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.

Pullback vector bundles as fibre products

Definition

Let $π : E \to M$ be a smooth vector bundle and let $f : N \to M$ be a smooth map. The pullback set of $E$ along $f$ is

$f^{*} E : = {(q, e) \in N \times E : f (q) = π (e)} .$

Its projection to $N$ is $(q, e) \mapsto q$ , and its fibre over $q \in N$ is canonically identified with $E_{f (q)}$ . The following theorem shows that this set carries a natural smooth vector-bundle structure.

Pullback vector bundles as fibre products

Definition

Depends on

Dependency tree · 2 levels

Sources