Totally Umbilical Hypersurfaces of Product Spaces

TL;DR

This paper characterizes totally umbilical hypersurfaces in product and warped product spaces, extending existing classifications to higher dimensions and more general settings using isoparametric families.

Contribution

It provides a comprehensive characterization of totally umbilical hypersurfaces in product and warped product manifolds, generalizing previous results to higher dimensions and broader classes.

Findings

Complete classification of hypersurfaces in $ ext{S}^n imes ext{R}$ and $ ext{H}^n imes ext{R}$.

Extension of results to arbitrary warped products involving spheres and hyperbolic spaces.

Identification of hypersurfaces as local graphs over isoparametric families.

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 hypersurfaces of $\mathbb{S}^2\times\mathbb{R}$ 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 $I\times_\omega\mathbb{H}^n.$