Einstein Hypersurfaces of Warped Product Spaces

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, for $n>3,$ 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 ${\mathbb {Q}}_\epsilon ^n.$