Lectures on Nonsmooth Differential Geometry

  • Nicola Gigli
  • Enrico Pasqualetto

Part of the SISSA Springer Series book series (SISSASS, volume 2)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Nicola Gigli, Enrico Pasqualetto
    Pages 1-31
  3. Nicola Gigli, Enrico Pasqualetto
    Pages 33-65
  4. Nicola Gigli, Enrico Pasqualetto
    Pages 67-96
  5. Nicola Gigli, Enrico Pasqualetto
    Pages 97-122
  6. Nicola Gigli, Enrico Pasqualetto
    Pages 123-142
  7. Nicola Gigli, Enrico Pasqualetto
    Pages 143-179
  8. Back Matter
    Pages 181-204

About this book


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.


Geometric Ananlysis Nonsmooth Calculus Lower Ricci Curvature Bounds Differential Geometry Differential Calculus

Authors and affiliations

  • Nicola Gigli
    • 1
  • Enrico Pasqualetto
    • 2
  1. 1.Department of MathematicsInternational School for Advanced Studies (SISSA)TriesteItaly
  2. 2.Department of Mathematics and StatisticsUniversity of JyväskyläJyväskyläFinland

Bibliographic information

  • DOI
  • Copyright Information Springer Nature Switzerland AG 2020
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-38612-2
  • Online ISBN 978-3-030-38613-9
  • Series Print ISSN 2524-857X
  • Series Online ISSN 2524-8588
  • Buy this book on publisher's site