## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

This paper is concerned with the study of [Formula: see text]-manifolds and Ricci solitons. It is shown that in a [Formula: see text]-spacetime, the fluid has vanishing vorticity and vanishing shear. It is found that in an [Formula: see text]-manifold, [Formula: see text] is an irrotational vector field, where [Formula: see text] is a non-zero smooth scalar function. It is proved that in a [Formula: see text]-spacetime with generator vector field [Formula: see text] obeying Einstein equation, [Formula: see text] or [Formula: see text] according to [Formula: see text] or [Formula: see text], where [Formula: see text] is a scalar function and [Formula: see text] is the energy momentum tensor. Also, it is shown that if [Formula: see text] is a non-null spacelike (respectively, timelike) vector field on a [Formula: see text]-spacetime with scalar curvature [Formula: see text] and cosmological constant [Formula: see text], then [Formula: see text] if and only if [Formula: see text] (respectively, [Formula: see text]), and [Formula: see text] if and only if [Formula: see text] (respectively, [Formula: see text]), and further [Formula: see text] if and only if [Formula: see text]. The nature of the scalar curvature of an [Formula: see text]-manifold admitting Yamabe soliton is obtained. Also, it is proved that an [Formula: see text]-manifold admitting [Formula: see text]-Ricci soliton is [Formula: see text]-Einstein and its scalar curvature is constant if and only if [Formula: see text] is constant. Further, it is shown that if [Formula: see text] is a scalar function with [Formula: see text] and [Formula: see text] vanishes, then the gradients of [Formula: see text], [Formula: see text], [Formula: see text] are co-directional with the generator [Formula: see text]. In a perfect fluid [Formula: see text]-spacetime admitting [Formula: see text]-Ricci soliton, it is proved that the pressure density [Formula: see text] and energy density [Formula: see text] are constants, and if it agrees Einstein field equation, then we obtain a necessary and sufficient condition for the scalar curvature to be constant. If such a spacetime possesses Ricci collineation, then it must admit an almost [Formula: see text]-Yamabe soliton and the converse holds when the Ricci operator is of constant norm. Also, in a perfect fluid [Formula: see text]-spacetime satisfying Einstein equation, it is shown that if Ricci collineation is admitted with respect to the generator [Formula: see text], then the matter content cannot be perfect fluid, and further [Formula: see text] with gravitational constant [Formula: see text] implies that [Formula: see text] is a Killing vector field. Finally, in an [Formula: see text]-manifold, it is proved that if the [Formula: see text]-curvature tensor is conservative, then scalar potential and the generator vector field are co-directional, and if the manifold possesses pseudosymmetry due to the [Formula: see text]-curvature tensor, then it is an [Formula: see text]-Einstein manifold.

To read the full-text of this research,

you can request a copy directly from the authors.

... The principal moto of this article is to investigate the curvature inheritance, Ricci solitons and collineations with different curvature tensors such as Ricci, conharmonic, projective curvature tensor. The different aspects of Ricci solitons has been recently studied by Ahsan et al. [1,13,14], Shaikh et al. [62][63][64][65][66][67], Blaga [17]. But Ricci solitons in various spacetimes remain to be investigated yet. ...

The purpose of the article is to investigate the existence of Ricci solitons and the nature of curvature inheritance as well as collineations on the Robinson-Trautman (briefly, RT) spacetime. It is shown that under certain conditions RT spacetime admits almost Ricci soliton, almost $\eta$-Ricci soliton, almost gradient $\eta$-Ricci soliton. As a generalization of curvature inheritance \cite{Duggal1992} and curvature collineation \cite{KLD1969}, in this paper, we introduce the notion of \textit{generalized curvature inheritance} and examine if RT spacetime admits such a notion. It is shown that RT spacetime also realizes the generalized curvature (resp. Ricci, Weyl conformal, concircular, conharmonic, Weyl projective) inheritance. Finally, several conditions are obtained, under which RT spacetime possesses curvature (resp. Ricci, conharmonic, Weyl projective) inheritance as well as curvature (resp. Ricci, Weyl conformal, concircular, conharmonic, Weyl projective) collineation.

The main goal of this paper is to study the properties of generalized Ricci recurrent perfect fluid spacetimes and the generalized Ricci recurrent (generalized Robertson–Walker (GRW)) spacetimes. It is proven that if the generalized Ricci recurrent perfect fluid spacetimes satisfy the Einstein’s field equations without cosmological constant, then the isotropic pressure and the energy density of the perfect fluid spacetime are invariant along the velocity vector field of the perfect fluid spacetime. In this series, we show that a generalized Ricci recurrent perfect fluid spacetime satisfying the Einstein’s field equations without cosmological constant is either Ricci recurrent or Ricci symmetric. An n-dimensional compact generalized Ricci recurrent GRW spacetime with almost Ricci soliton is geodesically complete, provided the soliton vector field of almost Ricci soliton is timelike. Also, we prove that a (GR)n GRW spacetime is Einstein. The properties of (GR)n GRW spacetimes equipped with almost Ricci soliton are studied.

