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.

Theorems & Definitions (27)

• Proposition 1.1
• Theorem 1.2
• Remark 1.3
• Corollary 1.4
• Corollary 1.5
• Remark 1.6
• Proposition 2.1
• Proposition 2.2
• Lemma 2.3
• Proposition 2.4