DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Flat charts for a distribution

Definition

Let $D$ be a rank- $k$ smooth distribution on an $n$ -manifold $M$ . A chart $φ : U \to R^{k} \times R^{n - k}$ is a flat chart for $D$ when

$D_{q} = d φ_{q}^{- 1} (R^{k} \times {0}) (q \in U) .$

Equivalently, in the coordinates $φ = (x, y)$ the distribution is spanned by the first $k$ coordinate vector fields.

Depends on

Frobenius local coordinate theorem

Used by

Plaques of a flat chart Definition
Existence and uniqueness of maximal connected integral manifolds Theorem
Regular foliations and integrable distributions correspond Theorem

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

Will J. Merry, Differential Geometry (standard reference, not scraped)