Tohoku Mathematical Journal
2019
December
SECOND SERIES VOL. 71, NO. 4

Tohoku Math. J.
71 (2019), 533-547

Title COMBINATORIAL RICCI CURVATURE ON CELL-COMPLEX AND GAUSS-BONNNET THEOREM

Author Kazuyoshi Watanabe

(Received June 16, 2017, revised December 7, 2017)
Abstract. 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.

Mathematics Subject Classification. Primary 05E45; Secondary 53B21.
Key words and phrases. cell complex, Ricci curvature, Gauss-Bonnet theorem.

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