Totally umbilical hypersurfaces of product spaces

Given a Riemannian manifold M, and an open interval I subset of R, we characterize nontrivial totally umbilical hypersurfaces of the product M x I-as well as of warped products I x(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 S-n x R and H-n x R the results by Souam and Toubiana on the classification of totally umbilical surfaces in S-2 x R and H-2 x R. It is also shown that an analogous classification holds for arbitrary warped products I x(omega) S-n and I x(omega) H-n.