Differential Geometry: Bundles, Connections, Metrics and Curvature (Oxford Graduate Texts in Mathematics) [Paperback]

Differential geometry is the branch of advanced mathematics that probably has more quality textbooks then just about any other. It has some true classics that everyone agrees should at least be browsed: Spivak's beautiful,lavishly illustrated and historically informed opus, John M. Lee's more topologically grounded but equally beautiful "trilogy",the more advanced tomes of Conlon and Jost,the more recent opuses by Jeffery Lee and Novikov, etc. It seems lately everyone and his cousin is trying to write The Great American Differential Geometry Textbook. It's really not hard to see why: The subject of differential geometry is not only one of the most beautiful and fascinating applications of calculus and topology,it's also one of the most powerful.The language of manifolds is the natural language of most aspects of both classical and modern physics- neither general relativity or particle physics can be correctly expressed without the concepts of coordinate charts on differentiable manifolds, Lie groups or fiber bundles. I was really looking forward to the finished text based on Cliff Taubes' Math 230 lectures for the first year graduate student DG course at Harvard, which he has taught on and off there for a number of years. A book by a recognized master of the subject is to be welcomed, as one can hope they bring their researcher's perspective to the material.
Well,the book's finally here and I'm sorry to report it's a bit of a letdown. On the positive side, it's VERY well written and covers virtually the entire current landscape of modern differential geometry.It has many good and well chosen examples in each section,something I feel is very important.It even covers material on complex manifolds and Hodge theory,which most beginning graduate textbooks avoid because of the technical subtleties of separating the strictly differential-geometric aspects from the algebraic geometric ones. So what's in here is very good.
Unfortunately,there are 3 problems with the book that make it a bit of a disappointment and they all have to do with what's NOT in the book. The first and most serious problem with Taubes' book is that it's not really a textbook at all,it's a set of lecture notes. It has ZERO exercises.Indeed-the book looks like Oxford University Press just took the final version of Taubes' online notes and slapped a cover on them. Not that that's a BAD thing,of course-some of the best sources there are on differential geometry (and advanced mathematics in general) are lecture notes (S.S.Chern and John Milnors's classic notes come to mind). But for coursework and something you want to pay considerable money for-you really want a bit more then just a printed set of lecture notes someone could have downloaded off the web for free. They're also a lot harder to use as a textbook since you need to look elsewhere for exercises. I don't think a corresponding set of exercises FROM THE AUTHOR to test your understanding is really too much to ask for in something you're spending 30-40 bucks on,is it? I'm sure Taubes has all the problem sets from the various sections of the original course-I'd STRONGLY encourage him to include a substantial set of them in the second edition.
The second problem-although this isn't as serious as the first-is that from a researcher of Taubes' credentials,you'd expect a little more creativity and insight into what all this good stuff is good for.Ok, granted,this is a beginners' text and you can't go too far off the basic playbook or it's going to be useless as a foundation for later studies. That being said,a closing chapter summarizing the current state of play in differential geometry using all the machinery that had been developed-particularly in the realm of mathematical physics-would help a lot to give the novice a exciting glimpse into the forefront of a major branch of pure and applied mathematics. He does digress sometimes into nice original material that's usually not touched in such books: The Schwarzchild metric, for instance. But he doesn't give any indication why it's important or it's role in general relativity.
Lastly-there's virtually no pictures in the book. NONE.ZERO.NADA. Ok,granted this is a graduate level text and graduate students really should draw their own pictures.But to me,one of the things that makes differential geometry so fascinating is that it's such a visual and visceral subject: One gets the feeling in a good classical DG course that if you were clever enough, you could prove just about everything with a picture. Giving a completely formal, non-visual presentation removes a lot of that conceptual excitement and makes it look a lot drier and less interesting then it really is. In that second edition, I'd consider including some visuals. You don't have to add many if you're a purist.But a few,particularly in the chapters on characteristic classes and sections of vector and fiber bundles,would clarify these parts immensely.
So the final verdict? A very solid source from which to learn DG for the first time at the graduate level,but it'll need to be supplemented extensively to fill in the shortcomings. Fortunately, each chapter comes with a very good set of references.Good supplementary reading and exercises can easily be selected from these. I would strongly recommend Guillemin and Pollack's classic as preliminary reading, the "trilogy" by John M.Lee for collateral reading and exercises,the physics-oriented text by Frankel for applications to physics and many good pictures and Wells' book for complex DG. With all these to compliment Taubes,you'll be in excellent shape for a year long course.

I bought this book for Kindle. The book is badly formatted. For mathematical symbols a square with a question mark appears in the page. Whoever at Oxford Press formatted thsis for Kindle screwed up. There are many equations that have unreadable( you can't understand) symbols. For a book on Diferrential
Geometry this is unacceptable. I was told that Amazon no longere sells the Kindle version of this book. Maybe when a new fixed version is on the way. By the way, the book can be read on Kindle for PC. Everything is fine under Kindle for PC