On Conformally at Almost Pseudo Ricci Symmetric Mani-folds

Abstract

The object of the present paper is to study conformally flat almost pseudo Ricci symmetric manifolds. The existence of a conformally flat almost pseudo Ricci symmetric manifold with non-zero and non-constant scalar curvature is shown by a non-trivial example. We also show the existence of an n-dimensional non-conformally flat almost pseudo Ricci symmetric manifold with vanishing scalar curvature.

On Conformally Flat Almost Pseudo-Ricci Symmetric Spacetimes

“…In a recent paper De and Gazi [5] proved that in a conformally flat A(PRS) n (n > 3), the vector field Q defined by g(X, Q) = B(X) is a concircular vector field. K. Yano [19] proved that in order that a Riemannian space admits a concircular vector field, it is necessary and sufficient that there exists a coordinate system with respect to which the fundamental quadratic differential form may be written in the form ds 2 = dx 1 2 + e q g * αβ dx α dx β , (6.1) where g * αβ = g δ αβ (x γ ) are the function of x γ only (α, β, γ, δ = 2, 3, .…”

“…From [5], we know that a conformally flat A(PRS) 4 is a quasi-Einstein spacetime. So a conformally flat A(PRS) 4 spacetime can be taken as a model of the perfect fluid spacetime in general relativity.…”

“…A non-flat semi-Riemannian manifold (M n , g) (n > 3), is called an almost pseudo-Ricci symmetric manifold [5] if its Ricci tensor S of type (0, 2) is not identically zero and satisfies the condition…”

Recurrent Z Forms on Riemannian and Kaehler Manifolds

“…Such a manifold is called almost pseudo-Ricci symmetric and denoted by A(PRS) n : A k and B k are non-null covectors. In [13] the properties of conformally flat A(PRS) n are studied. In the same paper the authors pointed out the importance of pseudo-Ricci symmetric manifolds in the theory of General Relativity.…”

“…(1.4) This kind of manifold is called pseudo-projective Ricci symmetric and denoted by (PPRS) n . Recently in [5] and [13] a further generalization of the condition of a (PRS) n manifold was considered. More precisely a manifold whose Ricci tensor 1250059-2…”