’s symplectic Hadamard question; without the full text the key upgrade step is uncheckable. the 1 major comments →

arxiv 2607.12940 v1 pith:GJZDUY2F submitted 2026-07-14 math.SG

The symplectic Hadamard question

Dan Cristofaro-Gardiner This is my paper

classification math.SG MSC 53D0553C2053C15

keywords symplectic geometryHadamard theoremcompatible metricsnonpositive curvaturesymplectomorphismsimply connected manifoldsMcDuff-Salamon question

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper answers a question of McDuff and Salamon by proving that simple connectivity, together with the existence of a complete Riemannian metric of nonpositive sectional curvature that is compatible with the symplectic form, forces the manifold to be globally symplectomorphic to Euclidean space with its standard symplectic structure. Compatibility means the metric is recovered from the symplectic form via an almost complex structure in the usual way. The result is a symplectic analogue of the classical Hadamard theorem, which says that a complete simply connected manifold of nonpositive curvature is diffeomorphic to Euclidean space; here the same geometric hypotheses, when the metric is compatible, upgrade the conclusion from diffeomorphism to symplectomorphism. A sympathetic reader cares because the theorem gives a purely geometric criterion that pins down the standard symplectic form on R^{2n} among all simply connected symplectic manifolds, without needing further dynamical or topological assumptions.

Core claim

Any simply connected symplectic manifold that admits a complete compatible metric of nonpositive sectional curvature is symplectomorphic to R^{2n} with its standard symplectic form. This settles the symplectic Hadamard question of McDuff-Salamon in the affirmative.

What carries the argument

A complete Riemannian metric compatible with the symplectic form (via an almost complex structure J so that g(u,v)=ω(u,Jv)) and of nonpositive sectional curvature. Compatibility links the metric geometry directly to the symplectic form, allowing the classical Hadamard diffeomorphism conclusion to be strengthened to a symplectomorphism.

Load-bearing premise

That a complete compatible metric of nonpositive curvature, plus simple connectivity, is already enough to produce a global symplectomorphism to standard R^{2n}.

What would settle it

An explicit simply connected symplectic manifold that admits a complete compatible metric of nonpositive sectional curvature yet is not symplectomorphic to standard R^{2n}.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

The standard symplectic structure on R^{2n} is the unique one (up to symplectomorphism) among simply connected symplectic manifolds that admit a complete nonpositively curved compatible metric.
Any simply connected symplectic manifold that is not standard R^{2n} cannot carry a complete compatible metric of nonpositive curvature.
The classical Hadamard theorem upgrades from a diffeomorphism to a symplectomorphism once the metric is required to be compatible with a symplectic form.
Existence of such a metric becomes a geometric obstruction that rules out exotic symplectic structures on simply connected manifolds.

Where Pith is reading between the lines

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

The same conclusion may hold under weaker curvature bounds (for instance nonpositive Ricci curvature) if compatibility is retained, offering a natural test of how much curvature control is essential.
The argument likely relies on producing a global Darboux chart that is also an isometry of the compatible metric; checking whether that chart can be constructed by integrating the almost complex structure would clarify the geometric bridge.
Analogous statements for contact manifolds with complete nonpositively curved compatible metrics would be a direct neighbouring question.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Abstract claims an affirmative answer to McDuff–Salamon’s symplectic Hadamard question; without the full text the key upgrade step is uncheckable.

read the letter

The one thing you need to know is that this abstract asserts a clean affirmative resolution of the symplectic Hadamard question: a simply connected symplectic manifold with a complete nonpositively curved compatible metric is symplectomorphic to standard R^{2n}. That would settle a named open problem of McDuff–Salamon and give a curvature-based characterization of the standard symplectic structure.

What is new is precisely that claim. Classical Hadamard already supplies a diffeomorphism to Euclidean space under the Riemannian hypotheses; the paper’s contribution is the upgrade to a symplectomorphism once the metric is compatible with ω. The statement is crisp, free of free parameters or circular normalizations, and directly answers the question as posed. Credit where due: framing and resolving a named problem in the literature is useful work if the argument holds.

