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

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.

Categories: math.DG