On a class of hypersurfaces in $\Sf^n\times \R$ and $\Hy^n\times \R$

Abstract

We give a complete description of all hypersurfaces of the product spaces $\Sf^n\times \R$ and $\Hy^n\times \R$ that have flat normal bundle when regarded as submanifolds with codimension two of the underlying flat spaces $\R^{n+2}\supset \Sf^n\times \R$ and $\Le^{n+2}\supset \Hy^n\times \R$. We prove that any such hypersurface in $\Sf^n\times \R$ (respectively, $\Hy^n\times \R$) can be constructed by means of a family of parallel hypersurfaces in $\Sf^n$ (respectively, $\Hy^n$) and a smooth function of one variable. Then we show that constant mean curvature hypersurfaces in this class are given in terms of an isoparametric family in the base space and a solution of a certain ODE. For minimal hypersurfaces such solution 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 $\Sf^n\times \R$ and $\Hy^n\times \R$, 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 $\Sf^2\times \R$ and $\Hy^2\times \R$. 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 particular of all Euclidean hypersurfaces whose unit normal vector field makes a constant angle with a fixed direction.