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

TL;DR

This paper classifies hypersurfaces with flat normal bundle in product spaces $ ext{S}^n imes ext{R}$ and $ ext{H}^n imes ext{R}$, describing their construction, constant mean curvature cases, and constant angle hypersurfaces, extending previous results.

Contribution

It provides a complete classification of hypersurfaces with flat normal bundle in these product spaces, including explicit descriptions and extensions of known results.

Findings

Hypersurfaces constructed via parallel hypersurfaces and a smooth function of one variable.

Constant mean curvature hypersurfaces characterized by isoparametric families and ODE solutions.

Classification of constant angle hypersurfaces extending previous surface results.

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.