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

TL;DR

This paper provides new characterizations of umbilic points in submanifolds within Riemannian and Lorentzian manifolds, leading to novel criteria for spheres and hyperbolic spaces, and extends classical results to Lorentzian geometry.

Contribution

It introduces new characterizations of umbilic points in both Riemannian and Lorentzian contexts, including Lorentzian analogs of classical theorems.

Findings

New characterizations of umbilic points in Riemannian and Lorentzian manifolds

Characterizations of spheres in Euclidean space

Characterizations of hyperbolic spaces in Lorentz-Minkowski space

Abstract

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.