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

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In this paper, we introduce a new definition of the Ricci curvature on cell-complexes and prove the Gauss-Bonnnet type theorem for graphs and 2-complexes that decompose closed surfaces. The differential forms on a cell complex are defined as linear maps on the chain complex, and the Laplacian operates this differential forms. Our Ricci curvature is defined by the combinatorial Bochner-Weitzenbock formula. We prove some propositionerties of combinatorial vector fields on a cell complex.