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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A vector bundle quotient by a subbundle is a smooth vector bundle

Statement

If $S \subseteq E$ is a smooth rank- $k$ subbundle of a smooth rank- $r$ vector bundle $E \to M$ , then the fibrewise quotient $E / S \to M$ is a smooth rank- $(r - k)$ vector bundle.

Facts & Assumptions

Given: A smooth vector bundle $E \to M$ and a smooth rank- $k$ subbundle $S \subseteq E$ .

[L1]

Locally, a subbundle is spanned by part of a frame of the ambient bundle (Vector subbundles).

Proof

technique · direct

1.1

Around each point of $M$ , choose a local frame $(e_{1}, \dots, e_{r})$ of $E$ such that $(e_{1}, \dots, e_{k})$ is a local frame of $S$ . Then the quotient classes of $e_{k + 1}, \dots, e_{r}$ form a basis of each quotient fibre $E_{p} / S_{p}$ .

L1givenchoose

2.1

Using the basis from step 1.1, identify the quotient fibre over $p$ with $R^{r - k}$ by reading the coefficients of the classes of $e_{k + 1}, \dots, e_{r}$ . If one changes to another adapted frame, the change-of-frame matrix has block upper-triangular form $[\begin{array}{cc} A & B \\ 0 & C \end{array}]$ , so the quotient coordinates transform by $C$ . Hence the quotient charts are smoothly compatible and define a smooth rank- $(r - k)$ vector bundle.

step 1.1algebra∎

Depends on

Used by

The fibrewise quotient of a vector bundle by arbitrary varying subspaces is a vector bundle False statement
Assuming countable choice, normal and conormal bundles are smooth vector bundles Proposition
The canonical map to a quotient bundle is a smooth bundle map Proposition

Dependency tree · two levels

10 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)