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.

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

Time-dependent vector fields and their evolution operators

Definition

Assume ${A C}_{ω}$ , so that $T M$ carries its canonical smooth structure. Let $M$ be a smooth manifold, and let $I \subseteq R$ be an interval. A time-dependent vector field on $M$ over $I$ is a smooth map

$X : I \times M \to T M$

such that $X (t, p) \in T_{p} M$ for every $(t, p) \in I \times M$ . One often writes $X_{t} (p) : = X (t, p)$ .

An evolution operator for $X$ is a family of maps

$Ψ_{t, s} : U_{s} \to U_{t}$

defined for pairs $(t, s)$ in some time domain, such that for each $p \in U_{s}$ the curve $r \mapsto Ψ_{r, s} (p)$ solves

$\frac{d}{d r} Ψ_{r, s} (p) = X_{r} (Ψ_{r, s} (p)), Ψ_{s, s} (p) = p .$

Depends on

A smooth vector field is a smooth section of the tangent bundle

Used by

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

Marius Crainic, Rui Loja Fernandes, and Ioan Marcut, Lectures on Poisson Geometry (standard reference, not scraped)
John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)