Mathematical properties of a class of four-dimensional neutral signature metrics

作者：Brooks D; Musoke N; McNutt D^*; Coley A

来源：Journal of Geometry and Physics, 2015, 97: 1-13.

DOI：10.1016/j.geomphys.2015.06.003

• Pseudo-Riemannian manifolds
• Holonomy
• Vanishing scalar curvature invariants
• Walker
• Kundt
• Equivalence problem

While the Lorentzian and Riemannian metrics for which all polynomial scalar curvature invariants vanish (the VSI property) are well-studied, less is known about the fourdimensional 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 holonomy algebras. The geometric implications of these 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.

• 出版日期2015-11