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.
Smooth partitions of unity subordinate to an open cover
Definition
Let be a smooth manifold and let be an open cover of . A family of smooth functions with is a smooth partition of unity subordinate to when:
- the family is locally finite;
- for every ; and
- for every .
Remarks
Because the support family is locally finite, the pointwise sum in (3) is locally a finite sum.
Depends on
Used by
- De rham cohomology is continuous homotopy invariant on smooth manifolds Corollary
- Hermitian metric and L² pairing on a compact Riemann surface Definition
- The maximal Dolbeault operator and its Hilbert adjoint on a compact Riemann surface Definition
- A continuous partition of unity need not be smooth False statement
- A locally finite positive smooth family normalizes to a partition of unity Lemma
- De Rham Mayer–Vietoris with boundary and an explicit partition lift Lemma
- Divisibility by a nowhere-vanishing one-form Lemma
- Extending a compact surface metric across its boundary Lemma
- General linear Stiefel approximations stabilize bounded constructible Hom Lemma
- Local finiteness near compact support Lemma
- Locally finite smooth partitions of unity on domains Lemma
- Sphere graph charts, surface density, and a finite partition Lemma
- Sheaves of smooth-function modules are cohomologically acyclic Theorem
- Smooth locally defined functions can be glued by a partition of unity Theorem
- Smooth partitions of unity exist on manifolds Theorem
- The measurable Riemann mapping theorem on the sphere Theorem
- The residue pairing for line-bundle cohomology Theorem
Dependency tree · two levels
7 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)
- Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)