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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 almost any other. It has some true classics that everyone agrees should at least be checked out. It seems like everyone and their cousin is trying to write The Great American Differential Geometry Textbook lately. 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, but also one of the most powerful. The language of varieties is the natural language of most aspects of both. Classical and Modern Physics: Neither general relativity nor particle physics can be properly expressed without the concepts of coordinate diagrams on differentiable manifolds, Lie groups, or fiber bundles. I was 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 for several years. A book by a recognized master of the subject is welcome, as he can be expected to bring his researcher’s perspective to the material.

Well, the book is finally here and I’m sorry to report that it’s a bit of a let down. The topics covered in the book are the usual suspects from a first-year graduate course, albeit treated at a slightly higher level than usual: smooth varieties, Lie groups, vector bundles, metrics on vector bundles, Riemannian metrics, geodesics on Riemannian manifolds, principal bundles, derivatives and covariant connections, holonomy, curvature polynomials and characteristic classes, Riemann curvature tensor, complex manifolds, holomorphic submanifolds of a complex manifold and Kähler metrics. On the plus side, it is VERY well written and covers pretty much the entire current landscape of modern differential geometry. The presentation is as much as possible, since in total the book is 298 pages long and consists of 19 small-sized chapters. . Professor Taubes provides detailed but concise proofs of the basic results, which demonstrates his authority in the field. So a huge amount is covered very efficiently yet quite clearly. Each chapter contains a detailed bibliography for further reading, which is one of the most interesting aspects of the book: the author discusses other works and how they have influenced his presentation. His hope is clearly that he will inspire his students to read the other works recommended at the same time as his, which shows excellent educational values on the part of the author. Unfortunately, this approach is a double-edged sword, as it goes hand-in-hand with one of the book’s faults, which we’ll touch on momentarily.

Taubes writes very well and fills his presentation with his many ideas. Also, it has many good and well-chosen examples in each section, which I think 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 from the algebraic geometric aspects. So what is here is very good. (Interestingly, Taubes credits his influence for the book with the late Rauol Bott’s legendary course at Harvard. So many recent textbooks and lecture notes on the subject credit Bott’s course with its inspiration: Loring Tu’s *An introduction to varieties*lecture notes by Ko Honda at USCD, Lawrence Conlon *Differentiable multiples *among the most prominent. It’s very humbling how an expert teacher can define a topic 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 what it is** no **in the book The first and most serious problem with Taubes’ book is that it’s not really a textbook, it’s a collection of lecture notes. Has **zero** exercises. In fact, the book looks like Oxford University Press just took the final version of Taubes’ online notes and gave them a cover. Not that this is necessarily a **bad** thing, of course: some of the best sources out there on differential geometry (and advanced math in general) are lecture notes (the classic SSChern and John Milnors notes come to mind). But for courses and something you want to pay considerable money for, you really want a little more than just a printed set of lecture notes that someone could have downloaded from the web for free.

They are also much more difficult to use as a textbook, as you have to look elsewhere for exercises. I do not believe a corresponding set of exercises *of the author who designed the text* Testing your understanding is really too much to ask for something you’re spending $30-$40 on, right? Is that the real motivation behind the very detailed and opinionated references for each chapter: students are not only encouraged to look at several of them simultaneously, but** required** to find your own exercises? If so, it really should have been specifically written and 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 students through the literature on what’s missing, it works well. In fact, it could be an even more exciting and productive course for students. But if you’re writing a textbook, it really needs to be completely self-contained, so any other references you suggest, strictly *optional*. Each course is different and if the book does not contain its own exercises, this greatly limits how dependent the course can be on the text. I’m sure Taubes has all the problem sets from the different sections of the original course – I would **strongly** encourage him to include a substantial set of them in the second edition.

The second problem, although not as serious as the first, is that from a researcher of Taubes’ credentials, one would expect a little more creativity and knowledge of what all this good stuff is for. Granted, this is a beginner’s text and you can’t stray too far from the core playbook or it will be useless as a basis for further study. That said, a closing chapter summarizing the current state of differential geometry using all the machinery that has been developed, especially in the realm of mathematical physics, would go a long way in giving the novice an exciting view of the cutting edge. of a large branch of pure and applied mathematics. He occasionally digresses into nice original material not usually touched on in these books: Schwarzchild’s metric, for example. But it gives no indication of why it is important or its role in general relativity.

Finally, there are virtually no pictures in the book. **none Zero. nothing ** Okay, admittedly this is an undergraduate 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: you get the feeling in a good classical DG course that if you were smart enough, you could prove almost all with a picture. . Making a presentation completely formal and non-visual takes away a lot of that conceptual excitement and makes it seem much drier and less interesting than it actually is. In this second edition, I would like to include some images. There’s no need to add many if you’re a purist. But a few, especially in the chapters on characteristic classes and sections of vector and fiber beams, would clarify these parts immensely.

So the final verdict? A very solid source for learning DG for the first time at graduate level, but will need to be supplemented extensively to cover gaps. Fortunately, each chapter comes with a very good set of references. Good supplementary reading and exercises can easily be selected from these. I highly recommend Guillemin and Pollack’s classic *Differential topology *as preliminary reading, John M. Lee’s “trilogy” for side readings and exercises, the fantastic 2-volume physics-oriented text *Fields of Geometry, Topology and Gauge *by Gregory Naber for connections and applications in physics, as well as many good pictures and concrete calculations. For a more in-depth presentation of complex differential geometry, try Wells’ classic and more recent text *Complex differential geometry *by Zhang. With all this to complement Taubes, you’ll be in excellent shape for a year’s worth of modern differential geometry.

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