Relation between Combinatorial Ricci Curvature and Lin-Lu-Yau's Ricci Curvature on Cell Complexes

Abstract

In this paper we compare the combinatorial Ricci curvature on cell complexes and Lin-Lu-Yau's Ricci curvature defined on graphs. On a cell complex, the combinatorial Ricci curvature is introduced by the Bochner-Weitzenb\"{o}ck formula. A cell complex corresponds to a graph such that the vertices are cells and the edges are vectors on the cell complex. We compare these two kinds of Ricci curvatures by the coupling and the Kantorovich duality.