Isoparametric hypersurfaces in ${\mathbb{S}}^n\times {\mathbb{R}}^m$ and ${\mathbb{H}}^n\times {\mathbb{R}}^m$

Abstract

We first show that every isoparametric hypersurface in ${\mathbb{S}}^n\times {\mathbb{R}}^m$ or ${\mathbb{H}}^n\times {\mathbb{R}}^m$ possesses a constant angle function with respect to the canonical product structure. Exploiting this rigidity, we achieve a complete classification of isoparametric and homogeneous hypersurfaces in these product spaces. Furthermore, we prove that an isoparametric hypersurface in ${\mathbb{S}}^n\times {\mathbb{R}}^m$ or ${\mathbb{H}}^n\times {\mathbb{R}}^m$ also has constant principal curvatures.

To appear in SCIENCE CHINA Mathematics