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

TL;DR

The paper establishes a constant-angle rigidity for isoparametric hypersurfaces in the product manifolds $\mathbb{S}^n\times\mathbb{R}^m$ and $\mathbb{H}^n\times\mathbb{R}^m$, and provides a complete classification of isoparametric and homogeneous hypersurfaces in these spaces. It shows that in these products the angle function with respect to the canonical product structure is constant for isoparametric hypersurfaces, and, conversely, that constant angle together with constant principal curvatures implies isoparametricity in products of real space forms. Building on prior work in lower-rank cases and exploiting a higher-dimensional generalization of the Urbano framework, the authors derive explicit families, including horizontal slices, vertical cylinders, and special immersed foliations, and prove the equivalence between the isoparametric condition and the joint constancy of angle and principal curvatures. The results extend the m=1 classification to higher Euclidean factors and set the stage for analogous treatments in remaining product types, with implications for the study of homogeneous and focal-submanifold structures in Riemannian products.

Abstract

We first show that every isoparametric hypersurface in $\mathbb{S}^{n}\times \mathbb{R}^{m}$ or $\mathbb{H}^{n}\times \mathbb{R}^{m}$ possesses a constant angle function with respect to the canonical product structure. Exploiting this rigidity, we achieve a complete classification of isoparametric and homogeneous hypersurfaces in these product spaces. Furthermore, in the product of any two real space forms, we prove that a hypersurface with both constant angle and constant principal curvatures must be isoparametric. Consequently, for hypersurfaces in $\mathbb{S}^{n}\times \mathbb{R}^{m}$ and $\mathbb{H}^{n}\times \mathbb{R}^{m}$, the conditions of having constant angle and constant principal curvatures are equivalent to being isoparametric.

Theorems & Definitions (20)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Corollary 1.7
• Corollary 1.8
• Proposition 3.1
• Remark 3.2