Hypersurfaces of $\mathbb{S}^3 \times \mathbb{R}$ and $\mathbb{H}^3 \times \mathbb{R}$ with constant principal curvatures
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 =...
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Zusammenfassung: | 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
$\mathbb{Q}^3\times\mathbb{R}$. |
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DOI: | 10.48550/arxiv.2409.07978 |