Lecture Notes On Differential Calculus on $\mathsf{RCD}$ Spaces

Abstract

These notes are intended to be an invitation to differential calculus on $\mathsf{RCD}$ spaces. We start by introducing the concept of an "$L^2$-normed $L^{\infty }$-module" and show how it can be used to develop a first-order (Sobolev) differential calculus on general metric measure spaces. In the second part of the manuscript we see how, on spaces with Ricci curvature bounded from below, a second-order calculus can also be built: objects like the Hessian, covariant and exterior derivatives and Ricci curvature are all well defined and have many of the properties they have in the smooth category.