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

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The ambient value of a Lie bracket is determined by the two pointwise vector values

Statement

For smooth vector fields $X$ and $Y$ , the ambient tangent vector $[X, Y]_{p}$ is determined by the pair of pointwise values $(X_{p}, Y_{p})$ .

Facts & Assumptions

Given: On $R^{2}$ at $p = (0, 0)$ , let $X : = \partial_{x}, Y : = x \partial_{y}, Z : = 0.$

[A1]

The fields $Y$ and $Z$ have the same value at $p$ .

Refutation

technique · direct

1.1

At the origin, $Y_{p} = 0 = Z_{p}$ , while $X_{p} = \partial_{x}$ .

given

1.2

Nevertheless, [given] $[X, Y] = \partial_{y} and [X, Z] = 0,$ so the Lie bracket at $p$ changes when one replaces $Y$ by another field with the same point value.

givenalgebra

1.3

Thus the ambient vector $[X, Y]_{p} \in T_{p} M$ is not determined by the two [given] pointwise vector values: first derivatives of the fields matter. This does not rule out quotient-valued constructions that use a fixed smooth distribution; it refutes only the ambient pointwise-value claim stated above.

given

2.1

Therefore the stated pointwise-determination claim is false. [given] ∎

given

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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)