DefinitionDefinition: AI-adaptedProof: 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Local and global flows generated by a vector field

Definition

Let $X$ be a smooth vector field on $M$ .

A local flow of $X$ consists of an open set $D \subseteq R \times M$ containing ${0} \times M$ and a smooth map $Φ : D \to M$ such that:

$Φ (0, p) = p$ for every $p \in M$ ;
for each $p$ , the fibre $D_{p} : = {t : (t, p) \in D}$ is an interval;
for each $p$ , the curve $t \mapsto Φ (t, p)$ is an integral curve of $X$ on $D_{p}$ ;
whenever both sides are defined, $Φ (t, Φ (s, p)) = Φ (t + s, p)$ .

If $D = R \times M$ , then $Φ$ is the global flow of $X$ .

Depends on

Used by

The flowout of an embedded submanifold by a vector field Definition
The fundamental theorem on flows Theorem

Dependency tree · two levels

4 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)