Totally Umbilical Hypersurfaces of Product Spaces

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 $I\times_ωM$ -- 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 $I\times_ω\mathbb{S}^n$ and $I\times_ω\mathbb{H}^n.$

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