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This paper introduces a description of a doubly warped product manifold, taking into account certain conditions related to the projective curvature tensor. We demonstrate that the factor manifolds of a projectively flat (symmetric) doubly warped product manifold possess constant sectional curvature. In the flatness scenario, a doubly warped product manifold reduces to a singly warped product manifold. We establish that the factor manifolds of a doubly warped product manifold with harmonic projective curvature tensor are Einstein manifolds and exhibit harmonic projective curvature tensor. In Sec. VI, we provide evidence that a projectively flat (symmetric) generalized Robertson–Walker space-time is both a perfect fluid and static. [ABSTRACT FROM AUTHOR]

Generalized Ricci recurrent spacetimes (GR) n are investigated in Gray's seven subspaces. It is proved that a (GR) n spacetime in all subspaces but one is an Einstein spacetime. The subspace ℐ cannot contain a (GR) n spacetime. Further, the subspaces ℐ ⊕ A and ℐ ⊕ B reduce to A and B , respectively. Next, we prove that a (GR) n spacetime is Ricci semi-symmetric if and only if either the spacetime is Einstein or the vector field Al is closed. Further, it is shown that the Ricci tensor of (GR) n is Riemann compatible if Al is closed. Finally, sufficient conditions are given on a (GR) n warped product manifold to guarantee that the factor manifolds are Einstein. Moreover, it is shown that a generalized Ricci recurrent GRW spacetime is an Einstein spacetime. [ABSTRACT FROM AUTHOR]

The object of the present paper is to characterize LP-Sasakian manifolds satisfying Ricci pseudosymmetry and Ricci generalized pseudosymmetry. Beside this we prove that if R(X; ξ):P = P(X; ξ):R holds, where R and P denote the curvature tensor and projective curvature tensor respectively, then the manifold becomes an Einstein manifold. Then we prove that divR = 0 and divC = 0 are equivalent if the scalar curvature is invariant under the characteristic vector field ξ, where 'div' denotes divergence. Finally, we characterize 3-dimensional LP-Sasakian manifolds admitting Yamabe solitons and prove that the scalar curvature is constant and the potential vector field V is Killing. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study N(k)-quasi Einstein manifolds. Existence of N(k)-quasi Einstein manifolds are proved by two non-trivial examples. Also a physical example of an N(k)-quasi-Einstein manifold is given. We study an N(k)-quasi-Einstein manifold satisfying the curvature conditions ˜Z(ξ,X)·S = 0, P(ξ,X)· ˜Z = 0, ˜Z(ξ,X)·P = 0, ˜Z(ξ,X)·C = 0 and P(ξ,X)· C = 0. Finally, we study Ricci-pseudosymmetric N(k)-quasi-Einstein manifolds. [ABSTRACT FROM AUTHOR]

The object of the present paper is to classify an LP-Sasakian manifold with a coefficient a satisfying certain curvature conditions. Also ℰ-concircularly flat and ø-concircularly flat LP-Sasakian manifold with a coefficient α have been studied. Finally, we have shown the applications of such a manifold in general relativity and cosmology. [ABSTRACT FROM AUTHOR]