False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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FALSE: the Lie bracket is C^infty-linear in each vector-field entry

Statement

False claim: the Lie bracket is $C^{\infty} (M)$ -linear in each vector-field entry.

Facts & Assumptions

Given: On $M = R$ , the smooth vector fields $X = Y = d / d x$ and the smooth function $f (x) = x$ .

[L1]

The Lie bracket satisfies $[X, f Y] = f [X, Y] + (X f) Y$ (Leibniz rules for the Lie bracket with function multiples).

Refutation

technique · direct

1.1

If the Lie bracket were $C^{\infty} (M)$ -linear in the second entry, one would have $[X, f Y] = f [X, Y]$ for every smooth function $f$ .

given

2.1

For the chosen $X$ and $Y$ , one has $[X, Y] = 0$ , so step 1.1 would give $[X, f Y] = 0$ . But [L1] gives $[X, f Y] = f [X, Y] + (X f) Y = 0 + 1 \cdot Y = Y \neq 0.$

L1step 1.1given

3.1

This contradiction shows that the Lie bracket is not even $C^{\infty} (M)$ -linear in its second vector-field entry, so the claim that it is $C^{\infty} (M)$ -linear in each vector-field entry is false.

step 1.1step 2.1discharge-contradiction∎

Depends on

Leibniz rules for the Lie bracket with function multiples

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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)