TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-31

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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.

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The pullback fibre product is a smooth vector bundle

Statement

If $π : E \to M$ is a smooth rank- $r$ vector bundle and $f : N \to M$ is smooth, then the fibre product $f^{*} E$ is a smooth rank- $r$ vector bundle over $N$ .

Facts & Assumptions

Given: A smooth rank- $r$ vector bundle $π : E \to M$ and a smooth map $f : N \to M$ .

[L1]

A vector bundle chart on $E$ over $U \subseteq M$ is a diffeomorphism $E ∣_{U} ≅ U \times R^{r}$ with transition functions of the form $(p, v) \mapsto (p, g_{β α} (p) v)$ (Vector bundle charts and transition functions).

[L2]

The restriction $E ∣_{U}$ is the same total space over the smaller open base (Restrictions of vector bundles).

Proof

technique · direct

1.1

Let $Φ_{α} : E ∣_{U_{α}} \to U_{α} \times R^{r}$ be a vector bundle chart. For $q \in f^{- 1} (U_{α})$ and $(q, e) \in f^{*} E$ , define ${\tilde{Φ}}_{α} (q, e) = (q, v)$ when $Φ_{α} (e) = (f (q), v)$ . This is a bijection $(f^{*} E) ∣_{f^{- 1} (U_{α})} \to f^{- 1} (U_{α}) \times R^{r}$ .

L1L2givenconstruct

2.1

On overlaps, ${\tilde{Φ}}_{β} \circ {\tilde{Φ}}_{α}^{- 1} (q, v) = (q, g_{β α} (f (q)) v)$ . These chart changes are smooth and fibrewise linear because the original transition functions are smooth. Therefore the pulled-back charts define a smooth rank- $r$ vector bundle over $N$ .

L1step 1.1algebra∎

Depends on

Used by

Pullback of the tautological line bundle along the antipodal cover Example
The pullback bundle is the set-theoretic inverse image of the total space False statement
Assuming countable choice, an ambient metric identifies the two normal bundles Proposition
Bundle maps over f are sections of the pulled-back Hom bundle Proposition
Pullback is functorial up to canonical bundle isomorphism Proposition

Dependency tree · two levels

14 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)