Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics) by Sigurdur Helgason (PDF)

Description

The study of homogeneous spaces provides 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.

User’s Reviews

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

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐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 Algebraand differential geometry,since this one is entirely unreadable,even by those who know and love the subjects.

⭐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

⭐Muito bom esse livro, recomendo.

⭐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.

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