On a Kaehlerian space-time manifold

TL;DR

This paper explores the properties of perfect fluid Kaehler space-time manifolds, demonstrating conditions under which they are Einstein manifolds, and analyzing their symmetry properties and scalar curvature implications.

Contribution

It provides new insights into the structure of perfect fluid Kaehler space-time manifolds, including conditions for Einstein manifolds and the non-existence of certain symmetric cases.

Findings

Perfect fluid Kaehler space-time manifolds are Einstein manifolds when pressure and energy density vanish.

Conformally flat perfect fluid Kaehler space-time manifolds are infinitesimally spatially isotropic.

Weakly Ricci symmetric perfect fluid Kaehler space-time manifolds with non-zero scalar curvature do not exist.

Abstract

In this paper, the theory of space-time in 4-dimensional Kaehler manifold has been studied. We have discussed the Einstein equation with cosmological constant in perfect fluid Kaehler space-time manifold and proved that the isotropic pressure, energy density and the energy momentum tensor vanish and such a space-time manifold is an Einstein manifold. We have shown also that a conformally flat perfect fluid Kaehler space-time manifold is infinitesimally spatially isotropic relative to the velocity vector field. In last two sections, we have studied weakly symmetric and weakly Ricci symmetric perfect fluid Kaehler space-time manifolds and it has been shown, either the manifold is of zero scalar curvature or the associated vector fields rho and alpha are related by g(rho,alpha) = 4. At last, we have proved that the weakly Ricci symmetric perfect fluid Kaehler space-time manifold of non-zero scalar curvature does not exist.