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Bochner's Method for Cell Complexes and Combinatorial Ricci Curvature

Abstract. In this paper we present a new notion of curvature for cell complexes. For each p , we define a p th combinatorial curvature function, which assigns a number to each p -cell of the complex. The curvature of a p -cell depends only on the relationships between the cell and its neighbors. In the case that p=1 , the curvature function appears to play the role for cell complexes that Ricci curvature plays for Riemannian manifolds. We begin by deriving a combinatorial analogue of Bochner's theorems, which demonstrate that there are topological restrictions to a space having a cell decomposition with everywhere positive curvature. Much of the rest of this paper is devoted to comparing the properties of the combinatorial Ricci curvature with those of its Riemannian avatar.

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Forman, . Bochner's Method for Cell Complexes and Combinatorial Ricci Curvature . Discrete Comput Geom 29, 323–374 (2003). https://doi.org/10.1007/s00454-002-0743-x

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  • DOI: https://doi.org/10.1007/s00454-002-0743-x

Keywords

  • Riemannian Manifold
  • Cell Complex
  • Ricci Curvature
  • Curvature Function
  • Positive Curvature