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

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Compactly supported smooth vector fields are complete

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Every compactly supported smooth vector field on a smooth manifold is complete.

Facts & Assumptions

Given: ${A C}_{ω}$ and a smooth vector field $X$ on $M$ with compact support $K$ . This assumption supplies the canonical smooth tangent bundle used in A smooth vector field is a smooth section of the tangent bundle.

[L1]

A vector field is complete if and only if its maximal flow is global (A vector field is complete if and only if its flow is global).

[L2]

The maximal flow exists on an open domain and its time slices are the maximal integral curves (The fundamental theorem on flows).

[L3]

The support of a section is the closure of the set where it is nonzero (Smooth sections, local sections, and support).

Proof

technique · direct

1.1L3given

Let $γ : I \to M$ be a maximal integral curve of $X$ through some point $p$ . If $γ (t_{0}) \notin K$ for some $t_{0} \in I$ , then $X_{γ (t_{0})} = 0$ by [L3], so the constant curve through $γ (t_{0})$ is an integral curve of $X$ . Uniqueness therefore forces $γ$ to be constant on the connected component of ${t \in I : γ (t) \notin K}$ containing $t_{0}$ . Thus every nonconstant part of $γ$ lies in $K$ .

2.1step 1.1given

Suppose $I$ had a finite right endpoint $b$ . Choose times $t_{n} ↑ b$ . If infinitely many $γ (t_{n})$ lie in $K$ , compactness of $K$ gives a subsequence converging to some $q \in K$ . Otherwise $γ (t_{n}) \notin K$ for all large $n$ , and step 1.1 makes those tail values constant on a neighbourhood of $b$ ; hence $γ (t_{n}) \to q$ for some $q \in M ∖ K$ .

3.1L2step 2.1

By [L2], there is a local flow through $q$ defined on some interval $(- ε, ε)$ . For $n$ large, $γ (t_{n})$ lies in its domain, so uniqueness of integral curves extends $γ$ past $b$ by flowing forward from $γ (t_{n})$ for time larger than $b - t_{n}$ . This contradicts maximality.

4.1L1step 3.1∎

The same argument excludes a finite left endpoint. Therefore every maximal integral curve is defined on all of $R$ , and [L1] implies that $X$ is complete.

Depends on

Used by

Every smooth vector field on a compact manifold is complete Corollary
The local Whitney move Definition
The middle-handle intersection matrix of an h-cobordism Definition
A local Whitney move in Euclidean space Example
Dual handle presentations of a genus-g surface Example
Adapted descending field near a compact morse band Lemma
Compact smooth manifolds have finite CW models under countable choice Lemma
Flow reparametrization realizes a level isotopy Lemma
Gradient-like perturbation separates adjacent critical levels Lemma
Handles of equal index can be attached on one level Lemma
Increasing reparametrization of finitely many critical levels Lemma
Moving a sphere off a lower-dimensional submanifold Lemma
Normalized gradient crosses a compact regular band in controlled time Lemma
A leafwise positive closed two-form calibrates a taut foliation Proposition
Dual elimination of top-index handles Proposition
First variation of volume for a normal variation Proposition
Relative Morse inequalities for a cobordism Proposition
A cobordism with no handles is a product Theorem
Adapted excellent Morse functions exist on compact cobordisms Theorem
Self-indexing Morse functions exist Theorem
The Whitney move removes a cancelling pair of intersection points Theorem

Dependency tree · two levels

18 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)