Next up: Lectures on Classical Differential Geometry, 2nd Ed by Dirk J. Struik. This is an oldie and a goodie. The text is currently available from Dover Publications, and you can pick it up from Amazon.com, too.

The original version of this book was published in 1950, and some material was added for a second edition in the 1980’s. This is a truly wonderful little book, but I can’t recommend it as a text for today’s students.

I bought my copy back in 1994 when I took an undergraduate differential geometry course at Ohio State from Neil Falkner. I felt like I couldn’t really absorb the book then, and I was one of the more successful students in that particular course. I am pretty sure that means it is an inappropriate choice for the average undergraduate. Looking at it again, I can’t see what caused me so much trouble, except that the language is old-fashioned. This add an extra layer between the student and the material.

I also know that I learned most by going to lecture and that I didn’t really study my textbooks. (I hadn’t learned how to do that, yet.) So maybe more of this is my fault than Struik’s, but I bet lots of students don’t really study their textbooks effectively. Absent taking time for direct instruction on how to read a math book, it falls on the instructor to pick something easier to digest.

The Chapter Headings

The book is organized into the following chapters:

• Curves
• Elementary Theory of Surfaces
• The Fundamental Equations
• Geometry on a Surface
• Some Special Subjects

There is also an appendix called “The Method of Pfaffians in the theory of curves and surfaces.” As was planned before, I read the beginnings of chapter 2.

Features of the Book

I have a lot to say about this book because I love it. It is like a little security blanket. As a slim little Dover volume, it has been my companion across campus for the dull minutes between meetings many times. (It always looks impressive when you carry a math book. I suggest it to all of you. You could carry one of those big yellow Springer books, but this one is easier on your arms.)

The material covered is roughly what one could do in a one-semester course, with a few advanced topics thrown in.

There are lots of references to the old literature. These appear throughout the text, but are summarized nicely in a bibliography that appears between the preface and the first chapter. These are kindly separated by the language in which they are written. Reading that list again makes me want to put some serious language study on my calendar. (My French is pretty rusty, and I have always wanted to learn some German and some Russian.) And his discussion in the preface places his book squarely between the research & organizational monographs of Darboux and Bianchi and the serious modern textbooks like do Carmo.

This book is later than Kreyszig’s, but is somehow of the same era. Struik’s work is streamlined and leaves tensor analysis by.

The writing is rather terse and the terminology is a bit unfamiliar for today’s students. There are examples and exercises, but reading this book and completing the exercises requires more independence of the reader than any of the other books I have read so far. The basic set of definitions for a surface and enough associated machinery to construct a normal and draw lots of pictures happens in just over three full pages. There is not a lot of discussion about the idea of a surface, you just lay out what you want and get going.

There are solutions to some of the exercises in the back. But it would be unfair to call them solutions, really it is more of an answer key.

There is a collection of advanced exercises near the end of the text. These are very challenging, and would make good projects for ambitious students.

Verdict

This is not a suitable text for today’s undergraduates. But if you want to have a selection of differential geometry texts as resources, this should be part of your collection. And at the price Dover Publications charges, it is hard to pass this one up.