Combinatorial Ricci curvature on cell-complex and Gauss-Bonnnet Theorem

Abstract

In this paper we present the Ricci curvature on cell-complexes and show the Gauss-Bonnnet type theorem on graphs and 2-complex that decomposes closed surface. The defferential forms on a cell complex is defined as linear maps on chain complex, and Laplacian operates this defferential forms. Then we construct the Bochner-Weitzenb\"ock formula and define the Ricci curvature. This curvature is determined by the combinatorial calculation. We show also the properties of combinatorial vector fields on a cell complex