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.
Flat charts for a distribution
Definition
Let be a rank- smooth distribution on an -manifold . A chart is a flat chart for when
Equivalently, in the coordinates the distribution is spanned by the first coordinate vector fields.
Depends on
Used by
- A nontransverse pullback need not reproduce the rank of a foliation Counterexample
- Plaques of a flat chart Definition
- A compact leafwise nullhomotopy persists under a transverse deformation Lemma
- A finite characteristic circuit has C² regular port traces Lemma
- A fixed leafwise cap gives a joint transverse product with exact collar data Lemma
- A leafwise path determines a germ of a transverse diffeomorphism Lemma
- A saddle polycycle has a smooth transverse family on either adjacent annulus Lemma
- A taut foliation of a compact connected manifold has a single closed transversal Lemma
- A vanishing cycle determines a nonzero limitwise-nullhomotopy class Lemma
- Fixed transverse fences and their finite crossing words Lemma
- Limitwise-nullhomotopy predicate descends to a normal subgroup Lemma
- The holonomy germ is independent of the foliation chart chain Lemma
- Transverse holonomy transport is well defined and equivariant on the model Lemma
- Existence and uniqueness of maximal connected integral manifolds Theorem
- Holonomy depends only on leafwise homotopy relative to endpoints Theorem
- Regular foliations and integrable distributions correspond 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
- Will J. Merry, Differential Geometry (standard reference, not scraped)