Differential geometry. Curves and surfaces (Q2879622)

This textbook is devoted to the differential geometry of curves and surfaces in three-dimensional Euclidean space. In addition, it also deals with non-Euclidean, Riemannian, fractal, and discrete geometries, and there are short notes on knots, soliton equations, and general relativity. It is a revised and extended edition of an earlier book by the same author in German with the same title.NEWLINENEWLINEA book on this subject would be expected to teach the basic ideas; additionally it might pursue special goals: striving for good didactics, giving pleasure with classification and pictures of curves and surfaces (the ``zoo'' of these objects), global aspects, theory of singularities, mappings aimed at cartography, other practical applications, excursions into history. The present book actually covers all these topics; it gives a good deal of material in only 263 pages. The titles of the chapters read: 1. Local theory of curves. 2. Plane curves. 3. Global theory of plane curves. 4. Local theory of surfaces. 5. Intrinsic surface geometry. 6. Special surfaces. 7. Mappings of surfaces. 8. Non-Euclidean geometry. 9. Global properties of surfaces. 10. Outlooks. 11. Sketch of the history of differential geometry.NEWLINENEWLINEHighlights are the integral theorem of Gauss-Bonnet, which is presented in several forms and is proved in the text, and other results of global character.NEWLINENEWLINEThe text is supplemented by 64 exercises the solutions of which are given at the end of the book. An appendix with explanations of basic notions, a subject index, and last not least a classified and extensive list of references and recommended literature. The book, being both comprehensible and modern, can be highly recommended to students of mathematics, sciences, and engineering as well as to more advanced readers interested in differential geometry. An English edition is desirable.