A Combinatorial Gauss-Bonnet Theorem
Robert W. Bell
Department of Mathematics and
Lyman Briggs School of Science
Michigan State University
The classical Gauss - Bonnet theorem for a closed surface S says thatintegral of the curvature over S depends only on the topological type ofS. For instance, although the unit sphere x2 + y2 + z2 = 1 and theellipsoid 3x2 + 5y2 + 7z2 = 1 are curved differently, if we integratetheir curvatures, we obtain the same value in both cases because thesphere and the ellipsoid are topologically the same surface.
We will prove a combinatorial generalization of the Gauss - Bonnet theoremfor two dimensional polyhedra. As a corollary, we will deduce theclassical theorem. No background is required for core of the talk;however, relating the combinatorial theorem to the classical one requiressome acquaintance with vector calculus.