DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Positive volume form on an oriented manifold

Definition

On an oriented smooth $n$ -manifold, a positive volume form is a nowhere-vanishing smooth top form $μ$ that evaluates positively on the chosen determinant ray. In a signed chart $μ = ρ d x^{1} \land \dots \land d x^{n}$ this means $σ_{ϕ} ρ > 0$ . In dimension zero it means $ε (p) μ (p) > 0$ . No metric is part of the data. Under ${A C}_{ω}$ such a form exists by Orientability is equivalent to a nowhere-vanishing top form: adjust the sign on each component to match the specified orientation.

Depends on

Used by

Divergence relative to a volume form Definition

Dependency tree · two levels

8 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

Nicolaescu §3.4.2 orientation-isomorphism paragraph p.120; Lee Proposition 16.6(c) (standard reference, not scraped)