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On a class of hypersurfaces in \(\mathbb{S}^n\times\mathbb{R}\) and \(\mathbb{H}^n\times\mathbb{R}\). (English) Zbl 1218.53061
Summary: We give a complete description of all hypersurfaces of the product spaces \(\mathbb S^n\times \mathbb R\) and \(\mathbb H^n\times \mathbb R\) that have flat normal bundle when regarded as submanifolds with codimension two of the underlying flat spaces \(\mathbb R^{n+2}\supset\mathbb S^n\times \mathbb R\) and \(\mathbb L^{n+2}\supset\mathbb H^n\times \mathbb R\). We prove that any such hypersurface in \(\mathbb S^n\times \mathbb R\) (respectively, \(\mathbb H^n\times \mathbb R\)) can be constructed by means of a family of parallel hypersurfaces in \(\mathbb S^n\) (respectively, \(\mathbb H^n\)) and a smooth function of one variable. Then we show that constant mean curvature hypersurfaces 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 \(\mathbb S^n\times \mathbb R\) and \(\mathbb H^n\times \mathbb R\), that is, hypersurfaces with the property that its unit normal vector field has a constant angle with the unit vector field spanning the second factor \(\mathbb R\). This extends previous results by Dillen, Fastenakels, Van der Veken, Vrancken and Munteanu for surfaces in \(\mathbb S^n\times \mathbb R\) and \(\mathbb H^2\times \mathbb R\). 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 has a constant angle with a fixed direction.

MSC:
53C40 Global submanifolds
53B25 Local submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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