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 vector bundles, rank, fibres, and trivial bundles

Definition

Let $π : E \to M$ be a smooth fibre bundle and let $r \in N$ .

A smooth vector bundle of rank $r$ is a smooth fibre bundle for which every fibre $E_{p} : = π^{- 1} (p)$ is an $r$ -dimensional real vector space and there is an open cover $(U_{α})$ of $M$ such that each restriction $π^{- 1} (U_{α}) \to U_{α}$ admits a local trivialization

$Φ_{α} : π^{- 1} (U_{α}) \to U_{α} \times R^{r}$

whose restriction on each fibre $E_{p}$ is a linear isomorphism $E_{p} \to R^{r}$ .

The fibre over $p \in M$ is called the fibre at $p$ . A vector bundle is trivial when it is globally isomorphic over $M$ to the product bundle $M \times R^{r} \to M$ .

Smooth vector bundles, rank, fibres, and trivial bundles

Definition

Depends on

Dependency tree · 2 levels

Sources