Book Review: Cliff Taubes Differential Geometry: Packages, 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 agrees should at least be consulted. It seems that lately everyone and their cousin are trying to write The Great American Textbook of Differential Geometry. It’s really not hard to see why: the topic of differential geometry is not only one of the most beautiful and fascinating applications of calculus and topology, it is 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 expressed correctly without the concepts of coordinate graphs on differentiable manifolds, Lie groups, or fiber bundles. I really wanted to see the finished text based on Cliff Taubes’ Math 230 lectures for the DG course for freshmen graduate students at Harvard, which he has taught on and off for several years. A book from a recognized teacher on the subject is welcome, as you can expect them to bring your 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 disappointment. The topics covered in the book are the usual suspects for a first-year graduate course, albeit covered at a somewhat higher level than usual: smooth varieties, Lie groups, vector packages, metrics in vector packages, metrics of Riemann, geodesics in Riemann manifolds, main beams, covariant derivatives and connections, holonomy, curvature polynomials and feature classes, Riemann curvature tensor, complex manifolds, holomorphic submanifolds of a complex manifold, and Kähler metrics. On the bright side, it is VERY well written and covers pretty much the entire current landscape of modern differential geometry. The presentation is as autonomous as possible, given that in total, the book has 298 pages and consists of 19 small chapters. . Professor Taubes gives detailed but concise tests of the basic results, demonstrating his authority on the subject. So a huge amount is covered very efficiently but quite clearly. Each chapter contains a detailed 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 clearly that it will inspire his students to read the other recommended works at the same time as his own, 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 flaws, which we’ll get to in a moment.

Taubes writes very well and spices up his presentation with his many ideas. Also, you have a lot of good and well chosen examples in each section, something that I think is very important. It even covers material on complex manifolds and Hodge’s theory, which most beginning graduate textbooks avoid due to the technical subtleties of separating strictly differential geometric aspects from algebraic geometric ones. So what is here is really very good. (Interestingly, Taubes attributes his influence to the book as the late Rauol Bott’s legendary course at Harvard. So many recent textbooks and lecture notes on the subject credit Bott’s course for inspiration: Loring Tu Introduction to collectors, Notes from Ko Honda’s USCD conference, Lawrence Conlon’s Differentiable collectors among the most prominent. It’s very humbling how an expert teacher can define a topic for a generation.)

Unfortunately there are 3 issues with the book that make it a bit disappointing and they all have to do with what not in the book. The first and most serious problem with the Taubes book is that it is not really a textbook, it is a set of class notes. Have zero training. In fact, the book looks like Oxford University Press just took the final version of Taubes’ online notes and covered them up. Not that that’s necessarily a bad Of course, some of the best sources out there on differential geometry (and advanced mathematics in general) are class notes (the classic notes by SSChern and John Milnors come to mind). But for course work and something you want to pay a hefty amount of money for, you really want a little more than just a printed set of class 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 don’t think a corresponding set of exercises of the author who designed the text Testing your understanding is too much to ask for something you’re spending $ 30- $ 40 on, right? It’s that the real motivation behind the very detailed and stubborn references for each chapter, not only are students encouraged to look at some of these at the same time, but required to find your own exercises? If so, it really should have been specifically stated and shows some laziness on the part of the author. When it comes to a set of class notes designed to frame an actual course where the instructor is there to guide students through the literature on what is missing, that 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 that any other reference you suggest is strictly Optional. Each course is different and if the book does not contain its own exercises, that greatly limits the dependence that the course can have on the text. I’m sure Taubes has all the problem sets from the various sections of the original course. strongly encourages you 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 the Taubes credentials, you would expect a little more creativity and understanding of what all these good things are for. Okay, this is a beginner’s text and you can’t stray too far from the basic playbook or it will be useless as a basis for further study. That said, a final chapter summarizing the current state of differential geometry using all the machinery that has been developed, particularly in the realm of mathematical physics, would go a long way toward giving the novice an exciting insight into the cutting edge of an important branch of science. pure and applied mathematics. Sometimes he veers off into nice source material that is generally not touched on in such books – Schwarzchild’s metric, for example. But it does not give any indication of why it is important or its role in general relativity.

Lastly, there are practically no images in the book. None. Zero. Nothing. Okay, since this is graduate level text and graduate students should really draw their own pictures. But for me, one of the things that makes differential geometry so fascinating is that it is such a visceral and visceral subject – in a good classic DG course, you get the feeling that if you were smart enough you could try almost all with an image. . Giving a completely formal, non-visual presentation removes much of that conceptual emotion and makes it seem much drier and less interesting than it actually is. In that second edition, I would consider including some images. You don’t have to add a lot if you are a purist. But some, particularly in the chapters on characteristic classes and sections of vector bundles and fibers, would greatly clarify these parts.

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

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