Lectures on Nonsmooth Differential Geometry

Nicola Gigli author Enrico Pasqualetto author

Format:Paperback

Publisher:Springer Nature Switzerland AG

Published:11th Feb '21

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Lectures on Nonsmooth Differential Geometry cover

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 acomprehensive 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.

“This is an excellent starting point for entering this highly active field of research, written by two of its leading researchers.” (M. Kunzinger, Monatshefte für Mathematik, Vol. 200 (2), 2023)

“The lecture notes in this book are a gentle introduction to the topic, intended for readers with no prior exposure to non-smooth analysis, and hint towards the most recent developments of the theory. … This book is written in a clear and accessible way. It is definitely a recommended introduction to a quickly developing subject.” (Daniele Semola, Mathematical Reviews, February, 2023)

“The book is written in a clear and precise style. The notions are well motivated and many examples are given. In the reviewer’s opinion, this monograph will be of great interest to Ph.D. students and researchers working in the field of nonsmooth differential geometry.” (Gabriel Eduard Vilcu, zbMATH 1452.53002, 2021)

ISBN: 9783030386153

Dimensions: unknown

Weight: unknown

204 pages

1st ed. 2020