DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-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 and exact differential forms

Definition

For the complex De rham cochain complex, put $Z^{k} (M) = \ker (d : Ω^{k} (M) \to Ω^{k + 1} (M))$ and $B^{k} (M) = im (d : Ω^{k - 1} (M) \to Ω^{k} (M))$ . A form is closed if it belongs to $Z^{k}$ and exact if it belongs to $B^{k}$ .

If $ω = d η$ , then $d ω = d^{2} η = 0$ by The exterior derivative squares to zero, so $B^{k} \subseteq Z^{k}$ . In particular $B^{0} = 0$ , since $Ω^{- 1} = 0$ . The zero form is both closed and exact in every degree.

Source locator

Lee, Introduction to Smooth Manifolds, 2nd ed., Chapter 17, pp.441–443, Proposition 17.2 and Corollary 17.3; local quotient calculations below.

Depends on

Used by

The closed angular form on the punctured plane is not exact Counterexample
De rham cohomology Definition
Every smooth form is either closed or exact False statement
The de rham cohomology class of a form is defined without closedness False statement
The poincare lemma says every closed form is globally exact False statement
Wedge with a closed form preserves exactness classes Lemma

Dependency tree · two levels

7 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)
Nigel Hitchin, Differentiable Manifolds (2014) (standard reference, not scraped)