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.
