DefinitionDefinition: AI-adaptedProof: Not applicableaudited 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.

Vector subbundles

Definition

Let $π : E \to M$ be a smooth rank- $r$ vector bundle, and let $k \in N$ satisfy $0 \leq k \leq r$ . A subset $S \subseteq E$ is a smooth vector subbundle of rank $k$ when:

$S \cap E_{p}$ is a $k$ -dimensional linear subspace of $E_{p}$ for every $p \in M$ , and
every point of $M$ has an open neighborhood $U$ with a local frame $(s_{1}, \dots, s_{r})$ of $E ∣_{U}$ such that $(s_{1}, \dots, s_{k})$ is a local frame of the fibrewise subsets $S \cap E_{p}$ : explicitly, $S \cap E_{p} = span {s_{1} (p), \dots, s_{k} (p)}$ for every $p \in U$ .

In particular, a vector subbundle has constant fibre dimension and is locally spanned by part of a frame of the ambient bundle.

Depends on

Used by

Quotient vector bundles by a subbundle Definition
The fibrewise quotient of a vector bundle by arbitrary varying subspaces is a vector bundle False statement
Assuming countable choice, an ambient metric identifies the two normal bundles Proposition
Assuming countable choice, normal and conormal bundles are smooth vector bundles Proposition
Constant-rank kernels and images of bundle maps over one base are subbundles Proposition
Orthogonal complements of subbundles are smooth subbundles Proposition
A vector bundle quotient by a subbundle is a smooth vector bundle Theorem

Dependency tree · two levels

11 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)