Search

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]

This article presents characterizations of warped product manifolds based on the flatness and symmetry of the conharmonic curvature tensor. It is proved that when a warped product manifold is conharmonically flat, both the base and fiber manifolds exhibit constant sectional curvatures. In addition, the specific forms of the conharmonic curvature tensor are derived for both the base and fiber manifolds. It is demonstrated that in a conharmonically symmetric warped product manifold, the fiber manifold has a constant sectional curvature, while the base manifold is both Cartan-symmetric and conharmonically symmetric. In this scenario, the form of the conharmonic curvature tensor on the fiber manifold is determined. We characterize the generalized Robertson–Walker (GRW) space-time through the flatness and the symmetry of the conharmonic curvature tensor. It is shown that a conharmonically flat (symmetric) GRW space-time is a perfect fluid. Finally, a conharmonically flat standard static space-time is investigated. [ABSTRACT FROM AUTHOR]

The section 1 of this paper deals with the definition for a pseudo symmetric spacetime. In section 2, it is proved that a (PZS)4 spacetime satisfying Codazzi type of Z-tensor does not exist. In which condition a (PZS)4 spacetime can be a perfect fluid has been found as a necessary and sufficient condition. After that, special properties are obtained if the (PZS)4 spacetime has harmonic conformal curvature tensor. [ABSTRACT FROM AUTHOR]

In this article, we initiate the study of conformal vector fields in paracontact geometry. First, we establish that if the Reeb vector field ζ is a conformal vector field on N(k)-paracontact metric manifold, then the manifold becomes a para-Sasakian manifold. Next, we show that if the conformal vector field V is pointwise collinear with the Reeb vector field ζ, then the manifold recovers a para-Sasakian manifold as well as V is a constant multiple of ζ. Furthermore, we prove that if a 3-dimensional N(k)-paracontact metric manifold admits a Killing vector field V, then either it is a space of constant sectional curvature k or, V is an infinitesimal strict paracontact transformation. Besides these, we also investigate conformal vector fields on para-Kenmotsu manifolds. [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]

In this study, we analyze almost pseudo-Ricci symmetric spacetimes endowed with Gray's decomposition, as well as generalized Robertson—Walker spacetimes. For almost pseudo-Ricci symmetric spacetimes, we determine the form of the Ricci tensor in all the O (n)-invariant subspaces provided by Gray's decomposition of the gradient of the Ricci tensor. In three cases we obtain that the Ricci tensor is in the form of perfect fluid and in one case the spacetime becomes a generalized Robertson—Walker spacetime. In other cases we obtain some algebraic results. Finally, it is shown that an almost pseudo-Ricci symmetric generalized Robertson—Walker spacetime is a perfect fluid 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]

We study the g-Tanaka-Webster connection associated to a Kenmotsu structure. With the help of g-Tanaka-Webster connection we characterize Kenmotsu manifolds and find certain curvature properties of this connection on Kenmotsu manifolds. Finally an illustrative example is given to verify some results. [ABSTRACT FROM AUTHOR]

In this paper we study ξ-projectively flat and ϕ-projectively flat 3-dimensional normal almost paracontact metric manifolds. Finally, an illustrative example is given. [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]

The object of the present paper is to study submanifolds satisfying Chen's equality in a Euclidean space. Also we study submanifolds satisfying the condition C.C = 0 and C.S = 0, where C denotes the concircular curvature tensor. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study a Kenmotsu manifold admitting Ricci solitons and gradient Ricci solitons. [ABSTRACT FROM AUTHOR]

In the present paper we prove that in an almost Kenmotsu manifold with generalized (κ,μ)'-nullity distribution the three conditions: (i) the Ricci tensor of M2n+1 is of Codazzi type, (ii) the manifold M2n+1 satisfies div C = 0, (iii) the manifold M2n+1 is locally isometric to Hn+1(-4) x Rn, are equivalent. Also we prove that if the manifold satisfies the cyclic parallel Ricci tensor, then the manifold is locally isometric to Hn+1(-4) x Rn. [ABSTRACT FROM AUTHOR]

We consider a conformally flat almost pseudo-Ricci symmetric spacetime. At first we show that a conformally flat almost pseudo-Ricci symmetric spacetime can be taken as a model of the perfect fluid spacetime in general relativity and cosmology. Next we show that if in a conformally flat almost pseudo-Ricci symmetric spacetime the matter distribution is perfect fluid whose velocity vector is the vector field corresponding to 1-form B of the spacetime, the energy density and the isotropic pressure are not constants. We also show that a conformally flat almost pseudo-Ricci symmetric spacetime is the Robertson-Walker spacetime. Finally we give an example of a conformally flat almost pseudo-Ricci symmetric spacetime with non-zero non-constant scalar curvature admitting a concircular vector field. [ABSTRACT FROM AUTHOR]

We classify Lorentzian para-Sasakian manifolds which satisfy P · C = 0, Z · C = LCQ(g;C); P · Z - Z · P = 0, and P · Z + Z · P = 0; where P is the v-Weyl projective tensor, Z is the concircular tensor, and C is the Weyl conformal curvature tensor. [ABSTRACT FROM AUTHOR]