The Geometric Meaning of Curvature: Local and Nonlocal Aspects of Ricci Curvature

Curvature is a concept originally developed in differential and Riemannian geometry. There are various established notions of curvature, in particular sectional and Ricci curvature. An important theme in Riemannian geometry has been to explore the geometric and topological consequences of bounds on those curvatures, like divergence or convergence of geodesics, convexity properties of distance functions, growth of the volume of distance balls, transportation distance between such balls, vanishing theorems for Betti numbers, bounds for the eigenvalues of the Laplace operator or control of harmonic functions. Several of these geometric properties turn out to be equivalent to the corresponding curvature bounds in the context of Riemannian geometry. Since those properties often are also meaningful in the more general framework of metric geometry, in recent years, there have been several research projects that turned those properties into axiomatic definitions of curvature bounds in metric geometry. In this contribution, after developing the Riemannian geometric background, we explore some of these axiomatic approaches. In particular, we shall describe the insights in graph theory and network analysis following from the corresponding axiomatic curvature definitions.