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.

Restrictions of vector bundles

Definition

Let $π : E \to M$ be a smooth vector bundle and let $U \subseteq M$ be open. The restriction of $E$ to $U$ is

$E ∣_{U} : = π^{- 1} (U)$

with projection $π ∣_{E ∣_{U}} : E ∣_{U} \to U$ .

The fibres of $E ∣_{U}$ are the same vector spaces $E_{p}$ for $p \in U$ , and the smooth structure on $E ∣_{U}$ is the one induced from the open subset $π^{- 1} (U) \subseteq E$ .

Restrictions of vector bundles

Definition

Depends on

Dependency tree · 2 levels

Sources