DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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 normal addition map for a Euclidean submanifold

Definition

Let $S \subseteq R^{m}$ be an embedded smooth submanifold. Using the Euclidean inner product, define its orthogonal normal bundle by $N^{⊥} S : = {(p, v) \in S \times R^{m} : v ⊥ T_{p} S} .$ Local slice charts and orthogonal projection onto $T_{p} S$ give this set its standard smooth rank- $(m - \dim S)$ vector-bundle structure. The normal addition map is $E : N^{⊥} S \to R^{m}, E (p, v) : = p + v .$ It restricts on the zero section to the inclusion $S ↪ R^{m}$ and is the basic model map used to build Euclidean tubular neighbourhoods.

Depends on

Dependency tree · 2 levels

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Tubular Neighborhoods