Characterizations of umbilic hypersurfaces in warped product manifolds

TL;DR

This paper provides new characterizations of umbilic hypersurfaces in various warped product manifolds, extending classical Euclidean results through integral formulas and inequalities.

Contribution

It introduces generalized characterizations of umbilic hypersurfaces in warped products, broadening the scope of classical theorems.

Findings

New integral formula for hypersurfaces

Generalizations of Jellet-Liebmann and Alexandrov theorems

Characterizations applicable to space forms and black hole spacetimes

Abstract

We consider closed orientable hypersurfaces in a wide class of warped product manifolds, which include space forms, deSitter-Schwarzschild and Reissner-Nordstr\"{o}m manifolds. By using a new integral formula or Brendle's Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These results can be viewed as generalizations of the classical Jellet-Liebmann theorem and the Alexandrov theorem in Euclidean space.