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Introduction to differential geometry and Riemannian geometry. English translation. (English) Zbl 0175.48101

Mathematical Expositions 16. Tornoto, Ont., Canda: University of Toronto Press. xii, 370 p. (1968).
An introduction to the differential geometry of curves and surfaces in Euclidean \(R^s\) and to \(n\)-dimensional Riemannian geometry. Chapter I contains preliminaries about topological and metric spaces, transformation groups, vector algebra and vector calculus. Chapters II–IV explain basic facts for curves and surfaces in Euclidean \(R^s\). Chapter V develops the tensor algebra for the purpose of further applications. The deeper parts of surface theory in Euclidean \(R^3\) are investigated in Chapter VI (formulae of Weingarten and Gauss), Chapter VII (geodesics), Chapters VIII–IX (isometric and other mappings of surfaces and their applications), Chapter X (global geometry), Chapter XI (absolute differentiation and connexion on surfaces). Chapter XII (minimal surfaces, modular surfaces, envelopes of families of surfaces etc.). Chapters XIII–XVI contain basic facts about \(n\)-dimensional Riemannian geometry (foundations, absolute differentiation and connexions, special properties, hypersurfaces).
The presentation is clear and good understandable. The general matter is always illustrated with convenient examples. There are also many interesting exercises and problems which are answered at the end of the book.

MSC:

53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
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