A visual introduction to Riemannian curvatures and some discrete generalizations

We try to provide a visual introduction to some objects used in Riemannian geometry: parallel transport, sectional curvature, Ricci curvature, Bianchi identities... We then explain some of the strategies used to define analogues of curvature in non-smooth or discrete spaces, beginning with Alexandrov curvature and δ-hyperbolic spaces, and insisting on various notions of generalized Ricci curvature, which we briefly compare. The first part of this text covers in a hopefully intuitive and visual way some of the usual objects of Riemannian geometry: parallel transport, sectional curvature, Ricci curvature, the Riemann tensor, Bianchi identities... For each of those we try to provide one or several pictures that convey the meaning (or at least one possible interpretation) of the formal definition. Next, we consider the problem of defining analogues of curvature for nonsmooth or discrete objets. This is in the spirit of “synthetic” or “coarse” geometry, in which large-scale properties of metric spaces are investigated instead of fine smallscale properties; one of the aims being to be impervious to small perturbations of the underlying space. Motivation for these non-smooth or discrete extensions often comes from various other fields of mathematics, such as group theory or optimal transport. From this viewpoint is emerging a theory of metric measure spaces, see for instance [Gro99]. Thus we cover the definitions and mention some applications of δ-hyperbolic spaces and Alexandrov curvature (generalizing sectional curvature), and of two notions of generalized Ricci curvature: displacement convexity of entropy introduced by Sturm and Lott–Villani (following Renesse–Sturm and others), and coarse Ricci curvature as used by the author. We mention new theorems obtained for the original Riemannian case thanks to this more general viewpoint. We also briefly discuss the differences between these two approaches to Ricci curvature. Note that scalar curvature is not covered, for lack of the author’s competence about recent developments and applications in this field. 1. Riemannian curvatures The purpose of this section is not to give a formal course in Riemannian geometry. We will therefore not reproduce formal definitions of the basic objects, but try to convey some pictorial intuition behind them. For a more formal introduction we 2010 Mathematics Subject Classification. Primary 53B20, 51K10.