TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The Pfaffian Frobenius criterion

Statement

If $θ^{1}, \dots, θ^{n - k}$ locally frame $D^{\circ}$ , then $D$ is involutive if and only if $d θ^{a} = \sum_{b} η_{b}^{a} \land θ^{b}$ locally for every $a$ ; equivalently, its annihilator ideal is differential.

Facts & Assumptions

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

[F1]

The preceding result states that A differential ideal is a graded wedge ideal $I \subseteq Ω^{*} (M)$ satisfying $d I \subseteq I$ . (A differential ideal in the algebra of forms).

Proof

technique · direct

1.1

For tangent fields $X, Y$ in $D$ , $d θ^{a} (X, Y) = - θ^{a} ([X, Y])$ ; thus involutivity forces each $d θ^{a}$ to vanish on $D$ and so to have the displayed coframe decomposition.

F1given

2.1

Conversely that decomposition vanishes on pairs from $D$ , so every $θ^{a} ([X, Y]) = 0$ and $[X, Y] \in D$ ; the frame-independent ideal statement is the same condition.

step 1.1∎

Depends on

Used by

The codimension-one Frobenius criterion Corollary
An integrable Pfaffian equation with a local first integral Example

Dependency tree · two levels

11 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)