CorollaryStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 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.

Angle-sum comparison for geodesic triangles

Statement

Assume the axiom of choice. Let $(M, g, J)$ be an oriented Riemannian surface and let $T \subseteq M$ be a positively oriented geodesic triangular disk of the kind considered in Gauss-Bonnet for a geodesic triangle, with interior angles $α, β, γ \in (0, 2 π)$ . Then

$α + β + γ > π, α + β + γ = π, α + β + γ < π$

according as the total Gaussian curvature $\int_{T} K d A$ is positive, zero, or negative. The comparison concerns the curvature integral and not merely the sign of $K$ at a single point.

Facts & Assumptions

Given: Full AC through the local disk Gauss–Bonnet supplier (The Axiom of Choice); A positively oriented geodesic triangular disk in a frameable neighbourhood with interior angles.

[F1]

For such a triangle $\int_{T} K d A = α + β + γ - π$ (Gauss-Bonnet for a geodesic triangle).

Proof

technique · rearrange the exact triangle identity and compare signs

1.1F1algebra

Subtracting $π$ from both sides of [F1] gives $α + β + γ - π = \int_{T} K d A$ .

2.1step 1.1algebra∎

By step 1.1, the real number $α + β + γ - π$ is positive, zero or negative exactly when the real number $\int_{T} K d A$ is positive, zero or negative; adding $π$ to each comparison gives the three stated alternatives.

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 9, printed pp. 165-172, notes the sign interpretation of the local formula: the curvature integral measures the excess of the angle sum over $π$ . Datar, Lectures on Riemannian Geometry, Lecture 2, printed pp. 10-13, records the same consequence. The rearrangement is immediate from the library theorem Gauss-Bonnet for a geodesic triangle.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)