A condition for a perfect-fluid space-time to be a generalized Robertson-Walker space-time

A perfect-fluid space-time of dimension n>3 with 1) irrotational velocity vector field, 2) null divergence of the Weyl tensor, is a generalised Robertson-Walker space-time with Einstein fiber. Condition 1) is verified whenever pressure and energy density are related by an equation of state. The contraction of the Weyl tensor with the velocity vector field is zero. Conversely, a generalized Robertson-Walker space-time with null divergence of the Weyl tensor is a perfect-fluid space-time.Comment: 7 pages. Misprint corrected in Sect II

A note on Almost Riemann Soliton and gradient almost Riemann soliton

The quest of the offering article is to investigate \emph{almost Riemann soliton} and \emph{gradient almost Riemann soliton} in a non-cosymplectic normal almost contact metric manifold $M^3$. Before all else, it is proved that if the metric of $M^3$ is Riemann soliton with divergence-free potential vector field $Z$, then the manifold is quasi-Sasakian and is of constant sectional curvature -$\lambda$, provided $\alpha,\beta =$ constant. Other than this, it is shown that if the metric of $M^3$ is \emph{ARS} and $Z$ is pointwise collinear with $\xi$ and has constant divergence, then $Z$ is a constant multiple of $\xi$ and the \emph{ARS} reduces to a Riemann soliton, provided $\alpha,\;\beta =$constant. Additionally, it is established that if $M^3$ with $\alpha,\; \beta =$ constant admits a gradient \emph{ARS} $(\gamma,\xi,\lambda)$, then the manifold is either quasi-Sasakian or is of constant sectional curvature $-(\alpha^2-\beta^2)$. At long last, we develop an example of $M^3$ conceding a Riemann soliton

On Almost Pseudo-Z-symmetric Manifolds

summary:The object of the present paper is to study almost pseudo-Z-symmetric manifolds. Some geometric properties have been studied. Next we consider conformally flat almost pseudo-Z-symmetric manifolds. We obtain a sufficient condition for an almost pseudo-Z-symmetric manifold to be a quasi Einstein manifold. Also we prove that a totally umbilical hypersurface of a conformally flat $A(PZS)_{n}$ ($n>3$) is a manifold of quasi constant curvature. Finally, we give an example to verify the result already obtained in Section 5

On Φ-recurrent N(k)-contact Metric Manifolds

In this paper we prove that a &#934;-recurrent N(k)-contact metric manifold is an &#951;-Einstein manifold with constant coefficients. Next, we prove that a 3-dimensional &#934;-recurrent N(k)-contact metric manifold is of constant curvature. The existence of a &#934;-recurrent N(k)-contact metric manifold is also proved.</p

Characterizations of Perfect fluid spacetimes obeying f(R)f(\mathcal{R})-gravity equipped with different gradient solitons

The prime object of this article is to study the perfect fluid spacetimes obeying $f(\mathcal{R})$-gravity, when $\eta$-Ricci solitons, gradient $\eta$-Ricci solitons, gradient Einstein Solitons and gradient $m$-quasi Einstein solitons are its metrics. At first, the existence of the $\eta$-Ricci solitons is proved by a non-trivial example. We establish conditions for which the $\eta$-Ricci solitons are expanding, steady or shrinking. Besides, in the perfect fluid spacetimes obeying $f(\mathcal{R})$-gravity, when the potential vector field of $\eta$-Ricci soliton is of gradient type, we acquire a Poisson equation. Moreover, we investigate gradient $\eta$-Ricci solitons, gradient Einstein Solitons and gradient $m$-quasi Einstein solitons in $f(\mathcal{R})$-gravity, respectively. As a result, we establish some significant theorems about dark matter era.Comment: 1