The soft spot is exactly the one the stress-test flags and it is load-bearing: we have only the abstract. No lemmas, no construction of the map, no indication whether the almost-complex structure is assumed integrable or how compatibility forces the form to be standard. The geometric bridge from NPC + complete + simply connected + compatible to global symplectomorphism is therefore uninspectable. That is not a manufactured flaw; it is simply the limit of an abstract-only look. If the full paper supplies a careful argument, the result is solid; if it leans on hidden decay or integrability assumptions, the claim weakens. We cannot tell yet.

This is for symplectic geometers who care about rigidity, classification under curvature, and the McDuff–Salamon list of questions. A serious referee should see the full text. I would send it to peer review rather than desk-reject; the question is important enough and the abstract is coherent enough to deserve that time. Whether I would cite it later depends entirely on the proof checking out.

Referee Report

1 major / 1 minor

Summary. The manuscript claims that a simply connected symplectic manifold admitting a complete Riemannian metric of nonpositive sectional curvature that is compatible with the symplectic form is symplectomorphic to Euclidean space R^{2n} with its standard symplectic structure. This is presented as an affirmative resolution of the symplectic Hadamard question of McDuff–Salamon. Only the abstract is available for the present review; no lemmas, constructions, or intermediate estimates can be inspected.

Significance. If the result holds as stated, it is a substantial contribution to symplectic geometry: it upgrades the classical Hadamard–Cartan diffeomorphism theorem to a global symplectomorphism under the natural compatibility hypothesis. The claim is clean, parameter-free, and answers a named open question in a standard reference. Those strengths cannot be fully credited without the body of the argument, but the formulation itself is of clear interest to the field.

major comments (1)

[Abstract] The classical Hadamard theorem already yields a diffeomorphism to R^{2n} from completeness, simple connectivity, and nonpositive sectional curvature. The load-bearing content of the paper is the upgrade of that diffeomorphism to a symplectomorphism using only compatibility of the metric with ω. The abstract asserts the upgrade but supplies no intermediate statements, no construction of the map, and no indication whether the almost-complex structure is assumed integrable or whether curvature-decay or other regularity hypotheses are used. With only the abstract available, this geometric bridge remains uninspectable and cannot be certified.

minor comments (1)

[Abstract] The phrase “compatible metric” is standard in the subfield but could be made fully explicit in the abstract (existence of an almost-complex structure J with g(u,v)=ω(u,Jv)) for broader readability.

Circularity Check

0 steps flagged

No circularity detectable; abstract states a pure classification theorem with no self-referential definitions, fits, or load-bearing self-citations visible.

full rationale

Only the abstract is available. It asserts that a simply connected symplectic manifold with a complete nonpositively curved compatible metric is symplectomorphic to standard R^{2n}, answering a question of McDuff-Salamon. No equations, no parameter fittings, no self-citations, no uniqueness theorems imported from the authors, and no ansatzes appear in the provided text. There is therefore no derivation chain that can be reduced by construction to its own inputs. Per the hard rules, circularity is claimed only when a specific quote exhibits an equivalence or forced prediction; none exists here. The result is presented as a standard mathematical implication from geometric hypotheses, which is self-contained against the (empty) circularity criteria. Residual uncertainty about the unseen proof does not constitute circularity under the stated standards.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Pure mathematical classification theorem. No free parameters or invented physical entities. The claim rests on standard domain definitions from symplectic and Riemannian geometry plus the stated topological and curvature hypotheses.

axioms (3)

domain assumption A metric compatible with a symplectic form ω is a Riemannian metric g induced by an almost-complex structure J via g(u,v)=ω(u,Jv).
Standard definition assumed by the phrase 'compatible metric' in the abstract.
domain assumption Nonpositive curvature means nonpositive sectional curvature of the Riemannian metric.
Standard Riemannian notion used in the Hadamard-type statement.
domain assumption The manifold is simply connected and the metric is complete.
Explicit hypotheses of the theorem, parallel to classical Hadamard–Cartan.

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0 comments

read the original abstract

We show that a simply connected symplectic manifold admitting a complete nonpositively curved compatible metric is symplectomorphic to R^{2n} with its standard symplectic form. This answers the "symplectic Hadamard question" of McDuff-Salamon in the affirmative.

The symplectic Hadamard question

Core claim

What carries the argument

Load-bearing premise

What would settle it

If this is right

Where Pith is reading between the lines

discussion (0)