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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.
##### MSC:
 05E45 Combinatorial aspects of simplicial complexes 53B21 Methods of local Riemannian geometry
##### Keywords:
cell complex; Ricci curvature; Gauss-Bonnet theorem
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##### References:
 [1] R. F. Arnord, The Discrete Hodge Star Operator and Poincaré Duality, Virginia Polytechnic Institute and State University, 2012. [2] R. Forman, Bochner’s method for cell complexes and combinatorial Ricci curvature, Discrete Comput. Geom. 29 (2003), no.3, 323-374. · Zbl 1040.53040 [3] R. Forman, Combinatorial Novikov-Morse theory, Internat. J. Math. 13 (2002), no.4, 333-368. · Zbl 1052.57053 [4] R. Forman, Combinatorial vector fields and dynamical systems, Math. Z. 228 (1998), no.4, 629-681. · Zbl 0922.58063 [5] R. Forman, Morse theory for cell complexes, Adv. Math. 134 (1998), no.1, 90-145. · Zbl 0896.57023 [6] A.T. Lundell and S. Weingram, The Topology of CW Complexes, The university series in higher mathematics, Springer New York, 2012. · Zbl 0207.21704 [7] P. M\textsuperscriptcCormick, Combinatorial Curvature of Cellular Complexes, The University of Melbourne, Department of Mathematics and Statistics, 2004.
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