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.

The total space of a rank-r bundle has dimension dim M + r

Statement

If $π : E \to M$ is a smooth vector bundle of rank $r$ and $\dim M = n$ , then $\dim E = n + r$ .

Facts & Assumptions

Given: A rank- $r$ smooth vector bundle $π : E \to M$ with $\dim M = n$ .

[L1]

Every point of $E$ lies in a vector bundle chart identified with $U \times R^{r}$ over an open set $U \subseteq M$ (Vector bundle charts and transition functions).

Proof

technique · direct

1.1L1given

Fix $e \in E$ with base point $p = π (e)$ . Choose a chart of $M$ sending a neighborhood of $p$ diffeomorphically onto an open subset of $R^{n}$ , and combine it with a vector bundle chart from [L1]. This identifies a neighborhood of $e$ in $E$ with an open subset of $R^{n} \times R^{r}$ .

2.1step 1.1algebra∎

Since $R^{n} \times R^{r} ≅ R^{n + r}$ , every point of $E$ has a chart of dimension $n + r$ . Therefore the total space $E$ is an $(n + r)$ -manifold.

Depends on

Used by

The Milnor sphere and disk bundles M_h,j and W_h,j Definition

Dependency tree · two levels

12 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)