Weakly Einstein hypersurfaces in space forms

Abstract

A Riemannian manifold $(M,g)$ is called \emph{weakly Einstein} if the tensor $R_{iabc}R_{j}^{~~abc}$ is a scalar multiple of the metric tensor $g_{ij}$. We give a complete classification of weakly Einstein hypersurfaces in the spaces of nonzero constant curvature (the classification in a Euclidean space has been previously known). The main result states that such a hypersurface can only be the product of two spaces of constant curvature or a rotation hypersurface.