TheoremStatement: 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.

Dual and Hom transition functions define smooth bundles

Statement

If $E \to M$ and $F \to M$ are smooth vector bundles, then $E^{*} \to M$ and $Hom (E, F) \to M$ are smooth vector bundles. In local bundle charts, the dual transition matrices are $(g_{β α}^{- 1})^{T}$ and the Hom transition matrices are $A \mapsto h_{β α} A g_{β α}^{- 1}$ .

Facts & Assumptions

Given: Smooth vector bundles $E \to M$ and $F \to M$ with local transition matrices $g_{β α}$ and $h_{β α}$ .

[L1]

Vector bundle chart changes are fibrewise linear and smooth (Vector bundle charts and transition functions).

[L2]

The matrix of the transpose linear map is the transpose matrix (In dual bases, the matrix of $T^{*}$ is the transpose of the matrix of $T$ ).

Proof

technique · direct

1.1

If $λ \in E_{p}^{*}$ has row-coordinate vector $ℓ$ in one dual basis, then after changing the primal basis by $g_{β α} (p)$ , the same functional has coordinate vector $ℓ (g_{β α} (p))^{- 1}$ . By [L2], the dual transition matrix is therefore $(g_{β α} (p)^{- 1})^{T}$ .

L1L2given

2.1

If $A : E_{p} \to F_{p}$ has matrix $B$ in one pair of local frames, then after changing frames by $g_{β α} (p)$ and $h_{β α} (p)$ , the same linear map has matrix $h_{β α} (p) B g_{β α} (p)^{- 1}$ . These formulas are smooth on overlaps because they are built from the smooth transition functions, so they define smooth bundle atlases on $E^{*}$ and $Hom (E, F)$ .

L1step 1.1algebra∎

Depends on

Used by

Sections of Hom are the same as smooth fibrewise linear maps Proposition

Dependency tree · two levels

19 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)