In this paper, we introduce the notion of recurrent conformal 2-forms on a pseudo-Riemannian manifold of arbitrary signature. Some theorems already proved for the same differential structure on a Riemannian manifold are proven to hold in this more general contest. Moreover other interesting results are pointed out; it is proven that if the associated covector is closed, then the Ricci tensor is Riemann compatible or equivalently, Weyl compatible: these notions were recently introduced and investigated by one of the present authors. Further some new results about the vanishing of some Weyl scalars on a pseudo-Riemannian manifold are given: it turns out that they are consequence of the generalized Derdzinski–Shen theorem. Topological properties involving the vanishing of Pontryagin forms and recurrent conformal 2-forms are then stated. Finally, we study the properties of recurrent conformal 2-forms on Lorentzian manifolds (space-times). Previous theorems stated on a pseudo-Riemannian manifold of arbitrary signature are then interpreted in the light of the classification of space-times in four or in higher dimensions.

Recurrent conformal 2-forms on pseudo-Riemannian manifolds

The structure of recurrently related operators of the curvature tensor and of its covariant derivatives for n-dimensional Riemannian spaces with arbitrary signature is examined. Applications to Einstein's theory of gravitation are given.

Structure of space-time curvature

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

Conformally recurrent pseudo-Riemannian manifolds of dimension n>4 are investigated. The Weyl tensor is represented as a Kulkarni-Nomizu product. If the square of the Weyl tensor is nonzero, a covariantly constant symmetric tensor is constructed, that is quadratic in the Weyl tensor. Then, by Grycak's theorem, the explicit expression of the traceless part of the Ricci tensor is obtained, up to a scalar function. The Ricci tensor has at most two distinct eigenvalues, and the recurrence vector is an eigenvector. Lorentzian conformally recurrent manifolds are then considered. If the square of the Weyl tensor is nonzero, the manifold is decomposable. A null recurrence vector makes the Weyl tensor of algebraic type IId or higher in the Bel - Debever - Ortaggio classification, while a time-like recurrence vector makes the Weyl tensor purely electric.

On conformally recurrent manifolds of dimension greater than 4

In this paper we study the properties of conformally recur- rent pseudo Riemannian manifolds admitting a proper conformal vector field with respect to the scalar field �, focusing particularly on the 4- dimensional Lorentzian case. Some general properties already proven by one of the present authors for pseudo conformally symmetric manifolds endowed with a conformal vector field are proven also in the case, and some new others are stated. Moreover interesting results are pointed out; for example, it is proven that the Ricci tensor under certain conditions is Weyl compatible: this notion was recently introduced and investigated by one of the present authors. Further we study conformally recurrent 4-dimensional Lorentzian manifolds (space-times) admitting a conformal vector field: it is proven that the covectorj is null and unique up to scaling; moreover it is shown that the same vector is an eigenvector of the Ricci tensor. Finally, it is stated that such space-time is of Petrov type N with respect toj.

http://koreascience.or.kr:80/article/JAKO201414938219890.pdf

A. H. Thompson

Papers

Second-order recurrent space-times in general relativity

Robinson-Trautmann solutions on compact manifolds