The curvature of almost Robinson manifolds

Abstract

An almost Robinson structure on an $n$-dimensional Lorentzian manifold $(\mcM,g)$, where $n=2m+ε$, $ε\in \{ 0 ,1 \}$, is a complex $m$-plane distribution $\mcN$ that is totally null with respect to the complexified metric, and intersects its complex conjugate in a real null line distribution $\mcK$, say. When $\mcN$ and its orthogonal complement $\mcN^\perp$ are in involution, the line distribution $\mcK$ is tangent to a congruence of null geodesics, and the quotient of $\mcM$ by this flow acquires the structure of a CR manifold. In four dimensions, such a congruence is shearfree. We give classifications of the tracefree Ricci tensor, the Cotton-York tensor and the Weyl tensor, invariant under i) the stabiliser of a null line, and ii) the stabiliser of an almost Robinson structure. For the Weyl tensor, these are generalisations of the Petrov classification to higher dimensions. Since an almost Robinson structure is equivalent to a projective pure spinor field of real index $1$, the present work can also be viewed as spinorial classifications of curvature tensors. We illustrate these algebraic classifications by a number of examples of higher-dimensional general relativity that admit integrable almost Robinson structures, emphasising the degeneracy type of the Weyl tensor in each case.