Pseudo-Riemannian VSI spaces II

Abstract

In this paper we consider pseudo-Riemannian spaces of arbitrary signature for which all of the polynomial curvature invariants vanish (VSI spaces). Using an algebraic classification of pseudo-Riemannian spaces in terms of the boost-weight decomposition we first show more generally that a space which is not characterised by its invariants must possess the ${\mathbf{S}}_1^G$-property. As a corollary, we then show that a VSI space must possess the ${\mathbf{N}}^G$-property (these results are the analogues of the alignment theorem, including corollaries, for Lorentzian spacetimes). As an application we classify all 4D neutral VSI spaces and show that these belong to one of two classes: (1) those that possess a geodesic, expansion-free, shear-free, and twist-free null-congruence (Kundt metrics), or (2) those that possess an invariant null plane (Walker metrics). By explicit construction we show that the latter class contains a set of VSI metrics which have not previously been considered in the literature.

19 pages. Forgot to put this on arxiv

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Cited by in corpus (11)

• Geometry of Killing spinors in neutral signature
• Special Geometry and Space-time Signature
• $\mathcal{I}$-degenerate pseudo-Riemannian metrics
• Wick rotations and real GIT
• Klein and Conformal Superspaces, Split Algebras and Spinor Orbits
• Mathematical Properties of a Class of Four-dimensional Neutral Signature Metrics
• Locally Homogeneous Kundt Triples and CSI Metrics
• Real GIT with applications to compatible representations and Wick-rotations
• Quantum Klein Space and Superspace
• Universal Walker metrics
• Constructing Bach Flat Manifolds of signature $(2,2)$ using the modified Riemannian extension