C1 Book
Lectures on Nonsmooth Differential Geometry (2020)

Gigli, N., & Pasqualetto, E. (2020). Lectures on Nonsmooth Differential Geometry. Springer. SISSA Springer Series, 2. https://doi.org/10.1007/978-3-030-38613-9

JYU authors or editors

Publication details

All authors or editors: Gigli, Nicola; Pasqualetto, Enrico

ISBN: 978-3-030-38612-2

eISBN: 978-3-030-38613-9

Journal or series: SISSA Springer Series

ISSN: 2524-857X

eISSN: 2524-8588

Publication year: 2020

Number in series: 2

Number of pages in the book: 204

Publisher: Springer

Place of Publication: Cham

Publication country: Switzerland

Publication language: English

DOI: https://doi.org/10.1007/978-3-030-38613-9

Publication open access: Not open

Publication channel open access: 

Abstract

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.

Keywords: differential geometry; differential calculus; metric spaces; measure theory; functional analysis

Free keywords: geometric analysis; nonsmooth calculus; lower Ricci curvature bounds; differential geometry; differential calculus

Contributing organizations

Ministry reporting: Yes

Reporting Year: 2020

JUFO rating: 2