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Lectures on Nonsmooth Differential Geometry

SISSA Springer Series 2

Format: Hardback
Publisher: Springer Nature Switzerland AG, Cham, Switzerland
Published: 11th Feb 2020
Dimensions: w 156mm h 234mm d 14mm
Weight: 485g
ISBN-10: 3030386120
ISBN-13: 9783030386122
Barcode No: 9783030386122
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Synopsis
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.

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"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)