Metric Structures for Riemannian and Non-Riemannian Spaces (Progress in Mathematics) [Hardcover]

Mikhail Gromov (Author), Jacques LaFontaine (Editor), Pierre Pansu (Editor), S. M. Bates (Translator), M. Katz (Contributor), P. Pansu (Contributor), S. Semmes (Contributor)

Book Description

ISBN-10: 0817638989 | ISBN-13: 978-0817638986 | Publication Date: December 22, 2006 | Edition: Corrected

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.

Editorial Reviews

Review

"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

Language Notes

Text: English (translation)
Original Language: French

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Product Details

• Hardcover: 586 pages
• Publisher: Birkhäuser Boston; Corrected edition (December 22, 2006)
• Language: English
• ISBN-10: 0817638989
• ISBN-13: 978-0817638986
• Product Dimensions: 9.5 x 6.4 x 1.3 inches
• Shipping Weight: 3.6 pounds (View shipping rates and policies)
• Average Customer Review: 4.7 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #2,091,250 in Books (See Top 100 in Books)

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23 of 25 people found the following review helpful:

5.0 out of 5 stars The nature of high dimensions: a geometric insight, February 26, 2000

By 

Vladimir Pestov (Ottawa, Ontario, Canada) - See all my reviews
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This review is from: Metric Structures for Riemannian and Non-Riemannian Spaces (Progress in Mathematics) (Hardcover)

Formally speaking, this is the second edition of a set of Paris lecture notes published by Gromov two decades ago in the French language. However, such a wealth of entirely new material has been added that in essence we are talking of a new book.

Among the additions, the bulky new chapter 3 1/2+ stands out, dealing with the phenomenon of concentration of measure on high-dimensional structures. This is a relatively recent discovery of modern analysis and geometry, tracing its origin to the work of Paul Levy and especially Vitali Milman. The essence of the phenomenon is that on many multidimensional structures, every `nice' function is constant with high probability. The manifestations of the phenomenon are many - from geometric functional analysis (Dvoretzky theorem) through information theory (blowing-up lemma) and probability (law of large numbers) to graph theory (superconcentrators) and topological dynamics. As Gromov stresses in his book, even deeper aspects of the concentration phenomenon have been long since discovered and are constantly explored in statistical physics in the context of phase transitions of various kind, and some of the first known examples where phase transitions appear in the context of geometry have been discovered by Gromov himself, e.g. for hyperbolic groups. Finding and exploring more instances of phase transitions in mathematics might well become a unifying heuristic principle across a large number of disciplines.

The mathematical setting for dealing with concentration and related issues is the concept of a metric space equipped with finite measure, what Gromov calls an mm-space. Apart from concrete objects (such as for instance spheres and cubes), there are `higher-level' examples of mm-spaces, for instance those whose elements are isomorphism classes of mathematical objects themselves (e.g. Riemanning manifolds or finitely generated groups). This leads to a probabilistic treatment of such objects. Of course Gromov's strength is that his treatment is always concrete and he never theorizes without having particular objects and applications in mind.

It is quite safe to claim that the full range and power of applications of the interaction between metric and measure are yet to be discovered, which is what makes this book so important. It is rich in open questions and suggested new research directions, but more than that, it helps the reader to develop a good intuitive feeling of where things are going these days, what things ought to be done, and what constitutes proper mathematics.

Even though I unexpectedly found myself among the privileged ones who received a copy of the book as a gift from the author, I would have certainly purchased it otherwise, as I firmly believe that every mathematical library in the world, be it that of a top-class University or just a modest, lovingly selected office collection of a humble mathematician, will be wanting without a copy of the monograph under review, which might well become one of the most important books in mathematical sciences for the early XXIst century.

5 of 7 people found the following review helpful:

5.0 out of 5 stars profoundly important mathematics, October 26, 2006

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Christina Sormani (NYC) - See all my reviews
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This review is from: Metric Structures for Riemannian and Non-Riemannian Spaces (Progress in Mathematics) (Hardcover)

This book (originally published in French and improved here) is a fundamentally important book opening up an entire field of mathematics. For a textbook based on this material and related topics try Burago-Burago-Ivanov's textbook on the subject which can be taught to first year graduate students. For mathematicians and advanced graduate students, Gromov's book is a masterpiece.

0 of 1 people found the following review helpful:

4.0 out of 5 stars Great, but..., April 24, 2010

By 

Narada "the wondering jew" (Princeton, NJ, USA) - See all my reviews
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Amazon Verified Purchase

This review is from: Metric Structures for Riemannian and Non-Riemannian Spaces (Progress in Mathematics) (Hardcover)

The french version (written by Pierre Pansu, based on Gromov's lectures) is a jewel which gives a perfect introduction to the subject (not too much detail, a lot of insight, etc). The new parts (chapter 3.5, Semmes' supplement) really unbalance the book, though if you think of them as Volume 2, the changes might be palatable...