Metric Structures for Riemannian and Non-Riemannian Spaces

This book is an English translation of the famous "Green Book" by Lafontaine and Pansu (1979). It has been enriched and expanded with new material to reflect recent progress.

This book is an English translation of the famous "Green Book" by Lafontaine and Pansu (1979). It has been enriched and expanded with new material to reflect recent progress. Additionally, four appendices, by Gromov on Levy's inequality, by Pansu on "quasiconvex" domains, by Katz on systoles of Riemannian manifolds, and by Semmes overviewing analysis on metric spaces with measures, as well as an extensive bibliography and index round out this unique and beautiful book.

From the reviews:

"The book gives genius insight into the connections between topology and Riemannian geometry, geometry and probability, geometry and analysis, respectively. The huge variety of progressive key ideas could provide numerous research problems in the next decades." Publicationes Mathematicae

"This book will become one of the standard references in the research literature on the subject. Many fascinating open problems are pointed out. Since this domain has dramatically exploded since 1979 and also is one which has many contact points with diverse areas of mathematics, it is no small task to present a treatment which is at once broad and coherent. It is a major accomplishment of Misha Gromov to have written this exposition. It is hard work to go through the book, but it is worth the effort." Zentralblatt Math

"The first edition of this book...is considered one of the most influential books in geometry in the last twenty years... Among the most substantial additions [of the 2/e]...is a chapter on convergence of metric spaces with measures, and an appendix on analysis on metric spaces... In addition, numerous remarks, examples, proofs, and open problems are inserted throughout the book. The original text is preserved with new items conveniently indicated... This book is certain to be a source of inspiration for many researchers as well as required reading for students entering the subject." Mathematical Reviews

This is a reprint of the 2001 edition of Gromov's by now classical book on metric structures. this work will continue to set the standard in the field for the foreseeable future. (M. Kunzinger, Monatshefte für Mathematik, Vol. 156 (4), April, 2009)

Contenu
Preface to the French Edition.- Preface to the English Edition.- Introduction: Metrics Everywhere.- Length Structures: Path Metric Spaces.- Degree and Dilatation.- Metric Structures on Families of Metric Spaces.- Convergence and Concentration of Metrics and Measures.- Loewner Rediscovered.- Manifolds with Bounded Ricci Curvature.- Isoperimetric Inequalities and Amenability.- Morse Theory and Minimal Models.- Pinching and Collapse.- Appendix A: 'Quasiconvex' Domains in Rn.- Appendix B: Metric Spaces and Mappings Seen at Many Scales.- Appendix C: Paul Levy's Isoperimetric Inequality.- Appendix D: Systolically Free Manifolds.- Bibliography.- Glossary of Notation.- Index.