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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]

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]

Let (M, g) be a non-Kenmotsu (κ, μ)′-almost Kenmotsu manifold of dimension 2n + 1. In this paper, we prove that if the metric g of M is a *-Ricci soliton, then either M is locally isometric to the product ℍn+1(−4)×ℝn or the potential vector field is strict infinitesimal contact transformation. Moreover, two concrete examples of (κ, μ)′-almost Kenmotsu 3-manifolds admitting a Killing vector field and strict infinitesimal contact transformation are given.

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]

In the present paper we study three dimensional cosymplectic manifolds admitting almost Ricci solitons. Among others, we prove that in a three dimensional compact orientable cosymplectic manifold M³ without boundary, an almost Ricci soliton reduces to a Ricci soliton under certain restriction on the potential function λ. As a consequence we obtain a corollary. Moreover, we study gradient almost Ricci solitons. [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]

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]