Hypersurfaces with constant principal curvatures in ${\mathbb{S}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$

Abstract

In this paper, we classify the hypersurfaces in ${\mathbb{S}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$, $n\ne 3$, with $g$ distinct constant principal curvatures, $g\in \{1,2,3\}$, where ${\mathbb{S}}^n$ and ${\mathbb{H}}^n$ denote the sphere and hyperbolic space of dimension $n$, respectively. We prove that such hypersurfaces are isoparametric in those spaces. Furthermore, we find a necessary and sufficient condition for an isoparametric hypersurface in ${\mathbb{S}}^n\times \mathbb{R}\subset {\mathbb{R}}^{n+2}$ and ${\mathbb{H}}^n\times \mathbb{R}\subset {\mathbb{L}}^{n+2}$ with flat normal bundle, having constant principal curvatures.

28 pages