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Characterizations of umbilic points of isometric immersions in Riemannian and Lorentzian manifolds. (English) Zbl 1357.53065
Summary: Several characterizations of umbilic points of submanifolds in arbitrary Riemannian and Lorentzian manifolds are given. As a consequence, we obtain new characterizations of spheres in the Euclidean space and of hyperbolic spaces in the Lorentz-Minkowski space. We also prove the Lorentzian version of a classical result by E. Cartan [Leçons sur la géométrie des espaces de Riemann. Paris: Gauthier-Villars (Cahiers scientifiques publiés sous la direction de G (1928; JFM 54.0755.01)].
MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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