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

TL;DR

This paper characterizes a special class of hypersurfaces in product spaces of two space forms, providing explicit constructions, classifications, and conditions for constant mean curvature and product angle functions.

Contribution

It introduces a new class of hypersurfaces with flat normal bundle, offers explicit constructions, and classifies those with constant mean curvature and constant product angle.

Findings

Explicit construction of hypersurfaces in class  with flat normal bundle.

Classification of constant mean curvature hypersurfaces in this class.

Conditions for hypersurfaces with constant product angle function.

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.