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.

Vector subbundles

Definition

Depends on

Dependency tree · 2 levels

Sources