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 zero section is a smooth embedding

Statement

For a smooth vector bundle $π : E \to M$ , the zero section $0_{M} : M \to E$ , $p \mapsto 0_{p}$ , is a smooth embedding.

Facts & Assumptions

Given: A smooth vector bundle $π : E \to M$ .

[L1]

In a vector bundle chart, the bundle is identified over the identity with $U \times R^{r}$ (Smooth vector bundles, rank, fibres, and trivial bundles).

[L2]

A smooth embedding is an injective immersion which is a homeomorphism onto its image with the subspace topology (Smooth embeddings).

Proof

technique · direct

1.1

In a local trivialization $Φ : π^{- 1} (U) \to U \times R^{r}$ , the zero section is represented by $p \mapsto (p, 0)$ . This map is smooth, injective, and its image is the slice $U \times {0}$ .

L1given

2.1

The coordinate slice $U \times {0}$ is an embedded submanifold of the product, so $p \mapsto (p, 0)$ is an immersion and a homeomorphism onto its image. Transporting this property through the bundle charts proves that $0_{M}$ is a smooth embedding by [L2].

L2step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

15 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)