Einstein Hypersurfaces of Warped Product Spaces

We consider Einstein hypersurfaces of warped products $$I\times _\omega {\mathbb {Q}}_\epsilon ^n,$$ where $$I\subset {R}}$$ is an open interval and $${\mathbb ^n$$ the simply connected space form dimension $$n\ge 2$$ constant sectional curvature $$\epsilon \in \{-1,0,1\}$$ . show that, for all $$c\in (resp. $$c>0$$ ), there exist rotational c in {H}}^n$$ {R}}^n$$ {S}}^n$$ provided that $$\omega $$ nonconstant. also gradient T height function any hypersurface \mathbb {Q} _\epsilon (if nonzero) one its principal directions. Then, we a particular type with non vanishing T—which call ideal—and prove $$n>3,$$ such $$\Sigma has either precisely two or three distinct curvatures everywhere. latter case, certain warping functions ,$$ whereas former case necessarily rotational, regardless .$$ characterize ideal no angle as local graphs on families isoparametric ^n.$$