CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-04

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.

Diffeomorphism pushforward preserves Lie brackets

Statement

If $F : M \to N$ is a diffeomorphism and $X, Y$ are smooth vector fields on $M$ , then

$F_{*} [X, Y] = [F_{*} X, F_{*} Y] .$

Facts & Assumptions

Given: A diffeomorphism $F : M \to N$ and smooth vector fields $X, Y$ on $M$ .

[L1]

For a diffeomorphism, the pushforward $F_{*} X$ is by definition the unique vector field $F$ -related to $X$ (Pushforwards and pullbacks of vector fields by a diffeomorphism).

[L2]

Related vector fields have related Lie brackets (Related vector fields have related Lie brackets).

Proof

technique · direct

1.1

By [L1], the vector fields $X$ and $F_{*} X$ are $F$ -related, and likewise $Y$ and $F_{*} Y$ are $F$ -related.

L1given

2.1

Applying [L2] to step 1.1 shows that $[X, Y]$ is $F$ -related to $[F_{*} X, F_{*} Y]$ . By the definition of pushforward in [L1], this means exactly that $F_{*} [X, Y] = [F_{*} X, F_{*} Y]$ .

L1L2step 1.1∎

Depends on

Used by

An involutive local frame can be reduced to one field plus commuting transverse fields Lemma

Dependency tree · two levels

8 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)