False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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$α \land d α$ vanishes for every one-form

Statement

The assertion that $α \land d α = 0$ for every one-form is false.

Facts & Assumptions

Given: The manifolds, forms, vector fields, maps, and coordinates explicitly named in the statement.

[F1]

The preceding result states that For a nowhere-zero one-form $α$ , the hyperplane distribution $\ker α$ is integrable if and only if $α \land d α = 0$ . (The codimension-one Frobenius criterion).

Refutation

technique · direct

1.1

For $α = d z - x d y$ , $d α = - d x \land d y$ .

F1given

2.1

Therefore $α \land d α = - d x \land d y \land d z \neq 0$ , disproving the universal assertion.

step 1.1∎

Depends on

The codimension-one Frobenius criterion

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)

α∧dαα∧dα vanishes for every one-form