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

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

2007-08-18

arxiv.org/abs/0708.2506v1

Mathematics

Geometric Topology

21 pages, 1 figure

Scientific paper

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.

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