 Metric Structures for Riemannian and NonRiemannian Spaces

Metric Structures for Riemannian and NonRiemannian Spaces Author(s) Misha Gromov Country United States Language English Genre(s) Mathematics Publisher Birkhäuser Boston, Inc. Publication date 1999 Media type Print Pages xx+585 pp ISBN 0817638989
Metric Structures for Riemannian and NonRiemannian Spaces is a book in geometry by Mikhail Gromov. It was originally published in French in 1981 under the title Structures métriques pour les variétés riemanniennes, by publisher CEDIC, Paris. The 1981 edition was edited by Jacques Lafontaine and Pierre Pansu. The English version, considerably expanded, was published in 1999 by Birkhäuser Verlag, with appendices by Pierre Pansu, Stephen Semmes, and Mikhail Katz. Since then, the book was reprinted several times.Contents
Reviews
Reviewer Igor Belegradek for MathSciNet wrote that the book
 is considered one of the most influential books in geometry in the last twenty years. Since then the boundary of the field has dramatically exploded. Reflecting this growth, the new English edition has almost quadrupled in size.^{[1]}
Reviewer Mircea Craioveanu for Zentralblatt wrote:
 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.^{[2]}
Contents
ContentsChapter 1
The first chapter deals with the notion of a length structure on a space and studies path metric spaces.
Chapter 2
The second chapter deals with the notions of degree and dilatation. After presenting elementary properties of dilatations for spheres, the chapter delves into the asymptotics of the number of homotopy classes of Lipschitz maps as a function of the Lipschitz constant.
Chapter 3
The third chapter deals with metric structures on families of metric spaces. It defines the notions of Lipschitz and Hausdorff convergence, studies the Hausdorff–Lipschitz metric, analyzes quasiisometries and the word metric.
Chapter 3 ^{1}/_{2}
Chapter is new as compared to the French edition of 1981. It deals with convergence and concentration of metrics and measures, in the context of mmspaces.
Chapter 4
The fourth chapter, entitled Loewner Rediscovered, outlines the history of systolic geometry since Charles Loewner. It outlines Loewner's torus inequality and Pu's inequality, poses some questions in dimensions 3 and greater, deals with norms in homology and on Jacobi varieties, and presents an application of geometric measure theory. A new section E_{+} deals with unstable systolic inequalities and filling.
Chapter 5
The fifth chapter deals with manifolds with bounded Ricci curvature. It deals with precompactness, growth of fundamental groups, obtains estimates on the first Betti number.
Chapter 6
The sixth chapter deals with isoperimetric inequalities and amenability.
Chapter 7
The seventh chapter deals with Morse theory and Minimal models.
Chapter 8_{+}
The new Chapter 8_{+} deals with curvature pinching and collapse phenomena.
Appendices
The English edition has 4 appendices, 2 of them new. Appendix A, by Pierre Pansu, deals with quasiconvex domains in Euclidean space. Appendix B, by Stephen Semmes, deals with metric spaces and mappings seen from within. Appendix C, by Gromov, deals with Paul Lévy's isoperimetic inequality. Appendix D, by Mikhail Katz, deals with systolically free manifolds.
Notes
1systoles of surfaces 1systoles of manifolds Higher systoles Categories: Riemannian geometry
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