![]() the normal curvatures of all the unit tangent vectors of any point on S are constant and identical), then S is either part of a plane or part of a sphere in R 3.I thought it was the coolest magic trick I’d ever seen. ![]() I remember proving for a homework problem set that if a surface S in R 3 consists entirely of umbilic points (i.e. Armed with basic calculus, the first and second fundamental forms and linear algebra in your toolbox, it seems if you’re clever enough, you can prove just about anything about any geometric structure in R n. I first studied classical differential geometry out of Do Carmo’s Differential Geometry of Curves and Surfaces and the 2 nd edition of O’Neill’s Elementary Differential Geometry. ![]() It always seemed to me to be an incredibly intuitive subject - especially the classical version in Euclidean space. As expected, there is quite a bit of overlap, but this book stands very well on its own.ĭifferential geometry has always been one of my favorite subjects. ![]() One cannot help but compare the two books, despite the more modest aims of Taimanov’s solo book. Any library of a practicing mathematician or physicist that does not include a copy will be very much poorer for it. MGSF has already been reviewed here and there’s no need to dwell on its level of excellence, but I will say that there is probably no text that better exemplifies the unity approach above better then this one. Prasolov and the more advanced and awesome tome, Modern Geometric Structures and Fields, coauthored with his mentor, S. His name may also be familiar to students, in that he has co-authored two popular textbooks that have been translated into English and published by the AMS: Geometry, coauthored with the eminent topologist V. The author’s name should be familiar - a doctoral student of Novikov, he has published many new results on dynamical systems theory. Which brings me in a roundabout way to the blue paperback before me titled Lectures On Differential Geometry by Iskander A. It would quickly die.” The partial reunification of theoretical physics and pure mathematics brought about in the last thirty years, and the resulting explosive developments in operator theory, noncommutative geometry, quantum algebra and deformation theory, have underscored Lebesgue’s assessment. I think Lebesgue stated my opinion on this matter best: “Reduced to general theories, mathematics would become a beautiful form without content. Cut off from the natural sciences, mathematics seems to become merely an exercise in logic splitting and diagram chasing. Having been trained in biochemistry with heavy leanings towards physical chemistry before being seduced into pure mathematics, I have a deep sympathy for this viewpoint. Mathematics is the part of physics where experiments are cheap.” Physics is an experimental science, a part of natural science. The Russian school has traditionally believed in the inseparability of the mathematical and empirical sciences, an approach summed up by the words of Arnold: “Mathematics is a part of physics. But even more significant is the overriding philosophy of the department, best exemplified by the their very name: Mechanics and Mathematics. Komologrov, Gelfand, Naimark, Petroskii, Landau, Fedeev, Postnikov, Shafarevich, Vinberg, Novikov, Arnold, Zorich - just to name a few. The names affiliated with this Mecca of scientific study are legendary. I have a profound fascination for the works of Russian mathematicians, particularly those by the faculty or students of the legendary Mechanics and Mathematics Department at Moscow State University. I’m going to begin this review by saying something completely ridiculous that is nevertheless true: I love Russian mathematics textbooks.
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