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Elementary differential geometry pressley10/19/2023 ![]() ![]() ![]() Gouvêa is the author of several books, including, most recently, Math through the Ages, written in collaboration with William Berlinghoff. (This may require some judicious skipping of earlier sections.) All in all, I was quite happy with the book.įernando Q. On the other hand, the book does include several versions of the Gauss-Bonnett theorem, allowing the professor to end the course with a bang. The section on geodesics has essentially nothing on parallel transport, which is a pity. Since my students did know what linear transformations were, I used that language on the other hand, since they had never seen the big theorem, I presented an easy proof. Pressley uses matrices and ends up appealing to a big theorem (self-adjoint operators have real eigenvalues), but he seems to avoid using linear transformations directly. (And that's what I did in class.) The other problem with this section is the strangely uneven use of linear algebra. I felt it would have been better to actually introduce the covariant derivative and define the Weingarten operator properly. There's no simple way to define this in Pressley's setup, so it ends up appearing only after quite a lot of buildup, and basically as a product of two matrices that just happens to include a lot of information. One place where this approach runs into problems is with respect to the Weingarten operator (aka the shape operator). Mention the Christoffel symbols very quickly, but don't do very much with them. Use moving frames without mentioning connections. Instead of covariant derivatives, use derivatives with respect to local coordinates. Pressley takes the simplest route with respect to all the technical setup: avoid all of it. When I taught the course this Fall, I used Andrew Pressley's newish book, Elementary Differential Geometry. At the same time, one should try to teach a course that will prepare students for future courses, that includes some points of contact that can help students deal with the heavier notation they may meet in future courses. It seems necessary to avoid most of this theoretical baggage in an undergraduate course. 55 ac G08) 09-782 ena nape om URL: mreapich am ‘rican ta Asotin Chance Vol No 1S ardebyKSand KW Hee Tree. Modern differential geometry is built upon a very elaborate theoretical framework (from differential forms all the way to connections and cohomology) and a correspondingly elaborate notation. Andrew Pressley Department of Mathematics, King’s College, The Strand, London WC2R 2LS, UK erin some reprecedy Kind prion of ‘Sch Spent Poe of be GaUS Mather and Sot yen 2380 SE, Kent Kangley Road, Male Vall, WA SHO, ‘USA. Whenever I do, however, I find I have to make up my mind on a very basic question. Though I am definitely not an expert on the subject, I find elementary differential geometry fascinating and I love to teach it. ![]()
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