Customer Reviews

Geometry, Topology and Physics (Graduate Student Series in Physics)

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2 star:		(1)
1 star:		(1)

 

 

I suppose I should preface this by saying that I read this book *after* reading similar books, so my ability to understand this book is probably better than others, but that said, I think that my comparative evaluation is free from this bias...

There seem to be a few books on the market that are very similar to this one: Nash & Sen, Frankel, etc. This one is at the top of its class, in my opinion, for a couple reasons:

(1) It's written like a math text that covers physics-related material, not a book about mathematics for physicists. I prefer this; you may not. As a consequence, this book is more rigorous than its alternatives, it relies less on physical examples, and it cuts out a lot of lengthy explanation that you may not need. Of course, there are drawbacks to all of these "features" -- you need to decide what you need and what's best for you.

(2) It's most comprehensive, with Frankel coming in second, and Nash & Sen least comprehensive (though they have quite a bit on Fibre bundles and related topics). Nakahara has a chapter on complex manifolds, which is absent from the other two. Nakahara also concludes with a nice intro to string theory, which is absent from the other two as well (though nothing you couldn't find in Polchinski or the like). Actually -- I modify this slightly. Frankel covers less subjects than Nakahara, but with more depth (though also more wordiness -- I quit Frankel about 2/3 through because it wasn't succinct enough and I got tired of it).

Depending on your tastes, I would recommend this book before the other two.

It presupposes that you have an understanding of algebra (groups, rings, fields, etc.) but it has an introduction to the necessary components of topology within. Frankel has presupposes both algebra and topology; Nash & Sen presupposes only algebra.

Nakahara is one of my favorite books. It gives the reader the necessary knowledge in differential geometry and topology to understand theoretical physics from a modern viewpoint. Each chapter in Nakahara would normally take a full semester mathematics course to teach, but the necesseties for a physicist are distilled with just the right amount of rigor so that the reader is neither bored from excessive proof nor skeptical from simple plausibility arguments.

The first few chapters (homotopy, homology) are rather dry, but the text picks up after that. The manifold chapter is really good, particularly the Lie groups section which gives a geometric viewpoint of the objects which get very little attention in a typical particle physics course. Unfortunately, nothing is said on representation theory, but that can be found in Georgi's book. The cohomology chapter is wonderfully quick and to the point. I found myself having to tell myself to slow down because of the excitement I had in reading it. The Riemannian geometry chapter reads wonderfully and serves as a great reference for all those general relativity formulae you always forget. The end of that chapter has an exquisite little bit on spinors in curved spacetime. The complex geometry chapter is also wonderful. I find myself going back to it all the time when reading Polchinski's string text. The chapters on fiber bundles seem a bit on the overly mathy side, but then again, all the pain is in the definitions which becomes well worth it in the end. I haven't read the last few chapters (spending all of my time in Polchinski!) but I definitely will when I have some spare time. The notation in Nakahara is also really self explanatory and standard. It is written with the physicist in mind who doesn't mind a bit of sloppiness or ambiguity in his notation.

With regards to Frankel, Nakahara is much more modular than Frankel. Each chapter of Nakahara is pretty much self contained whereas Frankel kinda needs to be read straight through. I find it very difficult to just look up a random thing in Frankel and learn about it on the spot, whereas this seems to work in Nakahara just fine. Frankel is a bit more respectful of proper mathematics which also makes it a harder text to read for physicists.

Nakahara is a great text. When I visited Caltech I noticed it on the bookshelf of every theorist that I talked to. Anyone who wants to understand how it is that geometry is so important in modern theoretical physics would do himself a favor in buying this book.

This is a very useful book for understanding modern physics. You absolutely need such a book to really understand general relativity, string theory etc. For instance, Wald's book on general relativity will make much more sense once you go through Nakahara's book. It is very complete, clearly written, comprehensive and easy to read. I would also recommend Morita's "Geometry of differential forms' and Dubrovin,Novikov and Fomeko's 3 volume monograph, if you can find it. All in all, Nakahara's book is one of the best buys, precious book.

This text acts as a well-rounded introduction and overview of much of the mathematics underlying modern physics. While far from rigorous, the physics student will come away with a good understanding of how to use a wide variety of mathematical tools. This book is a necessity for every theoretical physicist. When used in a course (probably advanced undergrad or beginning grad), it should definitely be supplemented with more thorough texts, such as Geometry of Physics by Frankel. After such a course, one should be fully prepared for texts such as Spin Geometry by Michelson & Lawson, and String Theory by Polchinski. As for the mathematics presented in the book, go to one of the many excellent intro books to algebraic topology (Fulton, Munkres, Massey, Bott & Tu) and fibre bundles (Steenrod, Husemoller) for proper treatments of the subjects.

