![]() … A distinctive feature of the book is that, apart from a large number of exercises, it also contains many original problems with outlines of solutions.” (A. Only a basic background in analysis and linear algebra is needed to follow the presentation. “This book, written by one of the Russian masters, offers a comprehensive introduction to classical differential geometry of curves and surfaces. "Toponogov’s ‘concise guide’ to elementary differential geometry has the potential to be a useful reference and/or review book …. ![]() Numerous illustrations make the reading enjoyable." (Wolfgang Kühnel, Mathematical Reviews, Issue 2006 m) … the book is very welcome since it is an original contribution in various aspects and gives a number of geometric insights …. … the book is rich in geometry and concrete examples. … It can be recommended for first-year graduate students and also for use in the classroom. "This book by the late author covers … the subjects which are normally taught in a course on the differential geometry of curves and surfaces. The author includes a number of examples, illustrations, and exercises making this book well-suited for students or for self-study. A distinctive feature of the book is a large collection (80 to 90) ofnonstandard andoriginalproblems that introduce the student into the real world of geometry. The book provides an excellent introduction to the differential geometry of curves, surfaces and Riemannian manifolds that should be accessible to a variety of readers. In the last case, the formulations are discussed in detail. Bernstein’s theorem about saddle surfaces. Pogorelov’s theorem about rigidity of convex s- faces, and S.N. Aleksandrov’s comparison theorem about the angles of a triangle on a convex 1 surface, formulations of A.V. The second stream contains more dif?cult and additional material and for- lations of some complicated but important theorems, for example, a proof of A.D. It follows that df can be interpreted as a section of T M. If f : M R is a smooth function, its differential dx f is a linear form on Tx M, so dx f Tx M. ![]() It includes the whole of Chapter 1 except for the pr- lems (Sections 1.5, 1.7, 1.10) and Section 1.11, about the phase length of a curve, and the whole of Chapter 2 except for Section 2.6, about classes of surfaces, T- orems 2.8.1–2.8.4, the problems (Sections 2.7.4, 2.8.3) and the appendix (S- tion 2.9). This is a vector bundle, denoted by T M or 1 M, whose fiber at x M is the dual Tx M (Tx M ) of the tangent space Tx M. ![]() And finally, to familiarize geometry-oriented students with analysis and analysis-oriented students with geometry, at least in what concerns manifolds. It contains a small number of exercises and simple problems of a local nature. Second, to illustrate each new notion with non-trivial examples, as soon as possible after its introduc tion. The ?rst stream contains the standard theoretical material on differential ge- etry of curves and surfaces. Differentialgeometrische Konzepte fr Dreiecksnetze, Master thesis, University of Karlsruche, 135 pp., 2004. The material is given in two parallel streams. This concise guide to the differential geometry of curves and surfaces can be recommended to ?rst-year graduate students, strong senior students, and students specializing in geometry. ![]()
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