Almost Robinson geometries

Abstract

We investigate the geometry of almost Robinson manifolds, Lorentzian analogues of almost Hermitian manifolds, defined by Nurowski and Trautman as Lorentzian manifolds of even dimension equipped with a totally null complex distribution of maximal rank. Associated to such a structure, there is a congruence of null curves, which, in dimension four, is geodesic and non-shearing if and only if the complex distribution is involutive. Under suitable conditions, the distribution gives rise to an almost Cauchy-Riemann structure on the leaf space of the congruence. We give a comprehensive classification of such manifolds on the basis of their intrinsic torsion. This includes an investigation of the relation between an almost Robinson structure and the geometric properties of the leaf space of its congruence. We also obtain conformally invariant properties of such a structure, and we finally study an analogue of so-called generalised optical geometries as introduced by Robinson and Trautman.

arXiv comment93 pages, v2: A number of errors fixed. Notation clarified. Some passages reformulated and restructured, notably sections 3.7 and 3.8. Some examples more explicit. Further references and acknowledgments; v3: 95 pages, final version accepted for publication in Letters in Mathematical Physics

References in corpus (26)

• All supersymmetric solutions of minimal supergravity in five dimensions
• General Kerr-NUT-AdS Metrics in All Dimensions
• All supersymmetric solutions of minimal supergravity in six dimensions
• Robinson-Trautman spacetimes in higher dimensions
• Type D Einstein spacetimes in higher dimensions
• Ricci identities in higher dimensions
• General Kundt spacetimes in higher dimensions
• A Bestiary of Higher Dimensional Taub-NUT-AdS Spacetimes
• On the intrinsic torsion of spacetime structures
• Almost Hermitian Geometry on Six Dimensional Nilmanifolds
• Robinson-Trautman spacetimes with an electromagnetic field in higher dimensions
• Exact solutions to quadratic gravity
• Killing-Yano tensors and multi-hermitian structures
• Supergravity vacua and lorentzian Lie groups
• A holonomy characterisation of Fefferman spaces
• The complex Goldberg-Sachs theorem in higher dimensions
• Twistor Bundles, Einstein Equations and Real Structures
• Pure spinors, intrinsic torsion and curvature in even dimensions
• Optical structures, algebraically special spacetimes, and the Goldberg-Sachs theorem in five dimensions
• Static and radiating dyonic black holes coupled to conformally invariant electrodynamics in higher dimensions
• A Goldberg-Sachs theorem in dimension three
• Geometry and symmetries of null G-structures
• Pure spinors, intrinsic torsion and curvature in odd dimensions
• Twistor Geometry of Null Foliations in Complex Euclidean Space
• The curvature of almost Robinson manifolds
• Witt pseudo-Riemannian connection

Cited by in corpus (2)

• Perturbations of Fefferman spaces over almost CR manifolds
• Perturbations of Fefferman spaces over CR three-manifolds