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.

Smooth distributions on a manifold

Definition

Assume ${A C}_{ω}$ , so that $T M$ carries the canonical smooth vector bundle structure supplied by Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure. Let $M$ be a smooth manifold and let $k \geq 0$ be an integer. A smooth distribution of rank $k$ on $M$ is a rank- $k$ smooth vector subbundle $D \subseteq T M$ .

Equivalently, to each $p \in M$ it assigns a $k$ -dimensional linear subspace $D_{p} \subseteq T_{p} M$ , with the dependence on $p$ smooth in the sense of vector subbundles.

Depends on

Used by

A bracket-closed variable-rank family outside regular Frobenius Counterexample
The standard contact plane field is not integrable Counterexample
Integral manifolds of a distribution Definition
The annihilator bundle of a distribution Definition
Vector fields tangent to a distribution Definition
The coordinate-plane distribution and its affine leaves Example
Every constant-dimensional family of tangent subspaces is a smooth distribution False statement
Every smooth distribution is integrable False statement
Frobenius applies to any variable-rank family of subspaces False statement
A smooth distribution is exactly a locally framed constant-rank family of tangent spaces Proposition
Integral manifolds have the distribution dimension Proposition
Local diffeomorphisms carry distributions and integral manifolds Proposition

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, 2nd ed. (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Keith Conrad, Local and global Frobenius theorems (standard reference, not scraped)