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Géométrie différentielle: variétés, courbes et surfaces. (Differential geometry: manifolds, curves and surfaces). (French) Zbl 0619.53001
Mathématiques. Paris: Presses Universitaires de France. III, 511 p.; FF 240.00 (1987).
This is an extension of the authors’ previous monograph ”Géométrie différentielle” (Paris 1972; Zbl 0251.53001). According to the demand which arose during the use of that first edition, two chapters on the theory of surfaces in Euclidean 3-space have been added. The first one deals with the standard material from the local theory of surfaces. This is illustrated by a lot of examples. The second additional chapter presents the highlights of the global theory of surfaces: geodesics, variational formulas, Gauss-Bonnet theorem, isoperimetric inequalities, rigidity theorems, curvature of fixed sign, minimal surfaces, surfaces of constant mean curvature, Weingarten surfaces, cyclides of Dupin etc. These two chapters have the character of a survey. For detailed proofs the reader generally has to consult other sources given by the authors in their long list of references.
Reviewer: Bernd Wegner

MSC:
53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
58-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis