In this paper, we classify the hypersurfaces in $\mathbb{ S}^{ n}\times \mathbb{ R}$ and $\mathbb{ H}^{ n}\times\mathbb{ R}$, $n\neq 3$, 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.
Categories: math.DG