Isoparametric Hypersurfaces of ${\mathbb{H}}^n\times \mathbb{R}$ and ${\mathbb{S}}^n\times \mathbb{R}$

We classify the isoparametric hypersurfaces and the homogeneous hypersurfaces of ${\mathbb{H}}^n\times \mathbb{R}$ and ${\mathbb{S}}^n\times \mathbb{R}$, $n\ge 2$, by establishing that any such hypersurface has constant angle function and constant principal curvatures.