DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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.

Smooth sections, local sections, and support

Definition

Let $π : E \to M$ be a smooth vector bundle.

A smooth section of $E$ is a smooth map $s : M \to E$ such that $π \circ s = {id}_{M}$ .

If $U \subseteq M$ is open, a smooth local section on $U$ is a smooth map $s : U \to E ∣_{U}$ with $π \circ s = {id}_{U}$ .

The support of a section $s$ is the closure of the set ${p \in M : s (p) \neq 0_{p}}$ . A section is compactly supported when this set has compact closure.

Depends on

Used by

Local and global frames of a vector bundle Definition
The trivial line bundle and its sections as functions Example
Smooth sections form a module over smooth functions Proposition
Smoothness of a section is equivalent to smooth local components Proposition

Dependency tree · two levels

7 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 (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)