PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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A smooth distribution is exactly a locally framed constant-rank family of tangent spaces

Statement

Let $p \mapsto D_{p} \subseteq T_{p} M$ be a rank- $k$ family of tangent subspaces on a smooth manifold $M$ . Then the following are equivalent:

$D$ is a smooth distribution.
Every point of $M$ has a neighborhood $U$ and smooth vector fields $X_{1}, \dots, X_{k}$ on $U$ such that the vectors $X_{1} (q), \dots, X_{k} (q)$ are linearly independent and span $D_{q}$ for all $q \in U$ .

Facts & Assumptions

Given: A rank- $k$ family $p \mapsto D_{p} \subseteq T_{p} M$ .

[A1]

In item 1, smoothness means that $D$ is a rank- $k$ smooth vector subbundle of $T M$ .

Proof

technique · direct

1.1

Assume $D$ is a smooth distribution. By the local description of a [given] subbundle, each point has a neighborhood $U$ and a frame of $T M ∣_{U}$ whose first $k$ members already frame $D ∣_{U}$ . Those first $k$ sections are smooth vector fields, pointwise independent, and span the prescribed subspaces.

given

1.2

Conversely, assume such local vector fields exist near every point. On a [given] neighborhood $U$ where $X_{1}, \dots, X_{k}$ are pointwise independent, their span is a rank- $k$ subbundle of $T M ∣_{U}$ , because in a local trivialization of $T M$ the columns formed by the $X_{i}$ have rank $k$ everywhere. Since that subbundle has fibres exactly $D_{q}$ , the family is a smooth distribution on $U$ .

givenalgebra

2.1

The two implications establish the equivalence. [given] ∎

given

Depends on

Used by

The annihilator bundle of a distribution Definition
Every constant-dimensional family of tangent subspaces is a smooth distribution False statement
Local sections of a distribution are freely generated by a local frame Proposition
The double annihilator recovers a finite-rank distribution Proposition
Frobenius local coordinate theorem Theorem

Dependency tree · two levels

13 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)
Keith Conrad, Local and global Frobenius theorems (standard reference, not scraped)