PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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Related vector fields have related Lie brackets

Statement

If smooth vector fields $X_{1}, X_{2}$ on $M$ are respectively $F$ -related to smooth vector fields $Y_{1}, Y_{2}$ on $N$ , then $[X_{1}, X_{2}]$ is $F$ -related to $[Y_{1}, Y_{2}]$ .

Facts & Assumptions

Given: A smooth map $F : M \to N$ , vector fields $X_{1}, X_{2}$ on $M$ , and vector fields $Y_{1}, Y_{2}$ on $N$ with each $X_{i}$ $F$ -related to $Y_{i}$ .

[L1]

$F$ -relatedness is equivalent to the intertwining identity on smooth functions (F-relatedness is equivalent to the derivation intertwining law).

[L2]

The commutator of smooth vector fields is again a smooth vector field (Smooth vector fields form a Lie algebra under the Lie bracket).

Proof

technique · direct

1.1L1given

For every $f \in C^{\infty} (N)$ , [L1] gives $X_{2} (f \circ F) = (Y_{2} f) \circ F and X_{1} (f \circ F) = (Y_{1} f) \circ F .$

2.1L1step 1.1

Apply $X_{1}$ to the first identity and $X_{2}$ to the second. Using [L1] again on the resulting target functions yields $X_{1} X_{2} (f \circ F) = (Y_{1} Y_{2} f) \circ F, X_{2} X_{1} (f \circ F) = (Y_{2} Y_{1} f) \circ F .$

3.1L1L2step 2.1

Subtracting the identities in step 2.1 gives $[X_{1}, X_{2}] (f \circ F) = ([Y_{1}, Y_{2}] f) \circ F$ for every smooth $f$ on $N$ . By [L2] both brackets are smooth vector fields, and [L1] identifies this identity as their $F$ -relatedness.

4.1step 3.1∎

Therefore related vector fields have related Lie brackets.

Depends on

Used by

Diffeomorphism pushforward preserves Lie brackets Corollary
Integrable distributions are involutive Proposition
Lie derivatives are natural for related vector fields Proposition
The pullback foliation under a transverse map Proposition
Differential of a Lie-group homomorphism is a Lie-algebra homomorphism Theorem

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)