LECTURES ON DIFFERENTIAL GEOMETRY (University Mathematics) by S S Chern (PDF)

Description

This book is a translation of an authoritative introductory text based on a lecture series delivered by the renowned differential geometer, Professor S S Chern in Beijing University in 1980. The original Chinese text, authored by Professor Chern and Professor Wei-Huan Chen, was a unique contribution to the mathematics literature, combining simplicity and economy of approach with depth of contents. The present translation is aimed at a wide audience, including (but not limited to) advanced undergraduate and graduate students in mathematics, as well as physicists interested in the diverse applications of differential geometry to physics. In addition to a thorough treatment of the fundamentals of manifold theory, exterior algebra, the exterior calculus, connections on fiber bundles, Riemannian geometry, Lie groups and moving frames, and complex manifolds (with a succinct introduction to the theory of Chern classes), and an appendix on the relationship between differential geometry and theoretical physics, this book includes a new chapter on Finsler geometry and a new appendix on the history and recent developments of differential geometry, the latter prepared specially for this edition by Professor Chern to bring the text into perspectives.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I like this monograph quite a bit. It is certainly an introduction, however, that does not imply that it is an easy read !With a willingness to dig in, Chern’s exposition proves to be a satisfying excursion. Well worth the effort ! I note that part one (topology) of Simmon’s text Introduction to Topology and Modern Analysis (1963, McGraw-Hill) is a helpful prerequisite to Chern (or, Buck’s Advanced Calculus). Allowing prerequisites, Chern comes together most beautifully !(1) Finsler Geometry (chapter eight, sixty pages). That is good ! Why is that good ? Chern writes “Solid State physics involves lattices and the geometry is naturally Finsler.” (page 266). How do we begin ? Chern also writes: “The starting point for Finsler geometry is the first notion of the integral calculus, the calculation of arc-lengths.”(2) You will need solid acquaintance with definitions and terminology, such as: homeomorphism, isomorphism, equivalence relation, open sets, automorphism, Jacobian matrix, partitions. Assimilation of this terminology carries one quite far !(3) I like to let S.S. Chern speak through his words: “Vector bundles and connections form the mathematical basis of gauge fields in physics.” (page 69, chapter four) and “the simplest device connecting the local and global properties of a manifold is the integration of exterior differential forms on the manifold.” (page 85) and “the relationship between integrals over a domain and integrals over its boundary is at the heart of calculus” (page 92), or “the concept of moving frames originates in mechanics.” (page 205), finally: “Topologically, the m-dimensional torus and the 2m-dimensional real torus are homeomorphic. Yet, the former has a complex-manifold structure and thus has richer contents.” (page 225). Every one of those statements is amplified aplenty throughout Chern’s marvelous discourse.(4) There is much here, and, of such continuing fascination, that it renders nearly impossible my the desire to isolate my favorite part(s) ! However, I would be remiss if I neglected to highlight Chern’s exposition of Submanifolds (pages 18-28). Beautiful ! Also, allow me to accentuate the discussion of “r-dimensional distributions” (page 83) where Chern starts from Frobenius Theorem (local) and proceeds to integral manifolds (global). Thus, another trek proceeding from the ‘local’ to the ‘global.'(5) You do not get far without proofs. The preface includes physicist K.S.Lam extolling the virtues of these lectures for the benefit of physicists (Lam helped Chern prepare the chapter of Finsler Geometry). While the lectures are beneficial to that audience (physicists), I fear that the proofs may well be unfamiliar territory. A glance at the proof of theorem #2.3 (page 190) provides evidence of Chern’s clarity of approach. That same clarity in the proof of theorem #1.1 (page 134): “There exists a Riemannian metric on any m-dimensional smooth manifold.” Chern writes: “In the context of fiber bundles, the existence of a Riemannian metric implies the existence of a positive-definite smooth section of the bundle of symmetric-covariant tensors of order-two on M.” Beautiful !(6) Concluding my review: Advanced undergraduates will find these lectures challenging. Card-carrying Physicists might even find these lectures challenging. With prerequisites firmly in hand (Simmons or Buck) along with a certain degree of mathematical sophistication (for the understanding of proofs), the lectures are inspiring !That is, the lectures are introductory, though, hardly elementary or necessarily easy !Yet, ever so satisfying and enriching !

⭐This book is written by famous authors alright. It may have their reason in the way they choose those materials and the way they are presented.The point is: as an introductory text, the various ideas and structures are not well motivated. They may be economical in the way of the presentation. However, it never seems natural from the point of view of a beginner. It is more natural to start with Riemannian geometry and then proceed to the more general concept of vector bundles and connections. It is in Riemannian geometry, that it is natural to first introduce the concept of a geodesic, and this leads, though a lot of books dont do it this way, to the concept of Levi -Civita connection and therefore holonomy and curvature. The general concept of vector bundles and connections before introducing the Riemannian geometry, makes a complex subject even more abstract and though maybe economical from the point of view of the writers, are formidable for a reader.Even the presentation of specific facts, the book should emphassize, for the benefit of the reader, the structrual (pictorial) aspects more than it does, to illuminate the essence of the formulas, for example, the way it introduces the theta forms on frame bundle omits entirely in mentioning that the essence of thse forms is simply the concept of a coframe. It merely constructs these forms using local coordinates, which seems to be quite tricky to get to its bottom.

⭐Let me begin by saying that I am biased. I worked as Mr. Chern’s assistant in a differential geometry class when I was a grad student. He was a great person to work for and his lectures were well organized. This book is a NOT aimed at the typical undergraduate. It is a major advance in comprehensability from the books from which I learned the covered material. Modern differential geometry does not yet have a great, easy for the novice, self-study friendly text that really covers the material – this book and the Russian trilogy by Dubrovin, et al. are major steps along the way.

⭐As many professors in China recommend, it is an excellent book by a great Geometrician. Though it may not be a beginning book, it should appear on your shelf as a classic one!

⭐I am reading this book now. It is as the other reviewers said,rather condensed. However, it would not be beyond comprehensionif the crucial pictures are established. It is my personal opinion that the first crucial place where it should be understood without any compromise is the section on the frame bundle. Later chapters build on this. Previous chapters aresynthesized here. To any readers who are interested, you are invited to discuss this book. My email address is topollogy@hotmail.com (Notice there are two “l” in “topollogy”)

⭐This is not a good book of Finsler geometry, it looks like some advanced text. World are waiting for a good introductory book in the field of Finsler geometry.

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