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Differential Geometry: Curves—Surfaces—Manifolds

Wolfgang Kühnel
American Mathematical Society
Publication Date: 
Number of Pages: 
Student Mathematical Library 77
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Jason M. Graham
, on

This text, published as part of the AMS “Student Mathematical Library” series, is the third edition of the English translation of Differentialgeometrie by Wolfgang Kühnel. The book provides an excellent introduction to the differential geometry of curves, surfaces and Riemannian manifolds that should be accessible to a variety of readers. The author includes a number of examples, illustrations, and exercises making this book well-suited for students or for self-study. Solutions to many of the exercises are provided in the final section of the text and there are several sections throughout the book that address topics related to special and general relativity. Frankly, this is a great book.

There are several features of Differential Geometry Curves—Surfaces—Manifolds that I really like. First, the author’s choice of, and use of notation is excellent. Second, the author strongly encourages the reader to think geometrically. In chapters 2, 3, and 4, concerned with the differential geometry of curves and surfaces, there are approximately fifty figures illustrating the geometry. Even in the later sections where it becomes more of a challenge to use figures, the author’s discussions provide geometric insight and intuition.

The third feature of this text that I would like to commend is that the vast majority of the definitions and results are boxed so that they stand out and are easy to find when flipping through the text. Last, the book is carefully written with rigorously stated definitions and results. Despite the fact that the author does not strive for complete generality at all times, one still gets the sense that one is learning modern differential geometry from this book. I feel confident that mastering the content of Differential Geometry Curves—Surfaces—Manifolds well prepares the reader to venture further into the subject.

As is true with any text, this book takes a particular perspective on the subject matter it covers. Being a book on differential geometry there is of course much discussion of curvature, metrics and tensors. However, the text does not develop the general calculus of tensors and differential forms, although the general Stokes Theorem is stated without proof and applied to derive the Gauss-Bonnet formula. I believe that the approach taken here is nonetheless appropriate for an introduction, since readers new to the subject will spend most of their time learning geometry, solving interesting problems, and seeing geometry as it relates to relativity theory. There is plenty of opportunity to learn more of the formal aspects of differential geometry in the references provided by Kühnel in the bibliography. Moreover, the content of the book will serve as an excellent source of motivation for studying further differential geometry.

I believe it would be very difficult to cover this entire book in a single-semester undergraduate course. Nevertheless, chapters 2, 3, and 4 alone offer a solid introductory course that should be accessible to advanced undergraduates, although it may be necessary to supplement the first chapter on prerequisite material. For example, in chapter 1 the implicit function theorem is stated but not proved, so that the treatment in the book may be a little terse for students seeing this background material for the first time. On the other hand, I do not believe it is necessary for students to have had a full semester of rigorous multivariable analysis to read this text, at least chapters 2 and 3. Note that after chapters 2 and 3, the reader will know about the theory of curves, the local theory of surfaces including the theory for minimal surfaces, and curves and surfaces in Minkowski space. So, even after only two chapters the reader is exposed to a lot of interesting mathematics. If you are looking to teach or learn differential geometry at an introductory level, Differential Geometry Curves—Surfaces—Manifolds is a great resource to have on hand.

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Jason M. Graham is an assistant professor in the department of mathematics at the University of Scranton, Scranton, Pennsylvania. His current professional interests are in teaching applied mathematics and mathematical biology, and collaborating with biologists specializing in the collective behavior of groups of organisms.

See the publisher's web page for this book.