Totally umbilical hypersurfaces of product spaces

Abstract

Given a Riemannian manifold M,  and an open interval $I\subset \mathbb {R},$ we characterize nontrivial totally umbilical hypersurfaces of the product $M\times I$—as well as of warped products $I\times _\omega M$—as those which are local graphs built on isoparametric families of totally umbilical hypersurfaces of M. By means of this characterization, we fully extend to $\mathbb {S}^n\times \mathbb {R}$ and $\mathbb {H}^n\times \mathbb {R}$ the results by Souam and Toubiana on the classification of totally umbilical surfaces in $\mathbb {S}^2\times \mathbb {R}$ and $\mathbb {H}^2\times \mathbb {R}.$ It is also shown that an analogous classification holds for arbitrary warped products $I\times _\omega \mathbb {S}^n$ and $I\times _\omega \mathbb {H}^n.$