Pseudo-Riemannian VSI spaces

TL;DR

This paper classifies pseudo-Riemannian VSI spaces, showing that spaces with the N-property are VSI and constructing metrics with specific geometric properties, including geodesic, expansion-free, shear-free, and twist-free null congruences.

Contribution

It introduces an algebraic classification using boost weight decomposition and defines the ${f S}_i$- and ${f N}$-properties, establishing a criterion for VSI spaces and constructing explicit metrics.

Findings

Spaces with the N-property are VSI.

Constructed VSI metrics have specific null congruence properties.

Discussed the relation to Walker metrics.

Abstract

In this paper we consider pseudo-Riemannian spaces of arbitrary signature for which all of their polynomial curvature invariants vanish (VSI spaces). We discuss an algebraic classification of pseudo-Riemannian spaces in terms of the boost weight decomposition and define the ${\bf S}_i$- and ${\bf N}$-properties, and show that if the curvature tensors of the space possess the ${\bf N}$-property then it is a VSI space. We then use this result to construct a set of metrics that are VSI. All of the VSI spaces constructed possess a geodesic, expansion-free, shear-free, and twist-free null-congruence. We also discuss the related Walker metrics.