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Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics): 34 Hardcover – 12 Jun 2001

by Sigurdur Helgason (Author)

6 ratings

ISBN-13: 978-0821828489 ISBN-10: 0821828487 Edition: UK ed.

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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.

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Product details

Hardcover: 641 pages
Publisher: American Mathematical Society; UK ed. edition (12 June 2001)
Language: English
ISBN-10: 0821828487
ISBN-13: 978-0821828489
Product Dimensions: 19 x 3.8 x 26.7 cm
Boxed-product Weight: 1.3 Kg
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"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

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Amazon.com: 6 reviews

Lucius Schoenbaum

Helgason Differential Geometry

24 January 2012 - Published on Amazon.com

Verified Purchase

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 and Lie theory out there (Warner, do Carmo, Lee, ...), many of which are quite good. It is filled with references and citations to original papers (some by the author) and is perhaps more connected to the historical genesis of the subject than other textbooks.

"A great book... a necessary item in any mathematical library." -S.S. Chern

4 people found this helpful

Acacio B. Neves

Four Stars

30 September 2014 - Published on Amazon.com

Verified Purchase

Muito bom esse livro, recomendo.

One person found this helpful

Physics Student

Very confusing to someone new to the material

1 December 2011 - Published on Amazon.com

Verified Purchase

Previous reviewers have praised this book for its precision and logical coherence, and these are accurate assessments, but not the whole story. When using this book for a course in Lie Groups, taught by Professor Helgason himself, I found this book severely lacking. Take for example Chapter I, which covers some basic differential geometry. The definition of a tangent vector is the standard algebraic definition (as derivations of functions on the manifold). This in itself is fine, but figuring out why such a strange looking object actually corresponds to the intuitive notion of a tangent vector is not explained. This caused me a great deal of confusion. Another example is that the section on affine connections is literally two pages long and, unsurprisingly given its brevity, devoid of insight. For comparison, in a differential geometry class I took, we spent a week or so on affine connections. Another telling example is that most of the exercises have solutions in the back, but even after reading the "solution," it often took me more than a few hours to solve a problem.

As one reviewer said, this is a graduate text, so a certain amount of mathematical maturity and background is expected. My complaint is that if you have the maturity and background to reasonably understand the text, then you probably didn't need to read the text in the first place. To someone who already knows differential geometry and wants to get another perspective, or needs to jog his memory, I am sure Helgason's treatment is fine, though.

Overall, I found the book very confusing, since it is very terse, does not give examples or even explain the intuition or context behind a slew of definitions and theorems, and assumes what I think is an unreasonable amount of background and mathematical maturity. Also, I found many of the proofs hard to follow. To those not already comfortable with the material, I suggest turning elsewhere. In particular, I have found Warner Foundations of Differentiable Manifolds and Lie Groups very good for understanding much of the material in Helgason on Lie Groups and manifolds.

(As a disclaimer, I have only read chapters I and II since that is what we covered, but I suspect the style does not differ significantly between other parts of the book.)

6 people found this helpful

Jim Curry

Unsurpassed, but demanding

28 May 2007 - Published on Amazon.com

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.

24 people found this helpful

Troy learns to design

An engineer's review-Not for starters

25 August 2012 - Published on Amazon.com

As an engineer who is currently applying the theory of symmetric spaces, I realize that there is no other book that can teach you the same stuff (O. Loos's book is far too algebraic and what I really need is (Pseudo-)Riemannian symmetric spaces, not those affine stuffs...).

I find the book difficult to follow from time to time, but I guess its because I barely finished Boothby's An intro to diff mfd and riemannian geometry. Yet as I hang on a little longer, I started to learn on multiple levels and even felt more confident about differential geometry as a result (before, I tried Nomizu and Kobayashi, but it didnt work for me). If you are also an engineer, I would highly recommend you to finish Boothby first (which would then require 1 or 2 companion books, such as Munkres and maybe Abraham, Marsden, Ratiu) and come back to read Ch 1&2. Ch3 on semi-simple lie algebra is a little demanding, so Humphrey might be a good first reading. The rest of the book is on symmetric spaces. I would recommend not to read them all but to read only what you need.

The only thing dissatisfying is that there are too few examples. But I guess its not a textbook anyway.

9 people found this helpful