# Book Review: Stoker’s Differential Geometry

James J Stoker’s Differential Geometry

The next text up for review is J.J. Stoker’s Differential Geometry. (These titles are going to have a theme.) You can see the author’s name is prominently displayed on the cover of my copy. When I had the book with me at the coffee shop this morning, my friend and colleague Doug Shaw quipped, “I didn’t know he wrote a book after Dracula!” Later in the day my wife made the same joke. Now you have all heard it twice, too.

The book is available from Wiley as part of their Wiley Classics Library. I can’t quite fathom why a fifty-year-old, print-on-demand book has an asking price of \$182. You can get it from Amazon.com for slightly cheaper, of course.

I first looked at this book two years ago. My helpful Wiley rep came by and asked what courses I would be teaching, and she sent me this shortly after our meeting. I got a copy free that way, so that might be part of why the book costs so much.

Let me say that I like this book. But for me. Not for undergraduate students. Yep. My reaction has caused me to write like that. In sentence fragments.

For an overview of structure let’s look at the chapter listings. By the way, the table of contents has a pretty detailed description of the individual section headings, too. That seems useful enough to locate any topic to within five pages or so. That would be good if you thought to use this as a reference somehow.

• Operations with Vectors
• Plane Curves
• Space Curves
• The Basic Elements of Surface Theory
• Some Special Surfaces
• The Partial Differential Equations of Surface Theory
• Inner Differential Geometry in the small from the Extrinsic Point of View
• Differential Geometry in the Large
• Intrinsic Differential Geometry of Manifolds. Relativity
• The Wedge Product and the Exterior Derivative of Differential Forms, with Applications to Surface Theory

### Points of Interest

Stoker has a very ambitious list of topics. This is much more material than can possibly be covered in a one semester introduction.

The assumed prerequisites are a bit higher than I had hoped. The preface mentions “Linear Algebra and basic elements of analysis,” which is more than I can assume of my students for sure. The introductory chapter on basic vector analysis topics might serve as a reasonable bridge, but, from the nature of the exposition, the intended audience is a more advanced student. Perhaps this is a good text for a senior in an analysis course, or a graduate student in a related discipline.

The second chapter on just plane curves is a bit of a novelty. And throughout, Stoker chooses to include topics that are not always encountered in a first course. The Jordan Curve Theorem for smooth plane curves makes an appearance in chapter II, for example.

The text is conversational and colloquial. You get the sense of a knowledgable mathematician explaining the material to a colleague over lunch. There is substantial attention paid to motivation and context and the why-do-we-consider-this-object-and-not-that-one and why-would-I-expect-that-to-be-important discussion. I really like this tone. The author’s love of the subject drips off the page. The down side is that it really does require a pretty sophisticated reader to keep up.

In keeping with a conversational approach, the Definition-Theorem-Proof style from many textbooks is avoided. Many results just happen in the run of the text. Some of the big ones are set in italics.

A figure on the Dupin Indicatrix

Sticking to the plan, I read Chapter IV: The Basic Elements of Surface Theory.

The most astonishing thing is the pace which Stoker sets. The chapter is 35 pages long, but in that space he gets these things in and more:
```The basic idea of a surface, the three fundamental forms, Gaussian and Mean curvatures, principal curvatures and principal directions, asymptotic lines and lines of curvature, the osculating paraboloid, the Dupin indicatrix, the characterization of a sphere as a surface consisting of only umbilics, orthogonal nets of coordinates, etc… ```
That is only some of it. It is just blistering. To do all of this and spend some time discussing a little bit of the “insider game,” something had to go. So what went? Examples. There are only three concrete examples in the chapter: the plane, the sphere, and the right circular cylinder. If you are feeling generous, you might say that surfaces realized as graphs $z = f(x,y)$ are implicitly covered, but since that is the subject of the first problem at the end of the chapter, I don’t feel so generous. These examples are very shallow, too. A standard parametrization is given and then components of the first fundamental form are recited.

### Verdict

No way. This is not an undergraduate text after all. I had a very positive feeling about this book when I first looked through it. I found the geometry first attitude and the inside scoop approach appealing, but the price off-putting. I guess those things are still true.

This cannot be an introductory undergraduate textbook at UNI. The level is all wrong.

### The MathSciNet Review

Mathematical Reviews also has a discussion of the relative merits of the text.