Differential Geometry of Curves and Surfaces

Mention the phrase “differential geometry” to a mathematician nowadays, and he or she will almost surely think of objects like manifolds, Riemannian metrics and tangent bundles. But it wasn’t always this way. Back in late 1960s and early 1970s, when I was an undergraduate, many university mathematics departments offered a course in differential geometry that focused on curves and surfaces in the plane and three-space. Textbooks like Differential Geometry of Curves and Surfaces by do Carmo, Elements of Differential Geometry by Millman and Parker, and Lectures on Classical Differential Geometry by Struik were standard texts for such courses.

Things seem different now. I suspect that far fewer colleges offer courses on classical differential geometry than was the case in my day, and that many — perhaps a substantial majority — of mathematics majors graduate from college without having ever heard of things like the Frenet frame or the Gauss-Bonnet theorem. Yet, there must still be some market for textbooks on the subject: the books by do Carmo and Struik are still in print (as Dover paperbacks), and the Pearson online catalog still lists Millman and Parker as available. Even better, new textbooks on the subject continue to be published (witness, for example, Differential Geometry of Curves and Surfaces by Banchoff and Lovett, McCleary’s Geometry from a Differentiable Viewpoint, A Differential Approach to Geometry by Borceux, O’Neill’s Elementary Differential Geometry, and Bär’s book of the same title, to name just a few). And now we have Tapp’s new book on the subject, a superb text that I think belongs at or near the top of the heap.

In terms of topic coverage, this book follows a pretty standard path. The first two chapters cover curves, both in the plane and in space; the first chapter takes us through the standard theory (unit speed curves, Frenet frame, invariance under rigid transformations) and the second is on “additional topics”, mostly global results that can, if deemed necessary by an instructor, be skipped. Examples include the four vertex theorem and Fenchel’s theorem.

The remaining four chapters are on surfaces. The author begins with some background multi-variable analysis (the derivative of a function from one Euclidean space to another), then proceeds to a definition of a regular surface and some basic properties. The next chapter introduces the various kinds of curvature, done via a linear-algebraic look at the Gauss map, and the second fundamental form. Chapter 5 is on geodesics and includes a proof of Gauss’ Theorema Egregium. The final chapter states and proves the Gauss-Bonnet theorem and, in a final section, surveys some other global results, including a classification (without proof) of complete surfaces with constant curvature.

The author has also followed the majority of textbook authors (O’Neill being the most well-known exception) in not using differential forms. For a first pass through the subject, I think this is a good decision — especially if, like Tapp, you are consciously trying to emphasize the geometric content of the subject.

If the choice of topics is fairly standard and therefore doesn’t readily distinguish this text from its competition, then what does? Several things, actually.

First, there is the physical appearance of the text. This is a visually appealing book, replete with many diagrams, lots of them in full color. I agree with the author that the use of color significantly helps to visualize things and assists in understanding the underlying mathematical ideas. Color is also used to block off, in boxes, definitions and statements of theorems. (I must confess to a certain cognitive dissonance here: the use of colored boxes in this way may have pedagogical benefits, but I do think that upper-level students should be taught to read mathematics without these trappings of more elementary texts.)

Second, the author’s writing style is extremely clear and well-motivated. Tapp is the author of several other books, including the very nice Matrix Groups for Undergraduates and the much more elementary Symmetry; he knows how to write in a way that students can understand. Differential geometry can be a very thorny subject, filled with cumbersome notation and technical ideas, but the author has strived, and (to the extent possible) succeeded, in making these ideas intuitively plausible to the student while still writing a book that is intellectually honest and rigorous. He has done so, in large part, by emphasizing the “geometry” in “differential geometry”.

Third, the author has attempted to keep things down to earth by mentioning concrete applications. These applications complement but do not overwhelm the development of the mathematical theory. They include, for example, the Foucault pendulum and Huygens’s tautochrone clock.

Finally, the selection of exercises is excellent. There are quite a few of them, both computational and proof-oriented, and they span a broad range of difficulty. Few if any are “busy work” problems; they are generally quite illuminating. Some develop new material; one exercise, for example, asks the reader to explain how a planimeter works and how it is related to Green’s theorem. No solutions are provided in the text (and there is, to my knowledge, no solutions manual for instructors).

My considerable enthusiasm for this book notwithstanding, I did find a few (relatively minor) nits to pick. For one thing, the bibliography is inadequate. What bibliography there is, consists of a handful of entries in a section called “Recommended Excursions” and lists books covering topics of a more advanced nature — knots, minimal surfaces, etc. There is, for example, no listing of other books on differential geometry at or near the level of this text. I understand that Macy’s does not advertise for Gimbel’s, but in the case of textbooks I strongly believe there is some value in having students peruse other books, just to see alternative approaches. And if one is going to list “recommended excursions”, then it would be helpful to have some annotated comments about what one is likely to find in the listed book.

Additionally, I think that some discussion (even a heuristic, expository one) of the abstract idea of a manifold would have been useful. Unfortunately, the word “manifold” does not, as far as I can tell, appear anywhere in this book. One of the values, for math majors, of studying classical differential geometry is to provide motivation for these more modern ideas. The last chapter of Millman and Parker, for example, is spent “motivating the abstract definitions of manifold theory so that if the reader wishes to study some of the more advanced works in the field he or she may do so without being attacked by a formalism which seems to be totally unmotivated.” I have always found this chapter to be a very useful one. It would have been nice for a student finishing this book to have some idea of how the notion of a surface can be generalized.

Notwithstanding these concerns, this is still the book I would use as a text for a beginning course on this subject. It would not surprise me if it quickly becomes the market leader.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.