PropositionStatement: AI-adaptedProof: AI-generatedprecheck passaudited 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.

Bundle maps over f are sections of the pulled-back Hom bundle

Statement

Let $Φ : E \to F$ be a smooth vector bundle map over a smooth map $f : M \to N$ . Then $Φ$ is naturally equivalent to a smooth section of the pulled-back Hom bundle $Hom (E, f^{*} F) \to M$ .

Facts & Assumptions

Given: Smooth vector bundles $E \to M$ , $F \to N$ , a smooth map $f : M \to N$ , and a bundle map $Φ : E \to F$ over $f$ .

[L1]

The pullback fibre product $f^{*} F$ is a smooth vector bundle over $M$ (The pullback fibre product is a smooth vector bundle).

[L2]

Sections of a Hom bundle are the same as fibrewise linear bundle maps over the identity (Sections of Hom are the same as smooth fibrewise linear maps).

Proof

technique · direct

1.1L1givenconstruct

For each $p \in M$ , the fibre map $Φ_{p} : E_{p} \to F_{f (p)}$ may be viewed as a linear map $E_{p} \to (f^{*} F)_{p}$ because $(f^{*} F)_{p}$ is canonically $F_{f (p)}$ . Hence $p \mapsto Φ_{p}$ defines a section of $Hom (E, f^{*} F) \to M$ .

2.1L1L2step 1.1algebra∎

In local trivializations the matrix of this section is exactly the local matrix of $Φ$ , so the section is smooth exactly when $Φ$ is smooth. Therefore [L2] applied to the bundles $E$ and $f^{*} F$ yields the required bijection.

Depends on

Used by

Dependency tree · two levels

17 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)