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

Abstract

We prove that the angle function associated with the canonical product structure is constant for an isoparametric hypersurface in ${\mathbb{S}}^n\times {\mathbb{S}}^m$, ${\mathbb{S}}^n\times {\mathbb{H}}^m$, or ${\mathbb{H}}^n\times {\mathbb{H}}^m$. This rigidity result enables us to provide a complete classification of isoparametric and homogeneous hypersurfaces in ${\mathbb{S}}^n\times {\mathbb{S}}^m$ and ${\mathbb{S}}^n\times {\mathbb{H}}^m$. Furthermore, we establish a geometric characterization in these two spaces: a hypersurface is isoparametric if and only if it has constant principal curvatures and a constant angle function.

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