For some years now, I, as well as a number of other contributors to this column, have on occasion expressed appreciation to Dover Publications for the service it provides to the mathematical community by re-issuing classic textbooks and making them available to a new generation at an affordable price. I now find myself in the position of once again thanking them, this time for publishing not just a re-issuance, but an actual new edition, of do Carmo’s classic textbook, first published by Prentice-Hall (now part of Pearson) in 1976.

Back in the day, it was fairly common for undergraduate mathematics departments to offer a course in differential geometry, which I suppose I should now refer to as “classical” differential geometry (curves and surfaces in the plane and three-space) to distinguish it from “modern” differential geometry (the study of differentiable manifolds). Of late, however, it seems to me (based on anecdotal evidence garnered from a highly unscientific survey) that not as many departments offer such a course. Yet, there must still be *some* market for books like this, because several have recently appeared, including a second edition of *Differential Geometry of Curves and Surfaces* by Banchoff and Lovett and another book with the same title by Kristopher Tapp.

Most books with titles like this offer similar content. This one is no exception. There are five chapters. The first two cover curves and surfaces, respectively, in three-space (and sometimes in the plane). The chapter on curve discusses both the local and (for plane curves) global theory. The local theory addresses things like the Frenet equations, while the global theory discusses curves “in the large”, including theorems like the Four-Vertex theorem and isoperimetric inequality. The definition of “regular surface” given in the second chapter is one that generalizes easily to manifolds, and much later in the text the author does discuss the notion of a differentiable manifold, first discussing “abstract surfaces” (i.e., 2-dimensional manifolds) and then noting that there was really nothing special about the number 2 and that the idea generalizes to *n*-dimensions. This chapter also introduces the first fundamental form, a quadratic form that can be used to compute a lot of information about the surface (length and angle, for example).

Chapter 3 introduces the Gauss map and explores its properties and applications. The derivative of the Gauss is not just any ordinary linear mapping; it turns out to be self-adjoint, and many of its algebraic features (eigenvalues, determinant, etc.) have geometric significance. This chapter explores these, introduces the second fundamental form, and ends with a few optional sections in which, among other things, ruled surfaces and minimal surfaces are discussed.

The intrinsic geometry of a surface is addressed in the next chapter. Intrinsic properties of a surface are those that depend only on the surface itself and do not depend on the way it is situated in the ambient Euclidean space. Another way to put this is that intrinsic geometry explores concepts that can be determined from an understanding of the first fundamental form. Highlights of this chapter are the Theorema Egregium of Gauss (good luck finding this in the Index, though; after fruitless searches under T and E, I finally found it under G for “Gauss theorem Egregium”) and the Gauss-Bonnet theorem, as well as the subjects of parallel transport and the covariant derivative.

The final chapter of the book is on global differential geometry, both of the surface and curves in three-space. This rather lengthy chapter is divided into eleven subsections, many independent of the others, each proving a “big” theorem in the subject; for example, the Hopf-Rinow theorem on geodesics. This chapter has a topological flavor, but the topological prerequisites (connectedness and compactness in Euclidean spaces) are largely summarized in an appendix to the chapter.

As the above summary of the contents of this book should make clear, there is a lot of material covered in this text, far more than can be covered in a single semester. There is probably enough for two semesters, in fact, assuming that there are universities that offer two full semesters of differential geometry as part of their course offerings. For those that don’t, the author offers several suggestions for using this book in a single semester.

One topic that is* not* covered in the text is differential forms. In choosing to omit this topic, the author has aligned himself with the vast majority of other undergraduate differential geometry textbook authors. Indeed, the only other text at this level that I can think of, off the top of my head, that *does* cover differential forms is O’Neill’s *Elementary Differential Geometry*; I’ve always viewed that text’s inclusion of the topic as one of the chief features distinguishing it from its competition. (There are, of course, books like Shurman’s recent *Calculus and Analysis in Euclidean Space* that cover these, but that’s not a book on differential geometry, although it has some nontrivial overlap with the subject.) I personally subscribe to the majority (and do Carmo’s) view, but instructors who wish to introduce the subject of differential forms might want to look at O’Neill rather than this book.

I don’t have a great deal of familiarity with the older edition of this book, but, as best as I can tell from a quick perusal of the two, the changes in this edition are local, rather than global. The only actual description of the changes occurs in the author’s brief preface to the new edition, where he states that he has included “many of the corrections and suggestions” sent to him by readers, so I assume those are the primary differences between the two editions. The tables of contents in both editions are substantially identical, differing only in page numbers; the new edition is about ten pages longer than the first.

