Abstract. We give a complete classification of umbilical submanifolds of arbitrary dimension and codimension of S n × R, extending the classification of umbilical surfaces in S 2 × R by Souam and Toubiana as well as the local description of umbilical hypersurfaces in S n ×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 S 1 ×R or S 2 ×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 S n × R and H n × 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 S n × R and H n × R having this property are rotational surfaces, and use this fact to improve some recent results by Alencar, do Carmo, and Tribuzy. We also obtain a Dajczer-type reduction of codimension theorem for submanifolds of S n × R and H n × R.

We give a complete classification of submanifolds with parallel second fundamental form of a product of two space forms. We also reduce the classification of umbilical submanifolds with dimension m ≥ 3 of a product Q n 1 k 1 × Q n 2 k 2 of two space forms whose curvatures satisfy k 1 + k 2 = 0 to the classification of m-dimensional umbilical submanifolds of codimension two of S n × R and H n × R. The case of S n × R was carried out in [13]. As a main tool we derive reduction of codimension theorems of independent interest for submanifolds of products of two space forms.MSC 2000: 53 B25, 53 C40.

We proved that a conformal immersion of M n 0 0 × M n 1