On pseudo-𝒦-symmetric semi-Riemannian manifolds and relativistic applications

Abstract

In this paper, we present a novel curvature tensor called the [Formula: see text]-curvature tensor, which is formed by combining the Weyl tensor and the [Formula: see text]-curvature tensor. It is proved that a semi-Riemannian manifold with traceless [Formula: see text]-curvature tensor is Einstein, while a [Formula: see text]-curvature flat semi-Riemannian manifold exhibits constant sectional curvature. We show that a space-time with traceless [Formula: see text]-curvature tensor or [Formula: see text]-curvature flat represents dark energy era. A semi-Riemannian manifold with divergence-free [Formula: see text]-curvature tensor is Weyl harmonic. In a space-time where the [Formula: see text]-curvature tensor is of divergence-free, both the isotropic pressure and energy density are constants. Under certain conditions, it is shown that a pseudo-[Formula: see text]-symmetric semi-Riemannian manifold reduces to a pseudo-Riemannian manifold. It is demonstrated that pseudo-[Formula: see text]-symmetric space-times can be regarded as generalized Robertson–Walker (GRW) space-times. Finally, a concrete example of pseudo-[Formula: see text]-symmetric semi-Riemannian manifolds is presented.