CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Closed differential forms are locally exact

Statement

Every closed smooth differential form of positive degree is locally exact.

Facts & Assumptions

Given: A closed $k$ -form $ω$ on a smooth manifold, $k \geq 1$ , and a point $p$ .

[F1]

Poincare lemma for differential forms on star shaped domains: Every closed smooth $k$ -form on a star-shaped open domain is exact for $k \geq 1$ . For centre $0$ , one primitive is $η_{x} (v_{1}, \dots, v_{k - 1}) = \int_{0}^{1} t^{k - 1} ω_{t x} (x, v_{1}, \dots, v_{k - 1}) d t$ .

[F2]

Pullback of forms is smooth functorial and preserves wedges: For a smooth map $F : M \to N$ , pullback sends smooth differential forms on $N$ to smooth differential forms on $M$ , is functorial, and satisfies $F^{*} (α \land β) = F^{*} α \land F^{*} β .$

[F3]

The exterior derivative commutes with pullback: For every smooth map $F : M \to N$ and every form $ω$ on $N$ , $d (F^{*} ω) = F^{*} (d ω) .$

Proof

technique · direct

1.1

Choose a coordinate neighbourhood $W$ of $p$ mapped diffeomorphically by $x$ onto a Euclidean open ball. In those coordinates $\tilde{ω} = (x^{- 1})^{*} (ω ∣_{W})$ is closed: naturality gives $d \tilde{ω} = (x^{- 1})^{*} d (ω ∣_{W}) = 0$ . The ball is star-shaped, so there is a smooth $(k - 1)$ -form $η$ there with $d η = \tilde{ω}$ .

F1F3given

2.1

Pulling back by $x$ , naturality of the exterior derivative gives $d (x^{*} η) = x^{*} d η = x^{*} (x^{- 1})^{*} (ω ∣_{W}) = ω ∣_{W}$ . Thus $x^{*} η$ is the requested local primitive. If the form is forced to be zero by dimension, the zero primitive works; on an empty manifold the assertion over all points is vacuous.

F2F3step 1.1∎

Source locator

Lee, Corollary 17.15, p.447, restricted explicitly to positive degree.

Depends on

Used by

Nothing in the library uses this result yet.

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, second edition (standard reference, not scraped)