![]() ![]() Is not enough, I'd move on to Kosinski's Differential Manifolds These two should get you through the basics. Isotopies, and classification of two dimensional surfaces. Map degree, intersection numbers, Morse theory, cobordisms, Transversality, vector bundles, tubular neighborhoods, collars, The topics covered include the basics of smooth manifolds,įunction spaces (odd but welcome for books of this class), Those of Lee and Hirsch contains many more typos than Lee. Also, the proofs are much more brief then This is more advanced then Lee and leans more Vector bundles, submersions, immersions, embeddings, Whitney'sĮmbedding theorem, differential forms, de Rham cohomology, Lieĭerivatives, integration on manifolds, Lie groups, and Lie algebras.Īfter finishing with Lee, I'd move on to Hirsch's Differential ![]() Topics covered include the basics of smooth manifolds, smooth It covers the basics in a modern, clear and rigorous manner. I'd start with Lee's Introduction to Smooth Manifolds. The books I've recommended, except possibly Aubin, aim for this. It is possible to do almost everything without them. They have no geometric meaning and just get in the way. One piece of advice: Avoid using local co-ordinates and especially those damn Christoffel symbols. He has a book on Riemannian geometry, but I don't know it very well. One that I also really like is "Riemannian Geometry" by Gallot, Hulin, Lafontaine.Īnd, back in the day, many of us also learned a lot by reading Thurston's notes on 3-manifolds.įor a more analysis-oriented book, check out Aubin's "Some Nonlinear Problems in Riemannian Geometry". I'm recommending only older books, because I haven't kept up with all the newer books out there. The rest of the book is great, of course.Īnother classic that ties in well with Lie groups is Cheeger and Ebin's "Comparison Theorems in Riemannian Geometry". And it's really about differential topology (that is the title after all) and not differential geometry.įor a really fast exposition of Riemannian geometry, there's a chapter in Milnor's "Morse Theory" that is a classic. I love Guillemin and Pollack, but it is just a rewrite for undergraduates of Milnor's "Topology from a Differentiable Viewpoint". Spivack is for me way too verbose and makes easy things look too complicated and difficult. Of course, this is a natural thing to do, while you're trying to work out your own proof anyway. So you'll go nuts, unless you have your own notation and you translate whatever you're reading into your own notation. Why? Because it appears that each differential geometer and therefore each differential geometry book uses its own notation different from everybody else's. Second, follow the advice of another former Harvard professor and develop your own notation. My interpretation of this is to look first at only the statements of the definitions and theorems and try to work out the proofs yourself. He would point to a book or paper and say, "You should know everything in here but don't read it!". One of them, Degeneration of Riemannian metrics under Ricci curvature bounds, is available on Amazon.įirst, follow the advice that a former Harvard math professor used to give his students. They lay the groundwork for his recent work on Ricci curvature. He is relying on notes he has written, which I can recommend, at least for a nice overview of the subject. So far, I like Petersen's book best.Īlso, as it happens, Cheeger is teaching a topics course on Ricci curvature. In particular, I wanted to do global Riemannian geometric theorems, up to at least the Cheeger-Gromoll splitting theorem. I also wanted to focus on differential geometry and not differential topology. I am teaching a graduate differential geometry course focusing on Riemannian geometry and have been looking more carefully at several textbooks, including those by Lee, Tu, Petersen, Gallot et al, Cheeger-Ebin. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |