JournalsprimsVol. 54, No. 4pp. 855–918

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

  • Nicola Gigli

    SISSA, Trieste, Italy
Lecture Notes On Differential Calculus on $\sf {RCD}$ Spaces cover
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Abstract

These notes are intended to be an invitation to differential calculus on RCD\sf {RCD} spaces. We start by introducing the concept of an "L2L^2-normed LL^\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.

Cite this article

Nicola Gigli, Lecture Notes On Differential Calculus on RCD\sf {RCD} Spaces. Publ. Res. Inst. Math. Sci. 54 (2018), no. 4, pp. 855–918

DOI 10.4171/PRIMS/54-4-4