PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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F-relatedness is equivalent to the derivation intertwining law

Statement

Let $F : M \to N$ be smooth, let $X$ be a smooth vector field on $M$ , and let $Y$ be a smooth vector field on $N$ . Then $X$ and $Y$ are $F$ -related if and only if

$X (f \circ F) = (Y f) \circ F$

for every $f \in C^{\infty} (N)$ .

Facts & Assumptions

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

[L1]

The differential satisfies $d F_{p} (v) (f) = v (f \circ F)$ for $v \in T_{p} M$ and $f \in C^{\infty} (N)$ (The differential of a smooth map).

[L2]

The action of a vector field on functions is defined pointwise by $(X h) (p) = X_{p} (h)$ (The action of a vector field on smooth functions).

[L3]

For a point inside an open set there is a smooth bump function equal to $1$ on a neighbourhood of that point and supported in the open set (A manifold bump for a compact set inside an open set).

Proof

technique · direct

1.1

Assume $X$ and $Y$ are $F$ -related. For any $p \in M$ and $f \in C^{\infty} (N)$ , [L1] and [L2] give $(X (f \circ F)) (p) = X_{p} (f \circ F) = d F_{p} (X_{p}) (f) = Y_{F (p)} (f) = ((Y f) \circ F) (p)$ .

L1L2given

1.2

Conversely, assume $X (f \circ F) = (Y f) \circ F$ for every smooth $f$ on $N$ . Fix $p \in M$ , and let $[g] \in C_{F (p)}^{\infty} (N)$ be a smooth germ. Choose an open neighbourhood $W$ of $F (p)$ on which $g$ is represented by a smooth function, and use [L3] to choose $χ : N \to [0, 1]$ that is $1$ on a neighbourhood of $F (p)$ and has support contained in $W$ . Let $h$ be the global smooth function obtained by extending $χ g$ by $0$ outside $W$ . Then $[h] = [g]$ at $F (p)$ . The hypothesis applied to $h$ gives $X_{p} (h \circ F) = Y_{F (p)} (h)$ , and [L1] identifies the left-hand side with $d F_{p} (X_{p}) ([h])$ . Thus $d F_{p} (X_{p}) ([g]) = Y_{F (p)} ([g])$ for every germ $[g]$ , so $d F_{p} (X_{p}) = Y_{F (p)}$ .

L1L2L3given

2.1

Therefore $X$ and $Y$ are $F$ -related exactly when the derivation intertwining law holds for every smooth target function.

step 1.1step 1.2∎

Depends on

Used by

Related complete vector fields have intertwined flows Proposition
Related vector fields have related Lie brackets Proposition

Dependency tree · two levels

9 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)