On a class of hypersurfaces of a product of two space forms

Abstract

We define hypersurfaces $f\colon M^n\to \mathbb{Q}_{c_1}^{k} \times \mathbb{Q}_{c_2}^{n-k+1}$ in class $\mathcal{A}$ of a product of two space forms as those that have flat normal bundle when regarded as submanifolds of the underlying flat ambient space. We provide an explicit construction of all of them in terms of parallel families of hypersurfaces of the factors, and show how such construction simplifies for the hypersurfaces within this class that have constant product angle function. We also show that hypersurfaces with constant mean curvature in class $\mathcal{A}$ are given in terms of parallel families of isoparametric hypersurfaces in each factor and a solution of a second order ODE. Finally, we classify hypersurfaces with constant mean curvature in class~$\mathcal{A}$ that have constant product angle function.