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Several characterizations of umbilic points of submanifolds in arbitraryRiemannian and Lorentzian manifolds are given. As a consequence, we obtain newcharacterizations of spheres in the Euclidean space and of hyperbolic spaces inthe Lorentz-Minkowski space. We also prove the Lorentzian version of aclassical result by Cartan.

Characterizations of umbilic points of isometric immersions in Riemannian and Lorentzian manifolds