Discrete curvatures motivated from Riemannian geometry and optimal transport: Bonnet-Myers-type diameter bounds and rigidity

This thesis gives an overview of three notions of Ricci curvature for discrete spaces, including Ollivier Ricci curvature (motivated from optimal transport), Bakry-mery curvature (from Bochner’s formula in Riemannian geometry) and Erbar-Maas entropic Ricci curvature (from optimal transport). The first part of the thesis provides background knowledge in optimal transport theory and Riemannian geometry which is essential to the understanding of generalized Ricci curvatures for metric measure spaces and the mentioned Ricci curvatures for graphs. For each of the three discrete curvature notions, discussed in their respective part of the thesis, we provide the definition of the curvature and use hypercubes as an example for the curvature calculation. We study various curvature results with an emphasis on upper bounds of diameter and lower bounds of the spectral gap for graphs with positive lower bound on the Ricci curvature. These results can be regarded as discrete analogues of the Bonnet-Myers theorem and the Lichnerowicz theorem in Riemannian geometry. In addition, we deeply investigate into the rigidity results (analogous to Cheng’s rigidity) in attempt to classify all graphs which yield the sharp diameter bound in the sense of Ollivier Ricci curvature and Bakry-mery curvature.