Mathematical Properties of a Class of Four-dimensional Neutral Signature Metrics

While the Lorenzian and Riemanian metrics for which all polynomial scalar curvature invariants vanish (the VSI property) are well-studied, less is known about the four-dimensional neutral signature metrics with the VSI property. Recently it was shown that the neutral signature metrics belong to two distinct subclasses: the Walker and Kundt metrics. In this paper we have chosen an example from each of the two subcases of the Ricci-flat VSI Walker metrics respectively. To investigate the difference between the metrics we determine the existence of a null, geodesic, expansion-free, shear-free and vorticity-free vector, and classify these spaces using their infinitesimal holonomy algebras. The geometric implications of the holonomy algebras are further studied by identifying the recurrent or covariantly constant null vectors, whose existence is required by the holonomy structure in each example. We conclude the paper with a simple example of the equivalence algorithm for these pseudo-Riemannian manifolds, which is the only approach to classification that provides all necessary information to determine equivalence.