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

2017/03/24 by Kazuyoshi Watanabe, Watanabe, Kazuyoshi · 1 citation
Mathematics · #05E45 #Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics #math.CO #math.DG #msc:05E45

paper · pdf · doi:10.48550/arxiv.1703.08409

arxiv created 2017/07/11 · arxiv updated 2017/07/12

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-Weitzenböck 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

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