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.
Inward, outward, and boundary-tangent vectors
Definition
At , use a boundary chart and the full -dimensional tangent space. A vector is inward, outward, or boundary-tangent if its last coordinate is respectively positive, negative, or zero. The boundary-preserving differential calculation makes these alternatives chart independent.
Depends on
Used by
- An inward field without a negative-time flow in the half-line Counterexample
- Induced boundary orientation Definition
- The closed half-space as a manifold with boundary Example
- Boundary orientation is independent of the outward vector field Proposition
- Boundary-defining functions exist locally and detect inward vectors Proposition
- The boundary tangent space is the boundary-tangent hyperplane Proposition
- A global inward-pointing boundary vector field exists Theorem
- Inward-pointing fields have local forward semiflows at the boundary Theorem
- Positive oriented atlases characterize orientations except for one-manifolds with boundary Theorem
Dependency tree · two levels
12 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)