Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics) [Hardcover]

Sigurdur Helgason (Author)

Book Description

ISBN-10: 0821828487 | ISBN-13: 978-0821828489 | Publication Date: June 12, 2001

The study of homogeneous spaces provides excellent insights into both differential geometry and Lie groups. In geometry, for instance, general theorems and properties will also hold for homogeneous spaces, and will usually be easier to understand and to prove in this setting. For Lie groups, a significant amount of analysis either begins with or reduces to analysis on homogeneous spaces, frequently on symmetric spaces. For many years and for many mathematicians, Sigurdur Helgason's classic Differential Geometry, Lie Groups, and Symmetric Spaces has been--and continues to be--the standard source for this material. <P>Helgason begins with a concise, self-contained introduction to differential geometry. He then introduces Lie groups and Lie algebras, including important results on their structure. This sets the stage for the introduction and study of symmetric spaces, which form the central part of the book. The text concludes with the classification of symmetric spaces by means of the Killing-Cartan classification of simple Lie algebras over $\mathbf{C}$ and Cartan's classification of simple Lie algebras over $\mathbf{R}$. <P>The excellent exposition is supplemented by extensive collections of useful exercises at the end of each chapter. All the problems have either solutions or substantial hints, found at the back of the book. <P>For this latest edition, Helgason has made corrections and added helpful notes and useful references. The sequels to the present book are published in the AMS's Mathematical Surveys and Monographs Series: Groups and Geometric Analysis, Volume 83, and Geometric Analysis on Symmetric Spaces, Volume 39. <P>Sigurdur Helgason was awarded the Steele Prize for Differential Geometry, Lie Groups, and Symmetric Spaces and Groups and Geometric Analysis.

Editorial Reviews

Review

"This book has been famous for many years and used by several generations of readers. It is important that the book has again become available for a general audience." -- European Mathematical Society Newsletter

"One of the most important and excellent textbooks and a reference work about contemporary differential geometry ..." -- Zentralblatt MATH

"Important improvements in the new edition of S. Helgason's book will turn it into a desk book for many following generations." -- Mathematica Bohemica

"A great book ... a necessary item in any mathematical library." -- S. S. Chern, University of California

"Written with unmatched lucidity, systematically, carefully, beautifully." -- S. Bochner, Princeton University

"Helgason's monograph is a beautifully done piece of work and should be extremely useful for several years to come, both in teaching and in research." -- D. Spencer, Princeton University

"A brilliant book: rigorous, tightly organized, and covering a vast amount of good mathematics." -- Barrett O'Neill, University of California

"Renders a great service in permitting the non-specialist, with a minimum knowledge of differential geometry and Lie groups, an initiation to the theory of symmetrical spaces." -- H. Cartan, Secretariat Mathématique, Paris

"The mathematical community has long been in need of a book on symmetric spaces. S. Helgason has admirably satisfied this need with his book, Differential Geometry and Symmetric Spaces. It is a remarkably well-written book ... a masterpiece of concise, lucid mathematical exposition ... it might be used as a textbook for "how to write mathematics"." -- Louis Auslander

"[The author] will earn the gratitude of many generations of mathematicians for this skillful, tasteful, and highly efficient presentation. It will surely become a classic." -- G. D. Mostow, Yale University

From the Publisher

The author was awarded "The Steele Prize" of the American Mathematical Society 1988 for this book, together with an accompanying volume, "Groups and Geometric Analysis" Academic Press, 1984 --This text refers to an alternate Hardcover edition.

Product Details

• Hardcover: 641 pages
• Publisher: American Mathematical Society (June 12, 2001)
• Language: English
• ISBN-10: 0821828487
• ISBN-13: 978-0821828489
• Product Dimensions: 10.3 x 7.3 x 1.5 inches
• Shipping Weight: 2.9 pounds (View shipping rates and policies)
• Average Customer Review: 3.6 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #771,049 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

17 of 19 people found the following review helpful:

4.0 out of 5 stars Unsurpassed, but demanding, May 27, 2007

By 

Jim Curry (MT) - See all my reviews
(VINE VOICE)   

This review is from: Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics) (Hardcover)

As I reviewed this book at Amazon, I found only one review, which I considered to be too harsh. You should understand that Helgason is writing a graduate textbook. Students will learn about "modules" in their graduate algebra course. They will learn De Rham's theorem in an introductory analysis course or sometimes even in a topology course (yes, it can happen). So, most of the language for which another reviewer criticized him would usually be covered in other graduate courses.

Helgason writes tersely but extremely precisely. I know of no other author who gives similar sophistication of point of view and quick, to the point, proofs. He is a "best of breed," and I suppose that is part of the reason he has been a core member of the faculty at M.I.T. for such a long time. A serious student cannot really avoid reading the entire progression of these texts, particularly the "Groups and Geometric Analysis" title, perhaps second in the Helgason manuscripts.

1 of 1 people found the following review helpful:

5.0 out of 5 stars Helgason Differential Geometry, January 24, 2012

By 

Lucius Schoenbaum - See all my reviews

Amazon Verified Purchase

This review is from: Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics) (Hardcover)

I'm not qualified to say much about this book, but I think it's excellent and thought it deserved a higher amazon rating. Besides being remarkably clear (much like the cold air of Helgason's home country of Iceland), I think it's a great, wonderful bridge between the original works in Lie theory and the more basic textbook treatments of DG out there (Warner, do Carmo, Lee, ...). It is filled with references and citations to original papers and is perhaps more connected to the historical genesis of the subject than other textbooks.

9 of 13 people found the following review helpful:

5.0 out of 5 stars Superb Treatise and Indispensible Reference, June 25, 2007

By 

Jason D. Brown - See all my reviews
(REAL NAME)   

This review is from: Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics) (Hardcover)

The mere thought or mention of the name Helgason inspires respect and awe. This book gets five stars all the way on its merit alone, regardless of who wrote it. Difficult as it is, the book starts from the fundamentals and works up in a coherent logical manner, there are no gaps in his presentation. The negative review below is completely unjustified. If anyone would like to at least see some of what this book is like go to ocw.mit.edu and download Helgason's notes which use excerpts in this book. Some of the topics in this book are covered in a more easy going way in "Lie Groups, Lie Algebras, and Some of Their Applications" by Robert Gilmore. (If I'm not mistaken Gilmore was a student of Helgason.) This book is mathematical exposition at it's absolute finest and I don't think but 1 in 1,000 people reading this page need me to tell them that much less need a review to persuade them. This book has quite a reputation.