Nonsmooth differential geometry - An approach tailored for spaces with Ricci curvature bounded from below

TL;DR

This paper explores the development of a differential calculus on metric measure spaces with Ricci curvature bounds, enabling the definition of second order geometric quantities like Hessians and Ricci curvature.

Contribution

It introduces a second order calculus framework for spaces with Ricci curvature bounded from below, extending first order differential structures to include Hessians and covariant derivatives.

Findings

Established a second order differential calculus on Ricci bounded spaces

Defined Hessian, covariant, and exterior derivatives in this setting

Provided tools for analyzing geometric properties of metric measure spaces

Abstract

We discuss in which sense general metric measure spaces possess a first order differential structure. Building on this, we then see that on spaces with Ricci curvature bounded from below a second order calculus can be developed, permitting to define Hessian, covariant/exterior derivatives and Ricci curvature.