Classification of complex and real, vacuum spaces of the type $[\text{N}]\otimes [\text{N}]$

Abstract

Complex and real, vacuum spaces with both self-dual and anti-self-dual parts of the Weyl tensor being of the type [N] are considered. Such spaces are classified according to two criteria. The first one takes into account the properties of the congruences of totally null, geodesic 2-dimensional surfaces (the null strings). The second criterion is based on investigations of the properties of the intersection of these congruences. It is proved that there exist six distinct types of the $[\text{N}]\otimes [\text{N}]$ spaces. New examples of the Lorentzian slices of the complex metrics are presented. Also, some type $[\text{N}]\otimes [\text{N}]$ spaces which do not posses Lorentzian slices are considered.

References in corpus (7)

• Twistor space for rolling bodies
• Self-Dual Conformal Gravity
• Null Killing vectors and geometry of null strings in Einstein spaces
• Classification of the Killing Vectors in Nonexpanding HH-Spaces with Lambda
• Proper conformal symmetries in SD Einstein spaces
• On twisting type $[\text{N}]\otimes [\text{N}]$ Ricci flat complex spacetimes with two homothetic symmetries
• Equations for Complex-Valued, Twisting, Type N, Vacuum Solutions, with one or two Killing/homothetic vectors

Cited by in corpus (2)

• Anti-Self-Dual Spacetimes, Gravitational Instantons and Knotted Zeros of the Weyl Tensor
• Hyperheavenly spaces and their application in Walker and para-Kähler geometries: Part I