Differential geometry is a branch of higher mathematics that may have more quality textbooks than any other textbook. It has some real classics, and everyone agrees that at least should be viewed. Recently it seems that everyone and his cousin are working hard to write the "American Difference Geometry Textbook." It's really hard to understand why: The subject of differential geometry is not only one of the most beautiful and fascinating applications in calculus and topology, it is also one of the most powerful. Manifold language is the natural language of most of the two. Classical and modern physics - If there is no distinguishable manifold, the concept of a graph on a Lie group or fiber bundle, general relativity or particle physics cannot be correctly expressed. I am very much looking forward to the completion of the Math 230 lecture based on Cliff Taubes at Harvard University for the first year graduate DG course, where he has been teaching for many years. A well-recognized master book should be welcomed because people can hope that they bring the researcher's point of view to the material.
Ok, this book is finally here, and I regret to report that it is a bit disappointing. The topics covered in this book are common suspects for the first year of graduate programs, although the coverage is slightly higher than usual: smooth manifolds, Lie groups, vector beams, metrics on vector beams, Riemann metrics, Riemannian manifolds Geodesic, main beam, covariant derivatives and connections, global, curvature polynomials and feature classes, Riemann curvature tensors, complex manifolds, full-mechanical manifolds of complex manifolds, and Kähler metrics. On the positive side, it is written very well and covers almost the entire current landscape of modern differential geometry. Although everyone said that the book has 298 pages, including 19 one-size chapters, it is as self-contained as possible. . Professor Taubs gave a detailed and concise proof of the basic results, proving his authority in this subject. Therefore, a lot of coverage is very effective but very clear. Each chapter contains a detailed bibliography, one of the most interesting aspects of the book - the author's comments on other works and how they affect his speech. His hope is obviously to motivate his students to read other recommended works at the same time, which shows the author's excellent educational value. Unfortunately, this method is a double-edged sword because it is closely related to a mistake in the book, and we will solve this problem temporarily.
Taubes wrote very well and he used his many insights to express his speech. In addition, it has many excellent and carefully selected examples in each section, which I think is very important. It even covers materials for complex manifolds and Hodge's theory, most of which have been avoided in graduate textbooks because of the subtle nuances of strict differential geometry and algebraic geometry. So the content here is very good. [Interestingly, Taubes thinks his influence on the book is the legendary Rauol Bott's legendary course at Harvard University. Recently, so many textbooks and lectures on this subject have been attributed to Bott's curriculum and their inspiration: Loring Tu& #39;s from
Profile of manifoldfrom
, Ko Honda's Handout at USCD, Lawrence Conlon from
Divisible manifold from
The most prominent of them. It is very humble for an expert teacher to define a theme for a generation. ]
Unfortunately, this book has three questions that make it a bit disappointing, and what are they related to? Do not In the book. The first and most serious problem in Taubes' book is that it is not a textbook at all, but a set of handouts. it has zero Exercise. In fact, the book looks like Oxford University Press has just accepted the final version of the Taubes online note and hit the cover. Not necessarily that Bad Of course, some of the best resources for differential geometry [and advanced mathematics] are handouts [the classic notes of S.S. Chern and John Milnors come to mind]. But for coursework, you want to pay a lot of money - you really want more, then just need a printed lecture note, someone can download it for free from the Internet.
They are also difficult to use as textbooks because you need to go elsewhere to find exercises. I don't think there is a corresponding exercise. from
Author from design text from
Testing your understanding is too much, asking for $30-40, is it? This is the real motivation behind each chapter's very detailed and self-righteous reference - not only encourages students to look at some of them at the same time, but need In order to find your own sport? If this is the case, it should indeed be specially spelled out, which indicates some laziness of the author. When it is a set of lecture notes designed to build an actual course, the teacher is there to guide students through the literature to understand what is missing, which is normal. In fact, it may bring more exciting and productive courses to students. But if you're writing a textbook, it really needs to be completely self-contained, so no matter what other references you make, it's strict. from
Optionalfrom
. Each course is different, and if the book does not contain its own exercises, it greatly limits the extent to which the course is dependent on the text. I am sure that Taubes has all the problem sets in all parts of the original course - I will very much Encourage him to add a lot of content in the second edition.
The second question - although this is not as serious as the first question - is a question from the Taubes certificate researcher, you will expect more creativity and insight into the benefits of all these good things. Ok, be awarded, this is a beginner's text, you can't be too far from the basic script, otherwise it will be useless as a basis for later research. That being said, using all the mechanisms that have been developed [especially in the field of mathematical physics] to summarize the end of the current state of differential geometry will help the novice to an exciting glimpse of the leading edge of pure mathematics and applied mathematics. He does sometimes put these raw materials in these books and usually does not involve: for example, the Schwarzchild indicator. But he did not explain why it is important in general relativity or its role.
Finally - there are almost no pictures in the book. No. zero. Nada. Ok, grant this is a graduate level text and graduate students really should draw their own photos. But for me, one of the reasons that makes differential geometry so fascinating is that it is such a visual and inner theme: the feeling that a person gets in a good classic DG course, if you are smart enough, you can use pictures Prove everything. Giving a fully formal, non-intuitive presentation can eliminate many conceptual excitements and make it look drier and more interesting. In the second edition, I considered including some visual effects. If you are a purist, you don't have to add a lot. However, a few, especially chapters on feature classes and vector and fiber bundle sections, will shed light on these sections.
So what is the final judgment? This is the first very reliable resource for learning DG at the graduate level, but it needs to be extensively supplemented to fill these shortcomings. Fortunately, each chapter comes with a good set of references. From these you can easily choose good supplementary reading and practice. I highly recommend the classics of Guillemin and Pollack from
Differential topology from
As a preliminary reading, John M. Lee's "Trilogy" is used for reading and practice, awesome 2 volumes of physics-oriented text from
Geometry, topology, and instrument fields from
Physic connections and applications are provided by Gregory Naber along with many good images and specific calculations. For a more in-depth demonstration of complex differential geometry, try using Wells classic and updated text from
Complex differential geometry from
By Zhang. By praising Taubes with all of this, you will have a one-year excellent course in modern differential geometry.
Orignal From: Book Review: Differential Geometry of Cliff Tauber: Bundling, Metrics, Connections, and Curvature
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