CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge 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.

Frobenius gives local first integrals

Statement

Let $D$ be an involutive rank- $k$ distribution on an $n$ -manifold $M$ . Then near every point there is a submersion $F = (f^{k + 1}, \dots, f^{n})$ onto an open set of $R^{n - k}$ such that

$D = \ker d F .$

Equivalently, the functions $f^{k + 1}, \dots, f^{n}$ are local first integrals for $D$ .

Facts & Assumptions

Given: An involutive rank- $k$ distribution $D$ and a point $p \in M$ .

[A1]

Choose Frobenius coordinates around $p$ .

Proof

technique · direct

1.1

In Frobenius coordinates $(x^{1}, \dots, x^{n})$ , the distribution is spanned by [given] $\partial_{x^{1}}, \dots, \partial_{x^{k}}$ . Define $F (x^{1}, \dots, x^{n}) : = (x^{k + 1}, \dots, x^{n}) .$ Its differential has rank $n - k$ , so $F$ is a submersion.

givenconstruct

1.2

A tangent vector lies in $\ker d F$ exactly when its last $n - k$ coordinate [given] components vanish, so precisely when it is a linear combination of $\partial_{x^{1}}, \dots, \partial_{x^{k}}$ . Hence $\ker d F = D$ .

givenalgebra

1.3

Therefore every involutive distribution is locally the common kernel of [given] $n - k$ smooth first integrals.

given∎

Depends on

Frobenius local coordinate theorem

Used by

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