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.

Depends on

Used by

Smooth vector bundles, rank, fibres, and trivial bundles Definition
Construction of a vector bundle from a smooth cocycle Theorem

Dependency tree · two levels

17 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)