Differential Geometry, Lie Groups, and Symmetric Spaces, Volume 80 (Pure and Applied Mathematics) [Hardcover]

Sigurdur Helgason (Author)

Book Description

ISBN-10: 0123384605 | ISBN-13: 978-0123384607 | Publication Date: January 11, 1979

The present book is intended as a textbook and reference work on three topics in the title. Together with a volume in progress on "Groups and Geometric Analysis" it supersedes my "Differential Geometry and Symmetric Spaces," published in 1962. Since that time several branches of the subject, particularly the function theory on symmetric spaces, have developed substantially. I felt that an expanded treatment might now be useful.

Editorial Reviews

Review

A great book. A necessary item in any mathematical library. -- S.S. Chern, Scripta Mathematica, 1963

An instant classic. -- A. Koranyi, American Scientist, 1985

Impressive skill and force in developing the needed background in manifold theory and algebra, giving crisp and accessible proofs of the major results--required reading for anyone interested in Lie group theory, pure or applied. -- R. Hermann, SIAM Review

The exposition is brilliant - might be used as a text for 'how to write mathematics'. -- L. Auslander, Bulletin of the American Mathematical Society, 1964

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

Product Details

• Hardcover: 628 pages
• Publisher: Academic Press (January 11, 1979)
• Language: English
• ISBN-10: 0123384605
• ISBN-13: 978-0123384607
• Product Dimensions: 9.5 x 6.5 x 1.2 inches
• Shipping Weight: 2.3 pounds
• Average Customer Review: 3.6 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #3,513,874 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.