A Characterization of the Angle Defect and the Euler Characteristic in Dimension 2 -- Preliminary Draft

TL;DR

This paper provides an axiomatic characterization of the angle defect and Euler characteristic for 2D simplicial complexes, clarifying their geometric and topological properties and generalizations.

Contribution

It introduces a novel axiomatic framework for the angle defect and Euler characteristic in 2D simplicial complexes, unifying existing approaches.

Findings

Characterization of the angle defect for simplicial surfaces

Extension of the characterization to generalized 2D complexes

A geometric local determination of the Euler characteristic

Abstract

The angle defect, which is the standard way to measure curvature at the vertices of polyhedral surfaces, goes back at least as far as Descartes. Although the angle defect has been widely studied, there does not appear to be in the literature an axiomatic characterization of the angle defect. We give a characterization of the angle defect for simplicial surfaces, and we show that variants of the same characterization work for two known approaches to generalizing the angle defect to arbitrary 2-dimensional simplicial complexes. Simultaneously, we give a characterization of the Euler characteristic on 2-dimensional simplicial complexes in terms of being geometrically locally determined.