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

Vector bundle charts and transition functions

Definition

Depends on

Dependency tree · 2 levels

Sources