Publications of the Research Institute for Mathematical Sciences

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Volume 54, Issue 4, 2018, pp. 855–918
DOI: 10.4171/PRIMS/54-4-4

Published online: 2018-10-18

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

Nicola Gigli[1]

(1) SISSA, Trieste, Italy

These notes are intended to be an invitation to differential calculus on $\sf {RCD}$ 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.

Keywords: Metric measure spaces, $\sf {RCD}$ spaces, differential calculus, Ricci curvature

Gigli Nicola: Lecture Notes On Differential Calculus on $\sf {RCD}$ Spaces. Publ. Res. Inst. Math. Sci. 54 (2018), 855-918. doi: 10.4171/PRIMS/54-4-4