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 $M \times R^{r} \to M$ .

Depends on

Used by

Dual and Hom vector bundles Definition
Pullback vector bundles as fibre products Definition
Restrictions of vector bundles Definition
Smooth bundle metrics Definition
Smooth sections, local sections, and support Definition
Vector bundle charts and transition functions Definition
Vector bundle maps over a smooth base map Definition
Vector subbundles Definition
Whitney sums of vector bundles Definition
Assuming countable choice, the tangent and cotangent bundles are smooth vector bundles Example
The trivial line bundle and its sections as functions Example
A vector bundle projection is a surjective submersion Proposition
Assuming countable choice, an ambient metric identifies the two normal bundles Proposition
The total space of a rank-r bundle has dimension dim M + r Proposition
The zero section is a smooth embedding Proposition

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)