Lectures on Nonsmooth Differential Geometry

This book provides an introduction to some aspects of the flourishing field of nonsmooth geometric analysis. In particular, a quite detailed account of the first-order structure of general metric measure spaces is presented, and the reader is introduced to the second-order calculus on spaces – known as RCD spaces – satisfying a synthetic lower Ricci curvature bound. Examples of the main topics covered include notions of Sobolev space on abstract metric measure spaces normed modules, which constitute a convenient technical tool for the introduction of a robust differential structure in the nonsmooth setting first-order differential operators and the corresponding functional spaces the theory of heat flow and its regularizing properties, within the general framework of “infinitesimally Hilbertian” metric measure spaces the RCD condition and its effects on the behavior of heat flow and second-order calculus on RCD spaces. The book is mainly intended for young researchers seeking a comprehensive and fairly self-contained introduction to this active research field. The only prerequisites are a basic knowledge of functional analysis, measure theory, and Riemannian geometry.

1. Preliminaries.- 2. Sobolev calculus on metric measure spaces.- 3. The theory of normed modules.- 4. First-order calculus on metric measure spaces.- 5. Heat flow on metric measure spaces.- 6. Second-order calculus on RCD spaces.- 7. Appendix A: Functional analytic tools.- 8. Appendix B: Solutions to the exercises.