Hypersurfaces of $\mathbb{S}^3 \times \mathbb{R}$ and $\mathbb{H}^3 \times \mathbb{R}$ with constant principal curvatures

TL;DR

This paper classifies hypersurfaces with constant principal curvatures in the product spaces $ ext{S}^3 imes ext{R}$ and $ ext{H}^3 imes ext{R}$, showing they are cylinders over isoparametric surfaces and providing a complete classification of homogeneous cases.

Contribution

It provides a complete classification of hypersurfaces with three distinct constant principal curvatures in $ ext{Q}^3 imes ext{R}$, filling a gap in the literature.

Findings

Hypersurfaces with three distinct constant principal curvatures are cylinders over isoparametric surfaces.

Hypersurfaces with constant principal curvatures are isoparametric.

Complete classification of extrinsically homogeneous hypersurfaces in $ ext{Q}^3 imes ext{R}$.

Abstract

We classify the hypersurfaces of $\mathbb{Q}^3\times\mathbb{R}$ with three distinct constant principal curvatures, where $\varepsilon \in \{1,-1\}$ and $\mathbb{Q}^3$ denotes the unit sphere $\mathbb{S}^3$ if $\varepsilon = 1$, whereas it denotes the hyperbolic space $\mathbb{H}^3$ if $\varepsilon = -1$. We show that they are cylinders over isoparametric surfaces in $\mathbb{Q}^3$, filling an intriguing gap in the existing literature. We also prove that the hypersurfaces with constant principal curvatures of $\mathbb{Q}^3\times\mathbb{R}$ are isoparametric. Furthermore, we provide the complete classification of the extrinsically homogeneous hypersurfaces in $\mathbb{Q}^3\times\mathbb{R}$.