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 Cartan.
"Characterizations of Umbilic Points of Isometric Immersions in Riemannian and Lorentzian Manifolds." Taiwanese J. Math. 20 (5) 1041 - 1052, 2016. https://doi.org/10.11650/tjm.20.2016.7383