Non-Walker Self-Dual Neutral Einstein Four-Manifolds of Petrov Type III

TL;DR

This paper characterizes a class of four-dimensional neutral Einstein manifolds of Petrov type III, revealing their local structure, curvature properties, and providing new examples that answer existing existence questions.

Contribution

It describes the local structure of non-Walker self-dual neutral Einstein 4-manifolds of Petrov type III, including their curvature homogeneity and Ricci-flatness conditions.

Findings

Manifolds are curvature homogeneous but not necessarily locally homogeneous.

Not all manifolds in this class are Ricci-flat.

The work provides examples answering an open existence question.

Abstract

The local structure of the manifolds named in the title is described. Although curvature homogeneous, they are not, in general, locally homogeneous. Not all of them are Ricci-flat, which answers an existence question about type III Jordan-Osserman metrics, raised by Diaz-Ramos, Garcia-Rio and Vazquez-Lorenzo (2006).