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Weakly Einstein conformal products

Andrzej Derdzinski, JeongHyeong Park, Wooseok Shin

TL;DR

This paper investigates 4‑dimensional weakly Einstein manifolds, focusing on those that are conformal to Riemannian products. It develops a framework using warped/conformal products and Weyl/Einstein tensors to construct and classify proper weakly Einstein metrics, notably producing new local cohomogeneity‑two examples and a simple coordinate description of the EPS space. The authors prove two main classification-type theorems: generically, proper weakly Einstein conformal products arise from explicit constructions, with nongeneric 3+1 cases treated separately, and they show that no proper weakly Einstein metric can have harmonic curvature. The results expand the known zoo of examples beyond EPS and certain Kahler surfaces, providing a structured approach to understanding conformal product geometries via spectral data of the Weyl and Einstein tensors.

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.

Weakly Einstein conformal products

TL;DR

This paper investigates 4‑dimensional weakly Einstein manifolds, focusing on those that are conformal to Riemannian products. It develops a framework using warped/conformal products and Weyl/Einstein tensors to construct and classify proper weakly Einstein metrics, notably producing new local cohomogeneity‑two examples and a simple coordinate description of the EPS space. The authors prove two main classification-type theorems: generically, proper weakly Einstein conformal products arise from explicit constructions, with nongeneric 3+1 cases treated separately, and they show that no proper weakly Einstein metric can have harmonic curvature. The results expand the known zoo of examples beyond EPS and certain Kahler surfaces, providing a structured approach to understanding conformal product geometries via spectral data of the Weyl and Einstein tensors.

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.

Paper Structure

This paper contains 22 sections, 17 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Warped products
  4. The EPS space
  5. Weakly Ein-stein curvature tensors
  6. Permutation groups
  7. Con-for-mal changes of product metrics
  8. Con-for-mal products and the Weyl tensor
  9. Examples
  10. Proof of Theorem \ref{['exatt']}
  11. Proof of Theorem \ref{['exaot']}
  12. Euclidean spheres and Lo-rentz-i-an pseu-do-spheres
  13. Surface metrics
  14. Geometric realizations with Kil-ling fields
  15. Geometric realizations via con-for-mal changes
  16. ...and 7 more sections

Key Result

Theorem 4.1

The simply connected complete model of an EPS space consists of $\,\mathrm{I\!R} ^4$ with the metric $\,g=d{\tau \iota }^2 +e ^{- {\tau \iota }} d\xi^2 +e ^{\tau \iota }(d\eta^2 +d\zeta^2)\,$ in the coordinates $\,{\tau \iota },\xi,\eta,\zeta$. Its full isom-e-try group is five-di-men-sion-al, acts depending on the parameters $\,c,p\in\mathrm{I\!R}\,$ and $\,w,q\in{ \hbox{$\mathrm{C}$}}\,$ with $

Theorems & Definitions (55)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Remark 3.1
  • Remark 3.2
  • ...and 45 more