  We give a complete classification of umbilical submanifolds of arbitrary
dimension and codimension of $\Sf^n\times \R$, extending the classification of
umbilical surfaces in $\Sf^2\times \R$ by Rabah-Souam and Toubiana as well as
the local description of umbilical hypersurfaces in $\Sf^n\times \R$ by Van der
Veken and Vrancken. We prove that, besides small spheres in a slice, up to
isometries of the ambient space they come in a two-parameter family of
rotational submanifolds whose substantial codimension is either one or two and
whose profile is a curve in a totally geodesic $\Sf^1\times \R$ or $\Sf^2\times
\R$, respectively, the former case arising in a one-parameter family. All of
them are diffeomorphic to a sphere, except for a single element that is
diffeomorphic to Euclidean space. We obtain explicit parametrizations of all
such submanifolds. We also study more general classes of submanifolds of
$\Sf^n\times \R$ and $\Hy^n\times \R$. In particular, we give a complete
description of all submanifolds in those product spaces for which the tangent
component of a unit vector field spanning the factor $\R$ is an eigenvector of
all shape operators. We show that surfaces with parallel mean curvature vector
in $\Sf^n\times \R$ and $\Hy^n\times \R$ having this property are rotational
surfaces. We also prove a Dajczer-type reduction of codimension theorem for
submanifolds of $\Sf^n\times \R$ and $\Hy^n\times \R$.

Differential Geometry