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The angle defect, which is the standard way to measure curvature at thevertices of polyhedral surfaces, goes back at least as far as Descartes.Although the angle defect has been widely studied, there does not appear to bein the literature an axiomatic characterization of the angle defect. We give acharacterization of the angle defect for simplicial surfaces, and we show thatvariants of the same characterization work for two known approaches togeneralizing the angle defect to arbitrary 2-dimensional simplicial complexes.Simultaneously, we give a characterization of the Euler characteristic on2-dimensional simplicial complexes in terms of being geometrically locallydetermined.

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