Book Review: Cliff Taubes’ Differential Geometry: Bundles, Metrics, Connections and Curvature

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. 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. The topics covered in the book are the usual suspects for a first year graduate course,albeit covered at a somewhat higher level then usual: smooth manifolds, Lie groups, vector bundles, metrics on vector bundles, Riemannian metrics, geodesics on Riemannian manifolds, principal bundles, covariant derivatives and connections, holonomy, curvature polynomials and characteristic classes, Riemannian curvature tensor, complex manifolds, holomorphic submanifolds of a complex manifold and K√§hler metrics. On the positive side, it’s VERY well written and covers virtually the entire current landscape of modern differential geometry.The presentation is as much as possible self-contained, given that all told, the book has 298 pages and consists of 19 bite-size chapters. Professor Taubes gives detailed yet concise proofs of basic results, which demonstrates his authority in the subject. So an enormous amount is covered very efficiently but quite clearly. Each chapter contains a detailed bibliography for additional reading, which is one of the most interesting aspects of the book-the author comments on other works and how they have influenced his presentation. His hope is clearly that it will inspire his students to read the other recommended works concurrently with his, which shows excellent educational values on the author’s part. Unfortunately,this approach is a double edged sword since it goes hand in hand with one of the book’s faults, which we’ll get to momentarily.

Taubes writes very well indeed and he peppers his presentation with his many insights. Also, 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 indeed. (Interestingly, Taubes credits his influence for the book to be the late Rauol Bott’s legendary course at Harvard. So many recent textbooks and lecture notes on the subject credit Bott’s course with their inspiration: Loring Tu’s An Introduction to Manifolds, Ko Honda’s lecture notes at USCD, Lawrence Conlon’s Differentiable Manifolds among the most prominent. It’s very humbling how one expert teacher can define a subject for a generation.)

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 necessarily 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 who designed the text to test your understanding is really too much to ask for in something you’re spending 30-40 bucks on, is it? Is that the real motivation behind the very detailed and opinionated references for each chapter-the students are not merely encouraged to look at some of these concurrently, but required in order to find their own exercises? If so, it really should have been specifically spelled out and it shows some laziness on the part of the author. When it’s a set of lecture notes designed to frame an actual course where the instructor is there to guide the students through the literature for what’s missing, that works fine. In fact, it might make for even more exciting and productive course for the students. But if you’re writing a textbook, it really needs to be completely self contained so that whatever other references you suggest, it’s strictly optional. Every course is different and if the book doesn’t contain it’s own exercises that limits enormously how dependent the course can be on the text. 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 Differential Topology as preliminary reading, the “trilogy” by John M.Lee for collateral reading and exercises, the awesome 2 volume physics-oriented text Geometry, Topology and Gauge Fields by Gregory Naber for connections and applications to physics as well as many good pictures and concrete computations. For a deeper presentation of complex differential geometry, try the classic by Wells and the more recent text Complex Differential Geometry by Zhang. With all these to compliment Taubes, you’ll be in excellent shape for a year long course in modern differential geometry.