Basic Elements of Differential Geometry and Topology

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Free Download Basic Elements of Differential Geometry and Topology by S. P. Novikov , A. T. Fomenko
English | PDF | 1990 | 499 Pages | ISBN : 0792310098 | 28.9 MB
For a number of years, beginning with the early 70's, the authors have been delivering lectures on the fundamentals of geometry and topology in the Faculty of Mechanics and Mathematics of Moscow State University. This text-book is the result of this work. We shall recall that for a long period of time the basic elements of modern geometry and topology were not included, even by departments and faculties of mathematics, as compulsory subjects in a university-level mathematical education. The standard courses in classical differential geometry have gradually become outdated, and there has been, hitherto, no unanimous standpoint as to which parts of modern geometry should be viewed as abolutely essential to a modern mathematical education. In view of the necessity of using a large number of geometric concepts and methods, a modernized course in geometry was begun in 1971 in the Mechanics division of the Faculty of Mechanics and Mathematics of Moscow State University. In addition to the traditional geometry of curves and surfaces, the course included the fundamental priniciples of tensor analysis, Riemannian geometry and topology. Some time later this course was also introduced in the division of mathematics. On the basis of these lecture courses, the following text-books appeared:

S.P. Novikov: Differential Geometry, Parts I and II, Research Institute of Mechanics of Moscow State University, 1972. S.P. Novikov and A.T. Fomenko: Differe
Institute of Mechanics of Moscow State University, 1974.
The present book is the outcome of a revision and updating of the above-mentioned lecture notes. The book is intended for the mathematical, physical and mechanical education of second and third year university students. The minimum abstractedness of the language and style of presentation of the material, consistency with the language of mechanics and physics, and the preference for the material important for natural sciences were the basic principles of the presentation.
At the end of the book are several Appendices which may serve to diversify the material presented in the main text. So, for the purposes of mechanical and physical education the information on elementary groups of transformations and geometric elements of variational calculus can be extended using these Appendices. For mathematicians, the Appendices may serve to enrich their knowledge of Lobachevsky geometry and homology theory. We believe that Appendices 2 and 3 are very instructive for those who wish to become acquainted with the simplest geometric ideas fundamental to physics. Appendix 7 includes selected problems and exercises for the course.
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