Weakly Einstein hypersurfaces in space forms

TL;DR

This paper classifies weakly Einstein hypersurfaces in space forms, showing they are either products of constant curvature spaces or rotation hypersurfaces, extending previous Euclidean results to nonzero curvature spaces.

Contribution

It provides a complete classification of weakly Einstein hypersurfaces in space forms, including nonzero curvature cases, which was not previously known.

Findings

Weakly Einstein hypersurfaces are either products of constant curvature spaces or rotation hypersurfaces.

The classification extends known Euclidean results to nonzero constant curvature spaces.

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.