Weakly Einstein conformal products

TL;DR

This paper classifies proper weakly Einstein four-manifolds conformal to Riemannian products, constructs new examples including cohomogeneity two cases, and proves the non-existence of proper weakly Einstein manifolds with harmonic curvature.

Contribution

It provides classification results for proper weakly Einstein metrics, introduces new examples with higher cohomogeneity, and clarifies the structure of known solutions like the EPS space.

Findings

Classified proper weakly Einstein conformal product manifolds.

Constructed new examples with cohomogeneity two.

Proved no proper weakly Einstein manifolds have harmonic curvature.

Abstract

One says that a Riemannian four-manifold is \emph{weakly Einstein} if the three-index contraction of its curvature tensor against itself equals a function times the metric. Since this includes all four-manifolds that are Einstein, or conformally flat and scalar-flat, the term \emph{proper} may be used for weakly Einstein manifolds (or metrics) not belonging to the latter two classes. We establish two classification-type results about proper weakly Einstein metrics conformal to Riemannian products. This includes constructions of new examples, among them -- some of (local) cohomogeneity two, in contrast with the two previously known narrow classes of examples, having cohomogeneity zero and one. We also exhibit a simple coordinate description of one of the known examples, the EPS space, which shows that it is a conformal product and constitutes a single local-homothety type. Finally, we prove that there exist no proper weakly Einstein manifolds with harmonic curvature.