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

TL;DR

This paper classifies isoparametric and homogeneous hypersurfaces in hyperbolic and spherical product spaces, showing they have constant angle functions and principal curvatures, thus advancing understanding of their geometric structure.

Contribution

It provides a complete classification of these hypersurfaces, revealing their constant geometric properties in the specified product spaces.

Findings

Hypersurfaces have constant angle functions.

Hypersurfaces have constant principal curvatures.

Classification results apply to both hyperbolic and spherical product spaces.

Abstract

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.