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.

Local and global frames of a vector bundle

Definition

Let $π : E \to M$ be a smooth rank- $r$ vector bundle and let $U \subseteq M$ be open.

A tuple of smooth local sections $(s_{1}, \dots, s_{r})$ on $U$ is a local frame when, for every $p \in U$ , the vectors $s_{1} (p), \dots, s_{r} (p)$ form a basis of the fibre $E_{p}$ .

When $U = M$ , the same data are called a global frame.

Depends on

Used by

A vector bundle is trivial if and only if it has a global frame Corollary
Vector subbundles Definition
The tautological line bundle over real projective space Example
Every vector in a fibre extends to a compactly supported smooth section Lemma
Local frames and local trivializations are equivalent data Proposition
Smoothness of a bundle map is equivalent to smooth local matrices Proposition
Smoothness of a section is equivalent to smooth local components Proposition
Every smooth vector bundle admits a smooth bundle metric Theorem

Dependency tree · two levels

15 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)