DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

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The flowout of an embedded submanifold by a vector field

Definition

Let $X$ be a smooth vector field on $M$ with local flow $Φ : D \to M$ , and let $S ↪ M$ be an embedded submanifold. If $O \subseteq D \cap (R \times S)$ is an open set, the image

$Φ (O)$

is called the flowout of $S$ along $X$ determined by $O$ .

When $O$ is a neighbourhood of ${0} \times S$ , this is the part of $M$ reached by flowing points of $S$ for small times.

Depends on

Used by

The flowout theorem Theorem

Dependency tree · two levels

6 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)