2021/10/27 by de Lima, Ronaldo F., Manfio, Fernando, Santos, João P. dos
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2110.14364
We consider Einstein hypersurfaces of warped products I×ω\mathbb Qεn, where I⊂\mathbb R is an open interval and \mathbb Qεn is the simply connected space form of dimension n≥ 2 and constant sectional curvature ε∈\-1,0,1\. We show that, for all c∈\mathbb R (resp. c>0), there exist rotational hypersurfaces of constant sectional curvature c in I×ω\mathbb Hn and I×ω\mathbb Rn (resp. I×ω\mathbb Sn), provided that ω is nonconstant. We also show that the gradient T of the height function of any Einstein hypersurface of I×ω\mathbb Qεn (if nonzero) is one of its principal directions. Then, we consider a particular type of Einstein hypersurface of I×ω\mathbb Qεn with non vanishing T -- which we call ideal -- and prove that such a hypersurface Σ has either precisely two or precisely three distinct principal curvatures everywhere. We show that, in the latter case, there exist such a Σ for certain warping functions ω, whereas in the former case, Σ is necessarily of constant sectional curvature and rotational, regardless the warping function ω. We also characterize ideal Einstein hypersurfaces of I×ω\mathbb Qεn with no vanishing angle function as local graphs on families of isoparametric hypersurfaces of \mathbb Qεn.