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.

Smooth sections, local sections, and support

Definition

Depends on

Dependency tree · 2 levels

Sources