  We prove that a Riemannian product of type M x R (where R denotes the
Euclidean line) admits totally umbilical hypersurfaces if and only if M has
locally the structure of a warped product and we give a complete description of
the totally umbilical hypersurfaces in this case. Moreover, we give a necessary
and sufficient condition under which a Riemannian three-manifold carrying a
unit Killing field admits totally geodesic surfaces and we study local and
global properties of three-manifolds satisfying this condition.
