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

TL;DR

This paper classifies hypersurfaces with a small number of constant principal curvatures in spherical and hyperbolic product spaces, establishing their isoparametric nature and providing conditions for such hypersurfaces with flat normal bundle.

Contribution

It provides a complete classification of hypersurfaces with up to three constant principal curvatures in these spaces, showing they are isoparametric and characterizing those with flat normal bundle.

Findings

Hypersurfaces with 1, 2, or 3 constant principal curvatures are isoparametric.

Necessary and sufficient conditions for isoparametric hypersurfaces with flat normal bundle.

Classification results extend to both spherical and hyperbolic product spaces.

Abstract

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.