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.

Vector bundle charts and transition functions

Definition

Let $π : E \to M$ be a smooth rank- $r$ vector bundle and let $Φ_{α} : π^{- 1} (U_{α}) \to U_{α} \times R^{r}$ be a local trivialization whose restriction on each fibre is a linear isomorphism. The pair $(U_{α}, Φ_{α})$ is a vector bundle chart.

For two vector bundle charts $(U_{α}, Φ_{α})$ and $(U_{β}, Φ_{β})$ , the overlap map

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

has the form

$Φ_{β} \circ Φ_{α}^{- 1} (p, v) = (p, g_{β α} (p) v),$

where $g_{β α} : U_{α} \cap U_{β} \to G L (r, R)$ is smooth. The map $g_{β α}$ is the transition function from chart $α$ to chart $β$ .

Depends on

Smooth vector bundles, rank, fibres, and trivial bundles

Used by

Vector bundle transition functions satisfy the cocycle identities Lemma
Isomorphic cocycles define isomorphic vector bundles Proposition
Local frames and local trivializations are equivalent data Proposition
The total space of a rank-r bundle has dimension dim M + r Proposition
Construction of a vector bundle from a smooth cocycle Theorem
Dual and Hom transition functions define smooth bundles Theorem
The pullback fibre product is a smooth vector bundle Theorem
Whitney sums are smooth vector bundles Theorem

Dependency tree · two levels

4 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)