Einstein Hypersurfaces of Warped Product Spaces

TL;DR

This paper classifies Einstein hypersurfaces in warped product spaces, revealing conditions for their curvature and principal directions, and introduces the concept of ideal hypersurfaces with specific curvature properties.

Contribution

It provides a comprehensive classification of Einstein hypersurfaces in warped products, including the existence of rotational hypersurfaces with constant curvature and the characterization of ideal hypersurfaces.

Findings

Existence of rotational hypersurfaces with constant sectional curvature in certain warped products.

Gradient of the height function is a principal direction for Einstein hypersurfaces.

Ideal Einstein hypersurfaces have either two or three principal curvatures, with specific geometric properties.

Abstract

We consider Einstein hypersurfaces of warped products $I\times_\omega\mathbb Q_\epsilon^n,$ where $I\subset\mathbb R$ is an open interval and $\mathbb Q_\epsilon^n$ is the simply connected space form of dimension $n\ge 2$ and constant sectional curvature $\epsilon\in\{-1,0,1\}.$ We show that, for all $c\in\mathbb R$ (resp. $c>0$), there exist rotational hypersurfaces of constant sectional curvature $c$ in $I\times_\omega\mathbb H^n$ and $I\times_\omega\mathbb R^n$ (resp. $I\times_\omega\mathbb S^n$), provided that $\omega$ is nonconstant. We also show that the gradient $T$ of the height function of any Einstein hypersurface of $I\times_\omega\mathbb Q_\epsilon^n$ (if nonzero) is one of its principal directions. Then, we consider a particular type of Einstein hypersurface of $I\times_\omega\mathbb Q_\epsilon^n$ with non vanishing $T$ -- which we call ideal -- and prove that such a hypersurface $\Sigma$ has either precisely two or precisely three distinct principal curvatures everywhere. We show that, in the latter case, there exist such a $\Sigma$ for certain warping functions $\omega,$ whereas in the former case, $\Sigma$ is necessarily of constant sectional curvature and rotational, regardless the warping function $\omega.$ We also characterize ideal Einstein hypersurfaces of $I\times_\omega\mathbb Q_\epsilon^n$ with no vanishing angle function as local graphs on families of isoparametric hypersurfaces of $\mathbb Q_\epsilon^n.$