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 $π^{- 1} (U) \subseteq E$ .

Depends on

Used by

The pullback fibre product is a smooth vector bundle Theorem

Dependency tree · two levels

13 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)