We essentially reproduced a large portion of the proof of Gauss' lemma (cf. So we conclude that $\left = 0$, which was our goal. Let $\phi_t$ be the flow of $X$ on $U$, and let $q=\exp_p\big)'(0) = X_p = 0, I have a "proof", but it doesn't use the fact that $X_p=0$, so I must be missing something. 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.Exercise 3.5b of do Carmo's Riemannian Geometry asks the reader to prove that given a Killing field $X$ on a manifold $M$, an isolated zero $p$ of $X$, and a normal neighborhood $U$ of $p$ in which $X$ has no other zeros, $X$ is tangent (in $U$) to the geodesic spheres centered at $p$. It really is a shame that I hadn’t.īut I can’t assign this to the undergraduates at UNI. I wish it had been my textbook and that I had read any of it before yesterday. The current text requires a more advanced reader. 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. But in the bibliography to O’Neill’s book, he has this little bit to say: I won’t include the MathSciNet review for this book, as it doesn’t say anything useful. (Maybe I have to exclude graph theory stuff.) What O’Neill Said I would bet that most of these theorems would be more recent than anything else in a typical undergraduate curriculum. 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. (Mashed potatoes and homemade gravy! Pumpkin Pie! Pecan Pie!) Certainly more than in the other books I have read so far. Many landmark results in differential geometry show up in the final chapter. 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. In each case the relevant properties are carefully discussed. 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. I especially like the fact that examples reappear. The discussions are clear and there are ample examples and exercises. 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.Ī figure on the construction of the tangent plane Assuming that students have seen the inverse function theorem in several variables is not workable at UNI by a long shot. This is too high a bar for most undergraduate classes in the United States. Points of Noteĭo Carmo assumes that his readers have some experience with advanced calculus, and even some basic analysis. I read chapter two, because that is where surfaces are introduced. 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. The essentials seem to be there, but do Carmo wants to move on and get to some surfaces. For me, it is a bit like going home for Thanksgiving.Īs you can see, curves get short shrift. A simple Google search also yields a lovingly compiled list of errata.īecause I used this books brother at a formative stage of my mathematical training, reading it is very comfortable. The text is published by Prentice Hall, which these days is an imprint of Pearson. (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.) 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. I know many people who were taught from it as undergraduates. This text is very popular with professional mathematicians, and it has been a standard for a long time. It is time to look at a classic: Manfredo do Carmo’s Differential Geometry of Curves and Surfaces. Do Carmo’s Differential Geometry of Curves and Surfaces
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