Albion College

Mathematics and Computer Science

Mathematics and Computer Science

COLLOQUIUM

A Combinatorial Gauss-Bonnet Theorem

Robert W. Bell

Assistant Professor

Department of Mathematics and

Lyman Briggs School of Science

Michigan State University

Assistant Professor

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 x^{2} + y^{2} + z^{2} = 1 and theellipsoid 3x^{2} + 5y^{2} + 7z^{2} = 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.

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.