LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-30

How 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.

Finite frameable decomposition of a regular disk region

Statement

Assume the axiom of choice. Let $(M, g)$ be an oriented Riemannian surface and let $D \subseteq M$ be a compact regular oriented disk region with finitely many ordinary corners. Then $D$ admits a finite face-to-face subdivision into regular triangular closed-disk pieces whose vertices, edges and faces form a finite regular CW structure on $D$ , such that each piece is contained in an oriented coordinate chart of $M$ , carries a smooth positively oriented $g$ -orthonormal frame, and has only ordinary corners after finite subdivision. Every new edge is piecewise smooth; the relative interior of each new edge that is not a supplied boundary subarc lies in $Int D$ , while its endpoints may lie on $\partial D$ .

Facts & Assumptions

Given: The oriented ambient Riemannian surface and compact regular disk region $D$ with ordinary corners. Full AC is assumed because [F1] uses the arbitrary-Jordan-curve planar suppliers (The Axiom of Choice).

[F1]

A compact regular cornered region has finite triangular closed-disk face, edge, and link data preserving its boundary; every face lies in a coordinate disk carrying a smooth orthonormal frame after choosing a local orientation (Finite curvilinear triangulation of a compact Riemannian surface).

[F2]

On an oriented surface, Gram–Schmidt applied to a positive coordinate basis gives a smooth positively oriented orthonormal frame (Oriented Riemannian surface and positive quarter-turn).

[F3]

The regular-region boundary convention gives ordinary sector corners and the outward-normal-first orientation (Regular oriented surface regions with corners).

Proof

technique · specialize the finite curvilinear face data to the oriented disk and orient each local frame positively

1.1F1F3given

Apply [F1] to $D$ in the supplied ambient surface. It gives finitely many face-to-face triangular closed disks, each in an ambient coordinate disk, with regular $C^{2}$ edges, ordinary sector corners, full-edge intersections, and the original boundary arcs retained up to subdivision. Every added edge has relative interior in $Int D$ .

2.1F1F2F3step 1.1∎

Since the ambient surface is oriented, choose the positive coordinate orientation on each face chart and apply [F2]. The resulting smooth positive $g$ -orthonormal frame is defined on an open neighbourhood of each closed face. Give each face the orientation induced from $D$ ; its sector corners remain ordinary by [F3]. The triangular closed cells of step 1.1 have full-edge or vertex intersections and nonzero sector links, so [F1]'s regular-CW conclusion applies to their vertices, edges and faces. Thus they form the asserted frameable disk decomposition. The exact full-AC use is inherited solely through [F1]'s planar Jordan/graph construction.

Source locator

Lee, Riemannian Manifolds, Chapter 9, printed pp. 165–172, proves the local formula in a positive chart frame and outlines finite subdivision in Problem 9-5. The actual finite corner-compatible subdivision is supplied by Finite curvilinear triangulation of a compact Riemannian surface; positive frame construction is Gram–Schmidt.

Depends on

Used by

Local Gauss-Bonnet for a frameable disk region Theorem
Local Gauss-Bonnet for an arbitrary disk region Theorem

Dependency tree · two levels

14 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, Riemannian Manifolds: An Introduction to Curvature (1997) (standard reference, not scraped)
Ved Datar, Lectures on Riemannian Geometry (standard reference, not scraped)