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 fibre bundles and local trivializations

Definition

Let $M$ and $E$ be smooth manifolds, let $F$ be a smooth manifold, and let $π : E \to M$ be a smooth surjection.

A smooth local trivialization with model fibre $F$ on an open set $U \subseteq M$ is a diffeomorphism

$Φ : π^{- 1} (U) \to U \times F$

over $U$ , meaning that ${pr}_{1} \circ Φ = π$ .

The map $π : E \to M$ is a smooth fibre bundle with model fibre $F$ when there is an open cover $(U_{α})$ of $M$ such that each restriction $π^{- 1} (U_{α}) \to U_{α}$ admits a smooth local trivialization.

Thus each point of $M$ has a neighborhood on which $E$ is smoothly identified with a product over the identity on the base.

Smooth fibre bundles and local trivializations

Definition

Depends on

Dependency tree · 2 levels

Sources