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## 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 than any other. It has some true classics that everyone should at least browse. It seems that lately everyone and his brother 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 nor particle physics can be correctly expressed without the concept of coordinate charts on differentiable manifolds, Lie groups or fiber bundles. I was really looking forward to a finished text based on Cliff Tubbs’ Math 230 lectures for the first-year graduate student DG course at Harvard, which he has taught there for many years. A book by a recognized master of the subject is welcome, as they can be expected to bring their researcher’s perspective to the material.

Well, the book is finally here and I’m sorry to report that it’s a bit underwhelming. The topics covered in the book are the usual suspects for a first-year undergraduate course, although covered at a slightly higher level than usual: smooth manifolds, Lie groups, vector bundles, matrices on vector bundles, Riemannian matrices, geodesics on Riemannian manifolds, principals. Bundles, covariant derivatives and connections, holonomy, curvature polynomials and characteristic classes, Riemannian curvature tensor, complex manifolds, holomorphic submanifolds of a complex manifold and Kähler metrics. On the positive side, it is very well written and covers almost the entire current landscape of modern differential geometry. The presentation is as self-contained as possible, as all told, the book has 298 pages and 19 bite-sized chapters. . Professor Taubes gives detailed but concise proofs of fundamental results, showing his authority on the subject. So a huge amount is covered very efficiently but very clearly. Each chapter includes an extensive bibliography for further reading, which is one of the most interesting aspects of the book—the author comments on other works and how they have influenced their presentation. His hope is evident that it will induce his students to read other recommended works with him, which show great academic value on the part of the author. Unfortunately, this approach is a double-edged sword because it goes hand-in-hand with one of the book’s faults, which we’ll get to in a moment.

Taubes writes very well indeed and he infuses his presentation with many of his insights. Also, it has very good and well-chosen examples in each section, which I think is very important. It also covers material on complex manifolds and Hodge theory, which are often overlooked by beginning undergraduate textbooks because of the technical subtleties that separate the strictly differential-geometric aspects from algebraic geometry. So what is here is really good. (Interestingly, Taubes credits his influence for the book as the late Raoul Bott’s legendary course at Harvard. Many recent textbooks and lecture notes on the subject credit Bott’s course as their inspiration: Loring To’s. *An Introduction to Manifolds*of Honda Lecture Notes at USCD, by Lawrence Colon *Differentiable manifolds *Among the most prominent. It’s humbling how an expert teacher can define a subject for a generation.)

Unfortunately, there are 3 problems with the book that make it a bit of a let down and they all have to do with something.** no **in the book. The first and most serious problem with Taubes book is that it is not really a textbook, it is a set of lecture notes. with this **zero** Practice. In fact- the book looks like Oxford University Press just took the final version of Taubes’ online notes and slapped a cover on them. That one is not necessary **bad** Of course – some of the best resources in differential geometry (and advanced math in general) are lecture notes (classic notes by S. Schern and John Milners come to mind). But for coursework and some you want to pay enough money – if you really want a bit more, a printed set of lecture notes can be downloaded from the web for free.

They are also very difficult to use as textbooks as you need to look elsewhere for exercises. I don’t think a uniform set of exercises *From the author who designed the lesson* It’s really a lot to ask that you’re spending 30-40 dollars to test your understanding, isn’t it? Is this the real motivation behind the many detailed and thoughtful references to each chapter – students are not only encouraged to look at some of these together, but** is necessary** Find your practice? If so, it must be spelled out really specifically and that 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 teacher is there to guide students through the literature for what’s missing, that works best. In fact, it can make the course more exciting and productive for students. But if you’re writing a textbook, it really needs to be completely self-contained so that the other references you suggest, it’s strictly. *Optional*. Every course is different and if the book doesn’t include its own exercises that limits how much the course can depend on the text. I’m sure Taubes has all the problems from the various sections of the original course – I wish **strongly** Encourage him to include a substantial set of them in the second edition.

The second problem—though not as serious as the first—is that from a researcher of Taubes’ credentials, you’d expect a bit more creativity and insight into what all this good stuff is good for. Well, granted, this is a beginner’s lesson and you can’t stray too far from the basic playbook or it’s going to be useless as a basis for later study. That being said, a concluding chapter summarizing the current state of play in differential geometry using all the machinery that has been developed – especially in the realm of mathematical physics – will do much to give the novice an exciting glimpse ahead of the major. Branches of pure and applied mathematics. He sometimes digresses into good original material not usually touched upon in such books: the Schwarzschild metric, for example. But he gives no indication as to why this is important or its role in general relativity.

Finally – there are virtually no pictures in the book. **None. zero Nada. ** Well, this is a graduate level text and graduate students really have to draw their own pictures. But for me, one of the things that makes differential geometry so fascinating is that it is such a visual and visceral subject: in a good classical DG course you realize that if you were clever enough, you could prove everything with a picture. . Giving a completely formal, non-visual presentation removes much of that conceptual stimulation and makes it seem drier and less interesting than it actually is. In that second edition, I will consider including some visuals. If you are a purist you don’t need to add much. But some, especially in the feature classes and the sections on vectors and fiber bundles, will make these parts much clearer.

But the final verdict? A very solid resource for learning DG for the first time at graduate level, but needs to be supplemented extensively to make up for weaknesses. Fortunately, each chapter comes with a very good set of references. Good supplementary reading and exercises can easily be selected from these. I strongly recommend Guillemin and Pollack’s classic *Differential topology *As initial reading, “Trilogy” by John M. Lee for collateral reading and exercises, a wonderful 2 volume physics-oriented text *Geometry, topology and gauge fields *Connections and Applications in Physics by Gregory Naber for lots of nice pictures and concrete calculations. For an in-depth presentation of complex differential geometry, try the classic and recent text by Wells. *Complex differential geometry *by Zhang. With all of this to compliment Taubes, you’ll be in great shape for a year-long course in modern differential geometry.

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