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

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Related complete vector fields have intertwined flows

Statement

Let $F : M \to N$ be smooth. If complete vector fields $X$ on $M$ and $Y$ on $N$ are $F$ -related, then for every $t \in R$ ,

$F \circ Φ_{t}^{X} = Φ_{t}^{Y} \circ F .$

Facts & Assumptions

Given: A smooth map $F : M \to N$ , complete vector fields $X$ and $Y$ , and their global flows $Φ^{X}, Φ^{Y}$ .

[L1]

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

[L2]

Complete vector fields have global flows (A vector field is complete if and only if its flow is global).

Proof

technique · direct

1.1

By [L2], the flows $Φ^{X}$ and $Φ^{Y}$ are defined for all real times. Fix $p \in M$ and define $α (t) : = F (Φ_{t}^{X} (p)), β (t) : = Φ_{t}^{Y} (F (p)) .$

L2given

2.1

The curve $β$ is a $Y$ -integral curve through $F (p)$ by definition of the flow. For $α$ , the chain rule encoded in [L1] shows that for every smooth function $f$ on $N$ , $\frac{d}{d t} f (α (t)) = X (f \circ F) (Φ_{t}^{X} (p)) = Y f (α (t)),$ so $α$ is also a $Y$ -integral curve through $F (p)$ .

L1step 1.1

3.1

The two curves $α$ and $β$ solve the same global initial-value problem for $Y$ , so uniqueness gives $α (t) = β (t)$ for every $t$ . This is exactly $F (Φ_{t}^{X} (p)) = Φ_{t}^{Y} (F (p)) .$

step 2.1

4.1

Therefore related complete vector fields have intertwined flows.

step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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)