DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.

Regular foliation atlases

Definition

A regular foliation atlas of codimension $n - k$ on an $n$ -manifold $M$ is an atlas of charts $φ_{α} = (x_{α}, y_{α}) : U_{α} \to R^{k} \times R^{n - k}$ such that on each overlap the transition map has the form

$(x_{β}, y_{β}) = (g_{β α} (x_{α}, y_{α}), h_{β α} (y_{α})) .$

Thus the second coordinates depend only on the old transverse coordinates, so plaques are sent to plaques.

Depends on

Plaques of a flat chart

Used by

Leaves of a regular foliation Definition
The Mobius-band line foliation Example
The product foliation Example
Regular foliations and integrable distributions correspond Theorem

Dependency tree · two levels

2 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)
Will J. Merry, Differential Geometry (standard reference, not scraped)