Isoparametric Hypersurfaces in product spaces of space forms

TL;DR

This paper classifies isoparametric hypersurfaces in product spaces of two-dimensional space forms with different curvatures, showing they have constant product angle functions and extending previous classifications by removing the constant principal curvature condition.

Contribution

It provides a complete classification of isoparametric hypersurfaces in certain product spaces, demonstrating they have constant product angle functions, thus generalizing prior results.

Findings

Hypersurfaces have constant product angle functions.

Classification extends previous work by removing the constant principal curvature condition.

Complete classification in specified product spaces.

Abstract

We give a complete classification of isoparametric hypersurfaces in a product space $M^2_{\kappa_1}\times M^2_{\kappa_2}$ of $2$-dimensional space forms for $\kappa_i\in \{-1,0,1\}$ with $\kappa_1\neq \kappa_2$. In fact we prove that any isoparametic hypersurface in such a space has constant product angle function, which enables us to remove the condition of constant principal curvatures from the classification obtained recently by J.B.M.dos Santos and J.P.dos Santos.