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

TL;DR

This paper proves a rigidity property of isoparametric hypersurfaces in product spaces involving spheres and hyperbolic spaces, leading to a complete classification and geometric characterization of such hypersurfaces.

Contribution

It establishes that the angle function is constant for these hypersurfaces, enabling a full classification and geometric characterization in the specified product spaces.

Findings

The angle function is constant for isoparametric hypersurfaces in the studied spaces.

Complete classification of isoparametric and homogeneous hypersurfaces in these spaces.

A hypersurface is isoparametric iff it has constant principal curvatures and a constant angle function.

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.