It is time to look at a classic: Manfredo do Carmo’s Differential Geometry of Curves and Surfaces. This text is very popular with professional mathematicians, and it has been a standard for a long time. I know many people who were taught from it as undergraduates.
The course I took did not use this, but I used do Carmo’s Riemannian Geometry as a text during my first year of graduate studies. (The two books are sometimes referred to as “baby do Carmo” and “grown up do Carmo” when it is not otherwise clear which one is being discussed.)
Because I used this books brother at a formative stage of my mathematical training, reading it is very comfortable. For me, it is a bit like going home for Thanksgiving.
The Chapter Headings
The book is organized like so:
- Regular Surfaces
- The Geometry of the Gauss Map
- The Intrinsic Geometry of Surfaces
- Global Differential Geometry
As you can see, curves get short shrift. The essentials seem to be there, but do Carmo wants to move on and get to some surfaces. And there is lots of good stuff on surfaces! Essentially all of the theorems I learned in a first year Riemannian geometry course in graduate school makes an appearance here in the special case of a surface. (Mmmmm, turkey.)
I read chapter two, because that is where surfaces are introduced.
Points of Note
do Carmo assumes that his readers have some experience with advanced calculus, and even some basic analysis. This is too high a bar for most undergraduate classes in the United States. Assuming that students have seen the inverse function theorem in several variables is not workable at UNI by a long shot. There is an appendix to the second chapter with a short review of the notions of continuity and differentiability for functions from one Euclidean space to another, but the author really just supposes that you are ready to talk at this level.
The writing is very good. The discussions are clear and there are ample examples and exercises. I especially like the fact that examples reappear. For example, the sphere shows up at least four times in the chapter: on page 55 to discuss its construction via overlapping coordinate patches; on page 86 to exhibit the idea of a tangent plane; on page 95 where we compute the first fundamental form; and on page 104 where we see it as an oriented surface. In each case the relevant properties are carefully discussed.
This book has been adopted by so many different classes at strong universities over the last forty years that I am sure you can find every exercise discussed in detail somewhere on the internet.
Many landmark results in differential geometry show up in the final chapter. (Mashed potatoes and homemade gravy! Pumpkin Pie! Pecan Pie!) Certainly more than in the other books I have read so far. I skimmed through the final chapter and noted Liebmann’s theorem on the rigidity of the sphere, the Hopf-Rinow theorem on complete surfaces, Bonnet’s theorem on the compactness of positively curved complete surfaces, theorems of Hadamard on simply connected surfaces having curvature bounds, Hilbert’s theorem on the impossibility of embedding the hyperbolic plane as a regular surface in Euclidean three-space, some global theorems on curves, and more. I would bet that most of these theorems would be more recent than anything else in a typical undergraduate curriculum. (Maybe I have to exclude graph theory stuff.)
What O’Neill Said
I won’t include the MathSciNet review for this book, as it doesn’t say anything useful. But in the bibliography to O’Neill’s book, he has this little bit to say:
The book by do Carmo is a clearly written exposition of differential geometry with a viewpoint similar to this one, but at a more advanced level.
I have to agree. The current text requires a more advanced reader. I like both books.
I love this book. I wish it had been my textbook and that I had read any of it before yesterday. It really is a shame that I hadn’t.
But I can’t assign this to the undergraduates at UNI. It would destroy them. Maybe undergraduates at Chicago could handle it, the strongest students at UNI could if they had had analysis already… but that is not my audience.