Abstract
In the present paper, we focused our attention to study pseudo-Ricci symmetric spacetimes in Gray’s decomposition subspaces. It is proved that PRS n spacetimes are Ricci flat in trivial, A , and B subspaces, whereas perfect fluid in subspaces I , I ⊕ A , and I ⊕ B , and have zero scalar curvature in subspace A ⊕ B . Finally, it is proved that pseudo-Ricci symmetric GRW spacetimes are vacuum, and as a consequence of this result, we address several corollaries.
Highlights
A pseudo-Ricci symmetric manifold (briefly (PRS)n) is a nonflat pseudo-Riemannian manifold whose Ricci tensor satisfies∇kRij 2AkRij + AiRkj + AjRik, (1)where A is a nonzero 1-form and ∇ indicates the covariant differentiation with respect to the metric g [1].e class of pseudo-Ricci symmetric manifolds is a subclass of weakly Ricci symmetric manifolds which were first introduced and studied by Tamassy and Binh [2]. ere has been much focus on the concept of (PRS)n manifolds; for instance, a sufficient condition on (PRS)n manifolds to be quasi-Einstein manifolds was introduced by De and Gazi [3]. (PRS)n manifolds whose scalar curvature satisfies ∇kR 0 have zero scalar curvature [1]
E class of pseudo-Ricci symmetric manifolds is a subclass of weakly Ricci symmetric manifolds which were first introduced and studied by Tamassy and Binh [2]. ere has been much focus on the concept of (PRS)n manifolds; for instance, a sufficient condition on (PRS)n manifolds to be quasi-Einstein manifolds was introduced by De and Gazi [3]. (PRS)n manifolds whose scalar curvature satisfies ∇kR 0 have zero scalar curvature [1]
E subspace is denoted by B. e Ricci tensors of manifolds in B are Codazzi; that is
Summary
E manifolds in the trivial subspace have parallel Ricci tensor; that is, ∇kRij 0. Generalized Robertson–Walker spacetimes are either Einstein or perfect fluid in Gray’s orthogonal subspaces except one in which the Ricci tensor is not restricted [12]. An n-dimensional Lorentzian manifold is said to be pseudo-Ricci symmetric spacetime if the Ricci tensor satisfies equation (1). It is proved that (PRS)n spacetimes in trivial, A, and B subspaces are Ricci flat, in subspaces I, I⊕A, and I⊕B are perfect fluid spacetimes, and in A⊕B have a zero scalar curvature. A different contraction of equation (1) with gij gives
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