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25 Remote Warehouse Mathematics- Differential Geometry

Metric Structures for Riemannian and Non-Riemannian Spaces (Modern Birkhuser Classics)

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Metric Structures for Riemannian and Non-Riemannian Spaces (Modern Birkhuser Classics) Cover

 

Synopses & Reviews

Publisher Comments:

Metric theory has undergone a dramatic phase transition in the last decades when its focus moved from the foundations of real analysis to Riemannian geometry and algebraic topology, to the theory of infinite groups and probability theory. The new wave began with seminal papers by Svarc and Milnor on the growth of groups and the spectacular proof of the rigidity of lattices by Mostow. This progress was followed by the creation of the asymptotic metric theory of infinite groups by Gromov. The structural metric approach to the Riemannian category, tracing back to Cheeger's thesis, pivots around the notion of the Gromov-Hausdorff distance between Riemannian manifolds. This distance organizes Riemannian manifolds of all possible topological types into a single connected moduli space, where convergence allows the collapse of dimension with unexpectedly rich geometry, as revealed in the work of Cheeger, Fukaya, Gromov and Perelman. Also, Gromov found metric structure within homotopy theory and thus introduced new invariants controlling combinatorial complexity of maps and spaces, such as the simplicial volume, which is responsible for degrees of maps between manifolds. During the same period, Banach spaces and probability theory underwent a geometric metamorphosis, stimulated by the Levy-Milman concentration phenomenon, encompassing the law of large numbers for metric spaces with measures and dimensions going to infinity. The first stages of the new developments were presented in Gromov's course in Paris, which turned into the famous "Green Book" by Lafontaine and Pansu (1979). The present English translation of that work 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.

Synopsis:

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.

Table of Contents

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.

Product Details

ISBN:
9780817645823
Author:
Gromov, Misha
Publisher:
Birkhauser
Editor:
LaFontaine, Jacques
Editor:
Pansu, Pierre
Author:
Katz, M.
Author:
Pansu, Pierre
Author:
Gromov, Mikhail
Author:
Bates, S. M.
Author:
Semmes, S.
Author:
Gromov, Mikhael
Author:
Pansu, P.
Author:
LaFontaine, Jacques
Subject:
Mathematical Analysis
Subject:
Geometry - Differential
Subject:
Riemannian manifolds
Subject:
Probability & Statistics - General
Subject:
Mathematics
Subject:
STRUCTURES
Subject:
diff.geometry
Subject:
diff.topology
Subject:
Manifolds
Subject:
Differential geometry
Subject:
Manifolds and Cell Complexes (incl. Diff.Topology)
Subject:
Algebraic topology
Subject:
Measure and Integration
Subject:
Analysis
Subject:
Mathematics-Differential Geometry
Copyright:
Edition Description:
1999. Corr. 2nd
Series:
Modern Birkhäuser Classics
Publication Date:
20061222
Binding:
TRADE PAPER
Language:
English
Illustrations:
Y
Pages:
606
Dimensions:
235 x 155 mm 1870 gr

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Related Subjects

Engineering » Engineering » General Engineering
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Science and Mathematics » Mathematics » Probability and Statistics » Statistics

Metric Structures for Riemannian and Non-Riemannian Spaces (Modern Birkhuser Classics) New Trade Paper
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Product details 606 pages Birkhauser - English 9780817645823 Reviews:
"Synopsis" by , 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.
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