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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Whitney sums are smooth vector bundles

Statement

If $E \to M$ and $F \to M$ are smooth vector bundles of ranks $r$ and $s$ , then $E \oplus F \to M$ is a smooth vector bundle of rank $r + s$ .

Facts & Assumptions

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

[L1]

On a common trivializing neighborhood, vector bundle charts identify $E$ and $F$ with $U \times R^{r}$ and $U \times R^{s}$ (Vector bundle charts and transition functions).

Proof

technique · direct

1.1L1givenconstruct

On a common trivializing neighborhood $U$ , use [L1] to identify $E ∣_{U} \oplus F ∣_{U}$ with $U \times (R^{r} \oplus R^{s}) ≅ U \times R^{r + s}$ . This gives a local trivialization of the Whitney sum.

2.1L1step 1.1algebra∎

If the transition matrices for $E$ and $F$ are $g_{β α}$ and $h_{β α}$ , then the transition matrix for $E \oplus F$ is the block diagonal matrix $diag (g_{β α}, h_{β α})$ , which is smooth on overlaps. Therefore these local trivializations define a smooth rank- $(r + s)$ bundle.

Depends on

Used by

Dependency tree · two levels

16 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)