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

TL;DR

This paper classifies isoparametric hypersurfaces in product spaces of spheres or hyperbolic spaces with Euclidean spaces, showing they have constant angle functions and principal curvatures, thus extending understanding of their geometric properties.

Contribution

It provides a complete classification of isoparametric and homogeneous hypersurfaces in these product spaces, revealing their constant angle functions and principal curvatures.

Findings

Every isoparametric hypersurface has a constant angle function.

Such hypersurfaces have constant principal curvatures.

The classification of these hypersurfaces is achieved in the paper.

Abstract

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.