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

The angle defect, which is the standard way to measure the curvatures 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. In this paper a characterization of the angle defect for simplicial surfaces is given, and it is shown that variants of the same characterization work for two known approaches to generalizing the angle defect to arbitrary 2-dimensional simplicial complexes. Simultaneously, a characterization of the Euler characteristic for 2-dimensional simplicial complexes is given in terms of being geometrically locally determined.