Weakly Einstein curvature tensors
Andrzej Derdzinski, JeongHyeong Park, Wooseok Shin
TL;DR
The paper addresses the problem of classifying weakly Einstein algebraic curvature tensors in oriented Euclidean 4-space by exploiting a pivotal equivalence $6 W e = - s e$. It develops a method to simultaneously diagonalize the Einstein tensor $e$ and the self-dual/anti-self-dual Weyl components $W^ ext{±}$, then performs a detailed case analysis based on eigenvalue multiplicities, yielding three disjoint five-parameter families that exhaust the non-Einstein weakly Einstein tensors. It provides explicit constructions in each case and shows how known geometric examples (EPS space and certain Kähler surfaces) fit within the framework. The results clarify the algebraic structure underpinning weakly Einstein 4-manifolds and connect algebraic classifications to concrete 4D geometries, enriching the understanding of non-Einstein, non-conformally-flat examples.
Abstract
We classify weakly Einstein algebraic curvature tensors in an oriented Euclidean 4-space, defined by requiring that the three-index contraction of the curvature tensor against itself be a multiple of the inner product. This algebraic formulation parallels the geometric notion of weakly Einstein Riemannian four-manifolds, which include conformally flat scalar-flat, and Einstein manifolds. Our main result provides a complete classification of non-Einstein weakly Einstein curvature tensors in dimension four, naturally dividing them into three disjoint five-dimensional families of algebraic types. These types are explicitly constructed using bases that simultaneously diagonalize both the Einstein tensor and the (anti)self-dual Weyl tensors, which consequently proves that such simultaneous diagonalizability follows from the weakly Einstein property. We also describe how the known geometric examples that are neither Einstein, nor conformally flat scalar-flat (namely, the EPS space and certain Kähler surfaces) fit within our classification framework.