In this paper we present some new results about n(≥ 4)-dimensional pseudo-Z symmetric space-times. First we show that if the tensor Z satisfies the Codazzi condition then its rank is one, the space-time is a quasi-Einstein manifold, and the associated 1-form results to be null and recurrent. In the case in which such covector can be rescaled to a covariantly constant we obtain a Brinkmann-wave. Anyway the metric results to be a subclass of the Kundt metric. Next we investigate pseudo-Z symmetric space-times with harmonic conformal curvature tensor: a complete classification of such spaces is obtained. They are necessarily quasi-Einstein and represent a perfect fluid space-time in the case of time-like associated covector; in the case of null associated covector they represent a pure radiation field. Further if the associated covector is locally a gradient we get a Brinkmann-wave space-time for n > 4 and a pp-wave space-time in n = 4. In all cases an algebraic classification for the Weyl tensor is provided for n = 4 and higher dimensions. Then conformally flat pseudo-Z symmetric space-times are investigated. In the case of null associated covector the space-time reduces to a plane wave and results to be generalized quasi-Einstein. In the case of time-like associated covector we show that under the condition of divergence-free Weyl tensor the space-time admits a proper concircular vector that can be rescaled to a time like vector of concurrent form and is a conformal Killing vector. A recent result then shows that the metric is necessarily a generalized Robertson–Walker space-time. In particular we show that a conformally flat (PZS)n, n ≥ 4, space-time is conformal to the Robertson–Walker space-time.

Pseudo-Z symmetric space-times with divergence-free Weyl tensor and pp-waves

A general discussion of the recurrence properties of the Riemann, Ricci and Weyl tensors is given Space-times possessing these properties are classified according to the Petrov type of the Weyl tensor and the Legre type of the Ricci tensor The proofs of some known theorems are shortened and some new results are given

https://iopscience.iop.org/article/10.1088/0305-4470/10/1/011/pdf

Recurrence conditions in space-time

It is shown that the validity of Huygens' principle for the conformally invariant scalar wave equation, Maxwell's equations or Weyl's neutrino equation on a Petrov type III space-time, implies that the space-time is conformally related to one on which the repeated principal null vector field of the Weyl tensor, is recurrent. Further, it is proven that if a certain mild assumption which is shown to be suggested by the necessary conditions, is imposed on the covariant derivative of the Weyl tensor, then there are no Petrov type III space-times on which any of the above equations satisfies Huygens' principle On demontre que la validite du principe de Huygens pour l'equation d'onde scalaire invariante conforme, les equations de Maxwell ou l'equation de Weyl pour les neutrinos sur un espace-temps de Petrov de type III impliquent que cet espace-temps est conformement relie a un espace ou le noyau principal repete du tenseur de Weyl est recurrent. On montre de plus sous une hypothese faible sur la derivee covariante du tenseur de Weyl suggeree par la condition necessaire qu'il n'existe pas d'espace-temps de Petrov de type III ou les equations ci-dessus satisfont le principe de Huygens

An explicit determination of the space-times on which the conformally invariant scalar wave equation satisfies Huygens' principle. Part III : Petrov type III space-times

We show that there exist no Petrov type D space-times on which the Weyl neutrino equation or Maxwell's equations satisfy Huygens' principle. In passing we give a new proof of the same result for the conformally invariant scalar wave equation which does not require the use of the seventh necessary condition On demontre qu'il n'existe aucun espace-temps de type D de Petrov sur lequel l'equation de neutrino de Weyl ou les equations de Maxwell satisfait au principe d'Huygens. On donne en passant une preuve nouvelle du meme resultat pour l'equation invariante conforme des ondes scalaires ou on n'emploie pas la septieme condition necessaire

An explicit determination of the Petrov type D space-times on which Weyl's neutrino equation and Maxwell's equations satisfy Huygens' principle

Two‐sided conformally recurrent four‐dimensional Riemannian manifolds are defined and analyzed from the point of view of the Petrov–Penrose classification. All two‐sided conformally recurrent four‐dimensional Riemannian manifolds of types D⊗D, N⊗N, and N⊗[−] are given.

Two‐sided conformally recurrent four‐dimensional Riemannian manifolds

Some similarities between 'P-P' wave space-times and complex recurrent and conformally symmetric space-times in general relativity are discussed.

G S Hall

Papers

Complex recurrent space-times