PropositionStatement: AI-adaptedProof: AI-generatedprecheck passjudge 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.

Sections of Hom are the same as smooth fibrewise linear maps

Statement

For smooth vector bundles $E, F \to M$ , smooth sections of $Hom (E, F)$ are in natural bijection with smooth vector bundle maps $E \to F$ over ${id}_{M}$ .

Facts & Assumptions

Given: Smooth vector bundles $E, F \to M$ .

[L1]

In local frames, a smooth bundle map over the identity is equivalent to a smooth matrix of coefficients (Smoothness of a bundle map is equivalent to smooth local matrices).

[L2]

The Hom bundle is built from the same local matrices (Dual and Hom transition functions define smooth bundles).

Proof

technique · direct

1.1

A section $s : M \to Hom (E, F)$ assigns to each $p \in M$ a linear map $s (p) : E_{p} \to F_{p}$ . Define $Φ_{s} : E \to F$ by $Φ_{s} (v) = s (π (v)) (v)$ . In a pair of local frames, the matrix entries of $s$ are exactly the matrix entries of $Φ_{s}$ .

L1L2givenconstruct

2.1

Conversely, a bundle map $Φ : E \to F$ over ${id}_{M}$ gives a section $s_{Φ} (p) = Φ ∣_{E_{p}} \in Hom (E_{p}, F_{p})$ . The constructions are inverse to one another, and [L1] together with [L2] shows that smoothness on either side is the same local matrix condition.

L1L2step 1.1algebra∎

Depends on

Used by

The graph of a bundle map as a subbundle of a Whitney sum Example
Bundle maps over f are sections of the pulled-back Hom bundle Proposition

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)