Combinatorial Ricci Curvature for Polyhedral Surfaces and Posets

2014/06/18 by Ethan D. Bloch, Ethan Bloch, Bloch, Ethan · 2 citations
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Primary 52B70 #Secondary 06A99 #Topological and Geometric Data Analysis #math.CO #msc:06A99 #msc:52B70

paper · pdf · doi:10.48550/arxiv.1406.4598

16 pages, 4 figures

arxiv created 2014/06/18 · openalex publication_date 2014/06/18 · arxiv updated 2014/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The combinatorial Ricci curvature of Forman, which is defined at the edges of a CW complex, and which makes use of only the face relations of the cells in the complex, does not satisfy an analog of the Gauss-Bonnet Theorem, and does not behave analogously to smooth surfaces with respect to negative curvature. We extend this curvature to vertices and faces in such a way that the problems with combinatorial Ricci curvature are mostly resolved. The discussion is stated in terms of ranked posets.

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