Book 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. ... Read more

Customer Reviews (6)

Great Summary
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.

an excellentbook!
As many professors in China recommend, itis 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!

Be Careful!
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.

To readers of this book
I am reading this book now. It is as the other reviewers said,
rather condensed. However, it would not be beyond comprehension
if 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 are
synthesized 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")

dense book
This book contains most important material in differential geometry in about 330 page. No exercise, few exaples make this book very dense, which is just the style of Chinese professors.
It deserves cautions that in chapter 8, Chern introduce the connection proposed by himself in 1948, which is the proper tools for finsler geometry. ... Read more

Book Description
From the reviews of the second edition: "The new methods of complex manifold theory are very useful tools for investigations in algebraic geometry, complex function theory, differential operators and so on. The differential geometrical methods of this theory were developed essentially under the influence of Professor S.-S. Chern's works. The present book is a second edition... It can serve as an introduction to, and a survey of, this theory and is based on the author's lectures held at the University of California and at a summer seminar of the Canadian Mathematical Congress....
The text is illustrated by many examples... The book is warmly recommended to everyone interested in complex differential geometry." #Acta Scientiarum Mathematicarum, 41, 3-4# ... Read more

Book Description
This volume is about the life and work of Shiing-Shen Chern (1911-), one of the leading mathematicians of this century. The book contains personal accounts by some friends, together with a summary of the mathematical works by Chern himself. Besides a selection of the mathematical papers the book also contains all his papers published after 1988. ... Read more

Book Description
Finsler geometry generalizes Riemannian geometry in the same sense that Banach spaces generalize Hilbert spaces.This book presents an expository account of seven important topics in Riemann-Finsler geometry, ones which have recently undergone significant development but have not had a detailed pedagogical treatment elsewhere. Each article will open the door to an active area of research, and is suitable for a special topics course in graduate-level differential geometry.The contributors consider issues related to volume, geodesics, curvature, complex differential geometry, and parametrized jet bundles, and include a variety of instructive examples. ... Read more

Book Description
<20 Century Mathematics, Volume 8> Contains the collected papers of ProfWei-Liang Chow, an original and versatile mathematician of the 20thCentury. For researchers in algebraic geometry. ... Read more

Book Description
Tight and taut manifolds form an important and special class of surfaces within differential geometry.This book contains in-depth articles by experts in the field as well as an extensive and comprehensive bibliography. This survey will open new avenues for further research and will be an important addition to any geometer's library. ... Read more

Book Description
...this volume contains selected papers presented at theconference...topics include canonical maps of Gorenstein 3-folds,fundamental groups of algebraic curves, Chen's interated integrals,algebraic fiber spaces and others ... Read more