Customer Reviews

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

5 star:		(2)
4 star:		(1)
3 star:		(0)
2 star:		(2)
1 star:		(0)

 

 

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.

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.

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.

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.)

I certainly hate being cheated.
This book is advance as a textbook for a course in Lie Algebra.
I can picture the man who wrote this book lecturing to the future great minds of MIT
and putting them to sleep.
The fellow is the worst sort of pedant.
On page one he mentions one of the more difficult theorems in modern Mathematics,
De Rham's theorem, then drops it like it was too hot to handle.
On page three he introduces Hausdorff's difficult separation axiom
without any explanation at all.
Throughout the book he beats you over the head with terms like "module"
without adequate definition or explanation of terms.
He literally expects you to have learned
what he is supposed to be teaching
before you take his course?
In short , anyone taking the course with this book as a text book
will be hunting for a good text on Lie AlgebraSemi-Simple Lie Algebras and Their Representations (Dover Books on Mathematics) Lie Groups, Lie Algebras, and Some of Their Applications
and differential geometry,
since this one is entirely unreadable,
even by those who know and love the subjects.