PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

A vector field is complete if and only if its flow is global

Statement

A smooth vector field $X$ on $M$ is complete if and only if its maximal flow domain is all of $R \times M$ .

Facts & Assumptions

Given: A smooth vector field $X$ with maximal flow $Φ : D \to M$ .

[L1]

The maximal flow domain is $D = {(t, p) : t \in I_{p}},$ where $I_{p}$ is the domain of the maximal integral curve through $p$ (The fundamental theorem on flows).

[L2]

A vector field is complete exactly when every maximal integral curve is defined on all of $R$ (Complete vector fields).

Proof

technique · direct

1.1

If $X$ is complete, then [L2] gives $I_{p} = R$ for every $p$ . By [L1], this means $D = R \times M$ .

L1L2given

1.2

Conversely, if $D = R \times M$ , then [L1] gives $I_{p} = R$ for every $p$ . Therefore [L2] says that $X$ is complete.

L1L2given

2.1

Hence $X$ is complete if and only if its maximal flow is global.

step 1.1step 1.2∎

Depends on

Used by

Constant vector fields have translation flows Example
The planar rotation field has the circle rotation flow Example
The radial vector field has the dilation flow Example
The vector field x² d/dx has finite-time escape Example
FALSE: every smooth vector field is complete False statement
Related complete vector fields have intertwined flows Proposition
Compactly supported smooth vector fields are complete Theorem

Dependency tree · two levels

5 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

Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)