Alphabeta MathHow 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 tubular neighbourhood theorem in a smooth ambient manifold
Statement
Let be a closed smooth embedded submanifold. Then has a tubular neighbourhood in .
Facts & Assumptions
Given: A closed smooth embedded submanifold .
The classical tubular neighbourhood theorem for manifolds says that every closed smooth embedded submanifold has a tubular neighbourhood in its ambient manifold.
The library definition of a tubular neighbourhood is the normal-bundle chart fixed on this page (Tubular neighbourhoods of embedded submanifolds).
Proof
By [F1], the embedded submanifold has a tubular neighbourhood in .
By [L1], this means there is an open neighbourhood of the zero section in the normal bundle and a diffeomorphism from onto an open neighbourhood of in that restricts to the inclusion on the zero section. This is exactly the claimed statement.
Depends on
Used by
Dependency tree · two levels
4 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, 2nd ed., Tubular Neighborhoods (standard reference, not scraped)