Isoparametric Hypersurfaces in Products of Simply Connected Space Forms

TL;DR

This paper proves that connected isoparametric hypersurfaces in products of simply connected space forms have constant angle functions and classifies those satisfying a one-point condition.

Contribution

It establishes the constant angle property for isoparametric hypersurfaces in these product spaces and provides a classification under specific conditions.

Findings

Connected isoparametric hypersurfaces have constant angle functions.

Classification of hypersurfaces satisfying a one-point condition.

Results apply to products of space forms with non-zero curvature sum.

Abstract

Let $\mathbb Q_{\epsilon_i}^{n_i}$ denote the simply connected space form of dimension $n_i\ge 2$ and constant sectional curvature $\epsilon_i$. We prove that any connected isoparametric hypersurface of $\mathbb Q_{\epsilon_1}^{n_1}\times\mathbb Q_{\epsilon_2}^{n_2}$ has constant angle function. We then use this property to classify the isoparametric and homogeneous hypersurfaces of $\mathbb Q_{\epsilon_1}^{n_1}\times\mathbb Q_{\epsilon_2}^{n_2}$, $|\epsilon_1|+|\epsilon_2|\ne 0$, that satisfy a one-point condition.