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 \({\mathbb {Q}}_\epsilon ^n.\)