Conformally quasi-recurrent pseudo-Riemannian manifolds

TL;DR

This paper explores conformally quasi-recurrent pseudo-Riemannian manifolds, revealing new geometric properties, including Weyl compatibility, quasi-Einstein conditions, and specific characteristics of 4D Lorentzian cases like Petrov type-N space-times.

Contribution

It introduces new results on the structure of CQR manifolds, including Weyl compatibility of Ricci tensors, conditions for quasi-Einstein manifolds, and detailed properties of 4D Lorentzian CQR space-times.

Findings

Ricci tensor and gradient of fundamental vector are Weyl compatible.

CQR manifolds with concircular fundamental vector are quasi-Einstein.

4D Lorentzian CQR manifolds have null, unique fundamental vectors that are Ricci eigenvectors and generate geodesics.

Abstract

Conformally quasi-recurrent (CQR)_n pseudo-Riemannian manifolds are investigated, and several new results are obtained. It is shown that the Ricci tensor and the gradient of the fundamental vector are Weyl compatible tensors (the notion was introduced recently by the authors and applies to significative space-times), (CQR)_n manifolds with concircular fundamental vector are quasi-Einstein. For 4-dimensional (CQR)$_4$ Lorentzian manifolds the fundamental vector is null and unique up to a scaling, it is an eigenvector of the Ricci tensor, and its integral curves are geodesics. Such space-times are Petrov type-N with respect to the fundamental null vector.