The first subject of the Great Elementary Differential Geometry Textbook Review of Winter 2012 is an older book by Erwin Kreyszig titled simply Differential Geometry.
The book is an old one, currently available from Dover Publications. It is relatively inexpensive, but still seems well made. You can also get it from Amazon.com. I don’t really understand how Amazon.com manages to undercut a Dover price. Just… wow.
As a quick overview, let me say that I really like this book. I am certainly glad to own it and I will read more of it. But I don’t think it is a good choice of textbook for an introduction unless the student is well-prepared and highly motivated.
Since you can get some sense of the plan of things, including the order of presentation and some of the choices made about what to include or exclude, this is the list of chapters:
- Theory of Curves
- Concept of a Surface. First Fundamental Form. Foundations of Tensor Calculus
- Second Fundamental Form. Gaussian and Mean Curvature of a Surface
- Geodesic Curvature and Geodesics
- Absolute Differentiation and Parallel Displacement
- Special Surfaces
Keeping to the general plan for completing my big review project, I read the chapter Concept of a Surface. First Fundamental Form. Foundations of Tensor Calculus.
Points in Favor of the Book
The writing is clear and careful. Examples are well-chosen to illustrate the basic concepts. Kreyszig made a good effort to discuss the very basic things that students can have trouble with in a direct and straightforward way. There is discussion of what it means to make a parametrized surface and several ways to think about them. There are some exercises that seem to be at an appropriate level of difficulty, but not enough of them.
The text is full of wonderful line drawings. There are over a hundred of them and they are stunning. The picture on page 81 showing the tangent plane to a surface is a nice example of the quality of the illustrations.
Points Against the Book
The language is old-fashioned. I think that sometimes this is just quaint and romantic, but other times it introduces extra cognitive load on the reader new to the subject.
The text presupposes some mathematical maturity and sophistication. Every text does, of course, make an assumption or two about the reader. I am unsure that the assumptions made here match the potential American college student. For instance, the level of algebraic facility assumed is a bit higher than I think my students will muster. On page 74, a general change of coordinates is discussed as being this kind of object:
That is all fine, except that it will make my students very uncomfortable. Students today are much more likely to have seen lots of advanced topics covered at a shallow level than to have a deep understanding of fundamental things in two and three dimensions. This kind of “change of coordinate” transformation assumes some facility with the notion of a function from the plane to the plane described in coordinates.
Other Things of Note
The exercises have solutions sketched at the end of the text.
Everything is done in three dimensions. That is, no object ever leaves . Of course, in Kreyszig’s older notation, this is written as .
The Big Choice
One major choice Kreyszig made was to use the tensor calculus. He prefers the more modern to Gauss’ for the components of the first fundamental form. Right after introducing the concept of a surface, there is an extended discussion of tensor notation, the summation convention, and the concepts of covariant and contravariant_ tensors.
I’ll let Kreyszig speak for himself on the matter. The last full paragraph of page 83 is:
We will investigate the differential geometry of surfaces by means of the tensor calculus. As we have already stressed in the introduction to this book, this method will cause no essential difficulties as soon as the reader becomes accustomed to the few rules which govern tensor calculus. A great advantage of this method lies in the fact that it can immediately be generalized to Riemannian spaces of higher dimension which have assumed increasing importance during the last few decades. In addition, many aspects of the theory of surfaces are simplified when treated with the aid of tensor calculus which thus leads to a better and deeper insight into several problems of differential geometry.
I agree with the list of advantages, but I disagree that they outweigh the disadvantages for my students.
The MathSciNet Review
Mathematical Reviews has a review of course. This is from the original edition, published in 1959.
This is a good source book, but not an appropriate text for undergraduates at UNI. Really, it is the tensor calculus decision that does it in. I am happy to supplement the text with more exercises, examples, and participate in discussions about what the language really means. (What else am I for?) But the extra burden of mastering basic tensor calculus gets in the way of enjoying the geometry.
Kreyszig notes in the preface:
In using tensor calculus one should never forget that the purpose of this calculus lies in its applications to certain problems; it is a tool only, albeit a very powerful one.
Exactly. And as a first course in differential geometry can be conducted without this tool, let us leave it by for now. Students who fall in love with geometry can be forced to master tensors later.
Oh, I am still in love with the diagrams. I want to learn to draw like that.