False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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Every nondegenerate two-form is symplectic

Statement refuted

Every nondegenerate two-form is symplectic.

Facts & Assumptions

Given: The proposed universal claim.

[F1]

A symplectic form must be both nondegenerate and closed. Symplectic form and symplectic manifold.

Refutation

technique · direct

1.1

On $R^{4}$ put $η = d x_{1} \land d y_{1} + e^{x_{1}} d x_{2} \land d y_{2}$ . Its square is $2 e^{x_{1}} d x_{1} \land d y_{1} \land d x_{2} \land d y_{2}$ , which never vanishes, so $η$ is nondegenerate.

F1algebra

2.1

But $d η = e^{x_{1}} d x_{1} \land d x_{2} \land d y_{2} \neq 0$ . Thus [F1] excludes $η$ from being symplectic, refuting the claim.

F1step 1.1algebra∎

Depends on

Symplectic form and symplectic manifold

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)