# Metric Curvatures and their Applications 2: Metric Ricci Curvature and Flow

## TL;DR

This paper reviews various metric approaches to Ricci curvature and flow on polyhedral surfaces and manifolds, introducing new discretizations based on Haantjes and Forman's curvatures, and analyzing their mathematical foundations.

## Contribution

It introduces novel discretizations of Ricci curvature using Haantjes and Forman's approaches, expanding the mathematical understanding of curvature discretization methods.

## Key findings

- Comparison of different discretization methods for Ricci curvature
- Introduction of Haantjes curvature as a geodesic curvature measure
- Proposal of a graph-based Forman Ricci curvature approach

## Abstract

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.