False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Every constant-dimensional family of tangent subspaces is a smooth distribution

Statement

Every constant-dimensional family of tangent subspaces is a smooth distribution.

Facts & Assumptions

Given: On $R^{2}$ , define $D_{(x, y)} : = {\begin{cases} span (\partial_{x}), & y > 0, \\ span (\partial_{x} + \partial_{y}), & y \leq 0. \end{cases}$

[A1]

This is a one-dimensional family of tangent lines.

Refutation

technique · direct

1.1

The family has constant dimension $1$ at every point.

given

1.2

If it were a smooth distribution near the origin, it would admit a local [given] nonvanishing smooth spanning field. Above the $x$ -axis that field would have to be tangent to $\partial_{x}$ , while below the axis it would have to be tangent to $\partial_{x} + \partial_{y}$ . Continuity at the origin would then force the two line directions to agree there, which they do not.

given

2.1

Therefore constant fibre dimension alone does not imply smoothness.

given∎

Depends on

Used by

Nothing in the library uses this result yet.

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

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