Metric curvatures and their applications 2: metric Ricci curvature and flow

In this second part of our overview of the different metric curvatures and their various applications, we concentrate on the Ricci curvature and flow for polyhedral surfaces and higher dimensional manifolds, and we largely review our previous studies on the subject, based upon Wald’s curvature. In addition to our previous metric approaches to the discretization of Ricci curvature, we consider yet another one, based on the Haantjes curvature, interpreted as a geodesic curvature. We also try to understand the mathematical reasons behind the recent proliferation of discretizations of Ricci curvature. Furthermore, we propose another approach to the metrization of Ricci curvature, based on Forman’s discretization, and in particular we propose on that uses our graph version of Forman’s Ricci curvature.

Keywords

metric curvatures, Ricci curvature

2010 Mathematics Subject Classification

Primary 52C26, 53C44, 68U05. Secondary 51K10, 57R40, 65D18.

Full Text (PDF format)

Received 12 July 2019

Published 15 July 2022