Isoparametric Hypersurfaces in product spaces of space forms

Abstract

We give a complete classification of isoparametric hypersurfaces in a product space $M^2_{κ_1}\times M^2_{κ_2}$ of $2$-dimensional space forms for $κ_i\in \{-1,0,1\}$ with $κ_1\neq κ_2$. 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.