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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.2000 Mathematics Subject Classification. Primary 52B70; Secondary 06A99. Key words and phrases. combinatorial Ricci curvature, poset. I would like to thank Sam Hsaio for his help with posets; and the Einstein Institute of Mathematics at the Hebrew University of Jerusalem, and especially Prof. Emanuel Farjoun, for their very kind hospitality during a sabbatical when parts of this paper were written.

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In a 1967 paper, Banchoff described a theory of critical points and curvature for polyhedra embedded in Euclidean space. For each convex cell complex K in R n , and for each linear map h : R n ! R satisfying a simple generality criterion, he defined an index for each vertex of K with respect to the map h, and showed that these indices satisfy two properties: (1) for each map h, the sum of the indices at all the vertices of K equals vðKÞ; and (2) for each vertex of K, the integral of the indices of the vertex with respect to all such linear maps equals the standard polyhedral notion of curvature of K at the vertex. In a previous paper, the author defined a different approach to curvature for arbitrary simplicial complexes, based upon a more direct generalization of the angle defect. In the present paper we present an analog of Banchoff 's theory that works with our generalized angle defect. (2000). Primary 52B99. Mathematics Subject Classification