Differential Geometry and Topology

Differential geometry and topology are two of the youngest but most developed branches of modern mathematics. They arose at the juncture of several scientific trends (among them classical analysis, algebra, geometry, mechanics, and theoretical physics), growing rapidly into a multibranched tree whose fruits proved valuable not only for their intrinsic contribution to mathematics but also for their manifold applications. Some of these applications are mentioned in this book. With such a lot of "parents," modern differential geometry and topology naturally inherited many of their features; being at the same time young areas of mathematics, they possess vivid individuality, the main characteristics being, perhaps, their universality and the synthetic character of the methods and concepts employed in their study. Here geometric concepts and descriptiveness, the language of algebra and functional and differential methods, and so on, are interlinked. This synthetic character of posing problems and finding their solution is, to a certain extent, in tune with the natural sciences of the Renaissance, when mathematics, mechanics, and astronomy were considered as the unique system of knowledge of the laws of the Universe. Though not claiming to be that all-encompassing, modern geometry enables us, nevertheless, to solve many applied problems of fundamental importance.