As far as correction of errors go, there were apparently a number of them in the first edition; Bjorn Poonen has posted a seven-page, single-spaced errata. A spot-check shows that while many of the errors noted here have been corrected in the new edition, quite a few remain, and there are also some errors not included in Poonen’s list. Current errors range from minor typos (e.g., the statement of problem 3(c) on page 8 omits the parentheses when identifying a point) to more substantive mathematical errors. As an example of the latter, for example, the author gives, as a characterization of what it means for a function \(F\) to be continuous at a point \(p\), the statement “points arbitrarily close to \(F(p)\) are images of points sufficiently close to \(p\)”. What about constant functions? Points arbitrarily close to \(F(p)\) aren’t the images of *anything*.

Unfortunately, the bibliography of the first edition has hardly been changed at all for the second. Only two items now listed in the bibliography have copyright dates past 1976, and both of these actually relate to items originally listed in the first edition: one of them is the book version of a set of notes that were mentioned in the first edition bibliography, and the other is volumes I to V of Spivak’s mammoth *A Comprehensive Introduction to Differential Geometry*; only the first two volumes were published as of 1976. (There is actually a redundancy now in the references, since item 22 references all five volumes and item 12 references the first.) There have been a lot of books and articles written on this subject in the last 40 years, and it seems like a missed opportunity not to mention some of them. The failure to mention up-to-date literature exists in the body of the text as well; the author gives as a reference, for anybody wanting to pursue the subject of minimal surfaces, the 1969 book *A Survey of Minimal Surfaces* by Osserman, but does not mention that that edition is no longer in print, but a revised edition was published, also by Dover, in 2014.

A few other quibbles: I’m less than enthusiastic about some of the choices made by the author with regard to notation and terminology. The vector cross product, for example, is called the vector product and is denoted by a wedge (\(\mathbf{u}\wedge\mathbf{v}\)) instead of the more familiar “\(\times\)“ notation. The word “differentiable”, as applied to curves, means what I would call “infinitely differentiable” or “\(C^\infty\)”. The author’s “connected” is what modern texts call “path connected” (this isn’t serious because the concepts are equivalent for regular surfaces, but may still cause confusion; it probably should at least have been mentioned that other books define the word differently.) Finally, in defining the torsion of a curve, the author uses a sign convention that, though used by some other authors, is, I think, not the most common used one.

Moreover, there are occasional moments of sloppiness. The author defines a parametrized differentiable curve to be a mapping into \(\mathbb{R}^3\) satisfying certain properties. Yet almost immediately thereafter, one of his first examples of one is a mapping into \(\mathbb{R}^2\), not \(\mathbb{R}^3\). Now, one can certainly view \(\mathbb{R}^2\) as sitting inside \(\mathbb{R}^3\), but this point goes unmentioned by the author. In addition, as noted in Poonen’s errata, the author’s definition of the tangent line to a curve \(\alpha\) containing both \(\alpha(t)\) and \(\alpha’(t)\) is incorrect; I assume he means the line containing \(\alpha(t)\) in the direction of \(\alpha’(t)\).

I hope that the previous recitation does not convey the impression that I do not like the book. In fact, I do. It is, in general, clearly, albeit fairly concisely, written, and there are many examples and figures. There are a large selection of exercises of various degrees of difficulty, some but not all of which are given hints or solutions in the back of the book. The original edition enjoyed widespread use, for decades, as an undergraduate text, and many professional mathematicians learned differential geometry from this book.

However, this book was written 40 years ago, at a time when college textbooks were somewhat more demanding of the reader than they tend to be now. And, of course, this book now faces stiffer competition than it did in 1976. Shortly after it was published, for example, Prentice-Hall also published Millman and Parker’s *Elements of Differential Geometry*, a book that makes heavy use of linear algebra in a very appealing way and has remained, over the years, my “go to” source whenever I wanted to read something about differential geometry. It also has an excellent chapter discussing the transition from the classical to the modern theory. Kristopher Tapp’s recent book also looks very attractive, and has many full-color illustrations. Another interesting book is McCleary’s *Geometry from a Differentiable Viewpoint*, which develops classical differential geometry in the context of the foundations of Euclidean geometry and the history and basic properties of non-Euclidean geometry. And I would be remiss if I failed to note that Ted Shifrin has generously posted his textbook-quality lecture notes on the AMS *Open Math Notes* site. Likewise, David Henderson’s interesting book on differential geometry (intended for self-study) is available for free, chapter-by-chapter download, courtesy of Project Euclid. Students may find these sources to be a bit easier to read and follow than do Carmo’s text.

For all the reasons expressed above, I would not say that this is the *best* book available from which to learn this material. Nevertheless, it remains a valuable book, one that can be honestly described as a classic text in the area. Like all classics, it is certainly worth owning, especially given that it is currently selling on amazon.com for less than twenty dollars. I will therefore end this review as I began it: thank you, Dover Publications.

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