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.
Embedded smooth submanifolds with boundary
Definition
An embedded smooth submanifold with boundary of is a subset supplied with a manifold-with-boundary smooth structure for which is a smooth embedding. In particular this definition does not assert .
Depends on
Used by
- A cycle has zero algebraic intersection with a bounding cycle Corollary
- A submanifold meeting the ambient boundary nonneatly Counterexample
- Curves and geometric intersection numbers on the marked disk Definition
- Dead-end components Definition
- Neat submanifolds of a manifold with boundary Definition
- Smooth cobordism triad for Morse theory Definition
- Bordant cycles have equal intersection numbers Lemma
- Disk bundles over compact bases are compact manifolds with boundary Lemma
- Green's second identity on a compact bordered domain of a Riemann surface Lemma
- Isotopy extension for a compact source with boundary Lemma
- Open manifolds admit exhaustions with no caps Lemma
- Open manifolds admit handle filtrations without top-index handles Lemma
- Opposite-index nondegenerate zeros cancel in a ball Lemma
- Regular exhaustion and Dirichlet solutions on relatively compact surface domains Lemma
- The index sum of an outward field is the Gauss degree Lemma
- The local index is additive under a transverse perturbation Lemma
- The outgoing boundary of a handle attachment trades the disk factors Lemma
- Two points avoiding a finite set lie in a common embedded ball Lemma
- Integration on an oriented embedded submanifold Proposition
- Boundary submanifolds of a boundaryless manifold have half-slice charts Theorem
- The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold Theorem
- Transverse preimages for maps from manifolds with boundary Theorem
Dependency tree · two levels
3 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
- Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
- Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)