Bull Braz Math Soc, New Series 41(2), 199-209 © 2010, Sociedade Brasileira de Matemática On a class of hypersurfaces in S n × R and H n × R Ruy Tojeiro Abstract. We give a complete description of all hypersurfaces of the product spaces S n × R and H n × R that have flat normal bundle when regarded as submanifolds with codimension two of the underlying flat spaces R n+2 ⊃ S n × R and L n+2 ⊃ H n × R. We prove that any such hypersurface in S n × R (respectively, H n × R) can be con- structed by means of a family of parallel hypersurfaces in S n (respectively, H n ) and a smooth function of one variable. Then we show that constant mean curvature hy- persurfaces in this class correspond to an isoparametric family in the base space and a smooth function that is explicitly determined in terms of the mean curvature function of the isoparametric family. As another consequence of our general result, we classify the constant angle hypersurfaces of S n × R and H n × R, that is, hypersurfaces with the property that its unit normal vector field makes a constant angle with the unit vector field spanning the second factor R. This extends previous results by Dillen, Fastenakels, Van der Veken, Vrancken and Munteanu for surfaces in S 2 × R and H 2 × R. Our method also yields a classification of all Euclidean hypersurfaces with the property that the tangent component of a constant vector field in the ambient space is a principal direction, in par- ticular of all Euclidean hypersurfaces whose unit normal vector field makes a constant angle with a fixed direction. Keywords: hypersurfaces of product spaces, flat normal bundle, constant angle hyper- surfaces, minimal and cmc hypersurfaces. Mathematical subject classification: 53 B25. 1 Introduction The study of hypersurfaces of the product spaces S n × R and H n × R has attracted the attention of several geometers in the last years. Here S n and H n denote the sphere and hyperbolic space of dimension n, respectively. A natural class of such hypersurfaces consists of those which have flat normal bundle when regarded as submanifolds with codimension two of the underlying flat spaces R n+2 ⊃ S n × R Received 18 September 2009.