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Combinatorial Ricci curvature on cell-complex and Gauss-Bonnnet theorem. (English) Zbl 1439.05236
Summary: 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-Weitzenböck formula. We prove some propositionerties of combinatorial vector fields on a cell complex.
05E45 Combinatorial aspects of simplicial complexes
53B21 Methods of local Riemannian geometry
Full Text: DOI Euclid
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