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Curvature in the eighties. (English) Zbl 0722.53001
This is an expository article reporting on several developments dealing with curvature in the eighties. Anything published before 1980 is considered as classic. In view of the vastness of the subject mainly topics are presented which are easily accessible to a reader unfamiliar with the field and which give a feeling for the various forms of curvature, both intrinsic and extrinsic. The results discussed by the author are mainly of two types: inequalities involving integrals of curvatures, and implications of one or another type of curvature being constant.
The first three chapters deal with curves and surfaces in 3-space. The most important integral inequalities are discussed in this context. Then surfaces of constant mean curvature are considered. Special attention is given to the approach to Alexandrov’s theorem initiated by A. Ros [J. Differ. Geom. 27, No.2, 215-220 (1988; Zbl 0638.53051)]. In chapter four a proof of this theorem is presented as it appeared in the paper by S. Montiel and A. Ros [Differential Geometry. A symposium in honour of Manfredo do Carmo, Proc. Int. Conf., Rio de Janeiro/Bras. 1988, Pitman Monogr. Surv. Pure Appl. Math. 52, 279-296 (1991; Zbl 0723.53032)]. Finally some further results on hypersurfaces of Euclidean spaces and of spheres are described, as well as some purely Riemannian matters. This includes a discussion of a breakthrough on a longstanding conjecture of Lichnerowicz which was achieved by Z. I. Szab√≥ [J. Differ. Geom. 31, No.1, 1-28 (1990; Zbl 0686.53042)].

MSC:
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C40 Global submanifolds
53C20 Global Riemannian geometry, including pinching
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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