Reading all the glowing reviews of this book, I wonder whether the reviewers actually tried to use the book to understand the material, or just checked the table of contents. There are so many misprints, throughout, that one wonders if the book was proofread at all. Some of the mistakes will be obvious to every physicist - for example, one of the Maxwell equations on page 56 is wrong - others are subtle, and will confuse the reader. The careful reader, who wants to really understand the material and tries to fill in the details of some of the derivations, will waste a lot of time trying to derive results that have misprints from intermediate steps which have different misprints! Some chapters are worse than others, but the average density of misprints seems to be more than one per page.
The book might be useful as a list of topics and a "road map" to the literature prior to 2003, but that hardly justifies the cost (or the paper) of a whole book.

A very nice blending of rigor and physical motivation with well chosen topics. Plenty of examples to illustrate important points. Especially noteworthy is its description of actions of lie algebras on manifolds : the best I have read so far.

Most of the topics are intepreted in terms of their topological/geomtrical structure (and the interplay between those two), but that's what the title of the book says. So you will learn things again in new ways, and gain a powerful new set of tools. If nothing else, it gives you a nice warm fuzzy feeling when you read other field/string theory books that glosses over the mathematics.

One minor rant : the notation of the book can be better. I personally uses indices to keep track of the type of objects (eg. greek index=components of tensors, no index=a geometrical object etc..), but Nakahara drops indices here and there "for simplicity". But that's my personal rant.

Good book. Buy it.

No doubt, the interplay of topology and physics has stimulated phenomenal research and breakthroughs in mathematics and physics alike.

Unfortunately, there is so much mathematics to master that the average graduate physics student is left bewildered.....until now.

The text is an excellent reference book. I emphasize reference. The book presupposes an acquaintance with basic undergraduate mathematics including linear algebra and vector analysis.

The author covers a wide range of topics from tensor analysis on manifolds to topology, fundamental groups, complex manifolds, differential geometry, fibre bundles etc.

The exposition in necessarily brief but the main theorems and IDEAS of each topic are presented with specific applications to physics. For example the use of differential geometry in general relativity and the use of principal bundles in gauge theories, etc.

Unfortunately, there are very few exercises necessitating the use of supplementary texts. However, to the author's credit appropriate supplementary texts are provided. The author goes to great lengths to show which texts inspired the chapters and follows the same line of presentation.

Perhaps the greatest attribute of the text is to take disparate branches of mathematics and coallate them under one text with applications to physics. In doing so one gains a better grasp of how the fields of mathematics interact in the domain of physics.

It's not the best way to learn geometry / topology for physics. It's better than nothing, though, if you are familiar with the topics already. There are many "holes" in Nakahara's book, which you would spend much more time and hard working in a "big" library. than you should to fill in. It's not worth that money and struggle. It's the last one you should consider about owning.

If you are a physics graduate who needs a nice guide to "understand" the aspects and skills of geo / top, I would recommend the following: (1) Milnor's Topology from the Differentiable Viewpoint, and (2) Kreysig's Differential Geometry. The first one was old, and so it does not assume much knowledge about the topic. The latter is a kind-of-Bible for the topic, and all solutions are provided for the problems. These two books will help you a lot if you care about the meaning, not only for those classroom exams or just showing off that you know something about it. Frankel is the next to put on your bookshelf as a detailed and rigorous development for your preparation to be a theoretical physicist.

If you have only a rough idea about topology, Hocking and Steen are the best choices, and they are Dover!!

Anyway, if I could find a cheap used Nakahara, I would get it as a reference.

Highly stimulating and extremely hard to read, written for mathematicians in physics. However, the chapter on Riemannian Geometry can be worked through, up to a point, without any knowledge of exterior differential forms, and is notable if for only one fact alone: a simple calculation is provided that explains explicitly that spheres in four and eight dimensions (3-spheres and 7-spheres) are flat with torsion! I don't know another reference that a physicist without special background in math can consult to understand this highly nonintuitive fact.

I bought this book to supplement my knowledge of mathematics which frequently is involved in understanding Particle physics concepts. The book is terse, but peppered with examples and insights about the definitions, and so far it is really fun to read. Seems like a good investment.