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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

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Riemann-Finsler geometry is a subject that concerns manifolds with Finsler metrics, including Riemannian metrics. It has applications in many fields of the natural sciences. Curvature is the central concept in Riemann–Finsler geometry. This invaluable textbook presents detailed discussions on important curvatures such as the Cartan torsion, the S-curvature, the Landsberg curvature and the Riemann curvature. It also deals with Finsler metrics with special curvature or geodesic properties, such as projectively flat Finsler metrics, Berwald metrics, Finsler metrics of scalar flag curvature or isotropic S-curvature, etc. Instructive examples are given in abundance, for further description of some important geometric concepts. The text includes the most recent results, although many of the problems discussed are classical. ... Read more

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

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Papers from the conference. Provides a forum for leading mathematicians to expound and discuss their views on new ideas in these fields, as well as trends in 21st century mathematics. For graduate students and researchers in geometry and topology. ... Read more

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This invaluable book contains the collected papers of Prof Wei-Liang Chow, an original and versatile mathematician of the 20th Century. Prof Chow’s name has become a household word in mathematics because of the Chow ring, Chow coordinates, and Chow’s theorem on analytic sets in projective spaces. The Chow ring has many advantages and is widely used in intersection theory of algebraic geometry. Chow coordinates have been a very versatile tool in many aspects of algebraic geometry. Chow’s theorem — that a compact analytic variety in a projective space is algebraic — is justly famous; it shows the close analogy between algebraic geometry and algebraic number theory. ... Read more

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Contains sections on Complex differential geometry, Partial differential equations, Homogeneous spaces, and Relativity. ... Read more

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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

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This volume features proceedings from the 1995 Joint Summer Research Conference on Finsler Geometry (Seattle, WA), chaired by S.S. Chern and co-chaired by D. Bao and Z. Shen.

The editors of this volume have provided comprehensive and informative "capsules" of presentations and technical reports. This was facilitated by classifying the papers into the following 6 separate sections---3 of which are applied and 3 are pure:

Finsler Geometry over the reals

Complex Finsler geometry

Generalized Finsler metrics

Applications to biology, engineering, and physics

Applications to control theory

Applications to relativistic field theory

Each section contains a preface that provides a coherent overview of the topic and includes an outline of the current directions of research and new perspectives. A short list of open problems concludes each contributed paper.

A number of photos are featured in the volume, for example, that of Finsler. In addition, conference participants are also highlighted. ... Read more

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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

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This is a translation of an 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, sought to combine 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 fibre 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 specifically for this edition by Professor Chern to bring the text into perspective. ... 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

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contains sections on Riemannian geometry, Submanifolds,Foliations, Algebraic and piecewise linear topology,Miscellaneous ... Read more

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In Riemannian geometry, measurements are made with both yardsticks and protractors. These tools are represented by a family of inner-products. In Riemann-Finsler geometry (or Finsler geometry for short), one is in principle equipped with only a family of Minkowski norms. So yardsticks are assigned but protractors are not. With such a limited tool kit, it is natural to wonder just how much geometry one can uncover and describe? It now appears that there is a reasonable answer. Finsler geometry encompasses a solid repertoire of rigidity and comparison theorems, most of them founded upon a fruitful analogue of the sectional curvature. There is also a bewildering array of explicit examples, illustrating many phenomena which admit only Finslerian interpretations. This book focuses on the elementary but essential items among these results. Much thought has gone into making the account a teachable one. David Bao is Professor of Mathematics and of the Honors College, at the University of Houston. He obtained his Ph. D. from the University of California at Berkeley in 1983, with Jerry Marsden as his advisor. Before coming to Houston, he did two years of post-doctoral studies at the Institute for Advanced Study in Princeton, New Jersey. Besides differential geometry, he is passionately curious about the ways cats and goldfish think. Shiing-Shen Chern is Professor Emeritus of Mathematics at the University of California at Berkeley, and Director Emeritus of the Mathematical Sciences Research Institute. He is also Distinguished Visiting Professor Emeritus at the University of Houston. Chern received his D. Sc. in 1936, as a student of W. Blaschke. He carried out his post-doctoral studies under E. Cartan. Chern has garnered a good number of distinctions to date. These include the Chauvenet Prize (1970), National Medal of Science (1975), the Humboldt Award (1982), the Steele Prize (1983), and the Wolf Foundation Prize (1983-84). Zhongmin Shen is Associate Professor of Mathematics at Indiana University Purdue University Indianapolis (IUPUI). He earned his Ph. D. from the State University of New York at Stony Brook in 1990 under Detlef Gromoll. He spent 1990-91 at the Mathematical Sciences Research Institute at Berkeley, and 1991-93 as a Hildebrandt Assistant Professor at the University of Michigan at Ann Arbor. ... Read more

Customer Reviews (1)

Mainly for experts in Finsler geometry
The authors claim to turn the subject of Finsler geometry with this book
into a more teachable one and to have a candid style of writing.
This is definitly true for the first 50 pages, where the concepts of
Finsler geometry are very well explained and the exercises
are manageable and perfectly interrelated with the text.

Then the Chern connection and the curvature tensor of Finsler geometry
drop out of the heaven without any explanation of the ideas leading to these
constructions. So one has to derive them alone. In doing so older
texts on Finsler geometry like the Grundlehren text of Rund are more
helpful than this volume.

But the book was apparently prepared with great care. The layout must be called
beautyful and it really facilitates reading this book. Many references
to the literature and classical papers of the subject are included. So beginners,
which want to get a first aquaintance in Finsler geometry, find at least some help.

... Read more