In this paper, we consider the Ricci curvature of a Ricci soliton. In particular, we have showed that a complete gradient Ricci soliton with non-negative Ricci curvature possessing a non-constant convex potential function having finite weighted Dirichlet integral satisfying an integral condition is Ricci flat and also it isometrically splits a line. We have also proved that a gradient Ricci soliton with non-constant concave potential function and bounded Ricci curvature is non-shrinking and hence the scalar curvature has at most one critical point.

We prove that in Robertson–Walker space-times (and in generalized Robertson–Walker spacetimes of dimension greater than 3 with divergence-free Weyl tensor) all higher-order gravitational corrections of the Hilbert–Einstein Lagrangian density F(R,□R,…,□kR)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(R,\square R, \ldots , \square ^k R)$$\end{document} have the form of perfect fluids in the field equations. This statement definitively allows to deal with dark energy fluids as curvature effects.

The present paper deals with the study of CR-submanifolds of (LCS)n-manifolds with respect to quarter symmetric non-metric connection. We investigate integrability of the distributions and the geometry of foliations. The totally umbilical CR-submanifolds of said ambient manifolds are also studied. An example is presented to illustrate the results.

In an-dimensional Friedmann-Robertson-Walker metric, it is rigorously shown that any analytical theory of gravity f(R,G), where R is the curvature scalar and G is the Gauss-Bonnet topological invariant, can be associated to a perfect-fluid stress-energy tensor. In this perspective, dark components of the cosmological Hubble flow can be geometrically interpreted
].

We show that an n-dimensional generalized Robertson-Walker (GRW) space-time with divergence-free conformal curvature tensor exhibits a perfect fluid stress-energy tensor for any f(R) gravity model. Furthermore we prove that a conformally flat GRW space-
time is still a perfect fluid in both f(R) and quadratic gravity where other curvature invariants are considered.

Generalized Robertson-Walker spacetimes extend the notion of Robertson-Walker spacetimes, by allowing for spatial non-homogeneity. A survey is presented, with main focus on Chen's characterization in terms of a timelike concircular vector. Together with their most important properties, some new results are presented.

We prove theorems about the Ricci and the Weyl tensors on generalized Robertson-Walker space-times of dimension $n\ge 3$. In particular, we show that the concircular vector introduced by Chen decomposes the Ricci tensor as a perfect fluid term plus a term linear in the contracted Weyl tensor. The Weyl tensor is harmonic if and only if it is annihilated by Chen's vector, and any of the two conditions is necessary and sufficient for the GRW space-time to be a quasi-Einstein (perfect fluid) manifold. Finally, the general structure of the Riemann tensor for Robertson-Walker space-times is given, in terms of Chen's vector. A GRW space-time in n = 4 with null conformal divergence is a Robertson-Walker space-time.

The object of the present paper is to introduce a new curvature tensor, named generalized quasi-conformal curvature tensor which bridges conformal curvature tensor, concircular curvature tensor, projective curvature tensor and conharmonic curvature tensor. Flatness and symmetric properties of generalized quasi-conformal curvature tensor are studied in the frame of (k, μ)-contact metric manifolds.

The object of the present paper is to study the invariant submanifolds of (LCS)n-manifolds. We study semiparallel and 2-semiparallel invariant submanifolds of (LCS)n-manifolds. Among others we study 3-dimensional invariant submanifolds of (LCS)n-manifolds. It is shown that every 3-dimensional invariant submanifold of a (LCS)n-manifold is totally geodesic.

A perfect-fluid space-time of dimension n ≥ 4, with (1) irrotational velocity vector field and (2) null divergence of the Weyl tensor, is a generalised Robertson-Walker space-time with an 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.

The object of the present paper is to study the second order parallel symmetric tensors and Ricci solitons on (LCS)n-manifolds. We found the conditions of Ricci soliton on (LCS)n-manifolds to be shrinking , steady and expanding respectively.

The present paper deals with a study of -pseudo symmetric and -pseudo Ricci symmetric -manifolds. It is shown that every -pseudo symmetric -manifold and -pseudo Ricci symmetric -manifold are -Einstein manifold.

We show new results on when a pseudo-slant submanifold is a LCS-manifold. Necessary and sufficient conditions for a submanifold to be pseudo-slant are given. We obtain necessary and sufficient conditions for the integrability of distributions which are involved in the definition of the pseudo-slant submanifold. We characterize the pseudoslant product and give necessary and sufficient conditions for a pseudo-slant submanifold to be the pseudo-slant product. Also we give an example of a slant submanifold in an LCS-manifold to illustrate the subject.

The object of the present paper is to study -manifolds. Several interesting results on a -manifold are obtained. Also the generalized Ricci recurrent -manifolds are studied. The existence of such a manifold is ensured by several non-trivial new examples.

We prove that a real hypersurface in a non-flat complex space form does not admit
a Ricci soliton whose potential vector field is the Reeb vector field. Moreover,
we classify a real hypersurface admitting so-called “$\eta$-Ricci
soliton” in a non-flat complex space form.

In this paper, we have proved that if a complete conformally flat gradient shrinking Ricci soliton has linear volume growth or the scalar curvature is finitely integrable and also the reciprocal of the potential function is subharmonic, then the manifold is isometric to the Euclidean sphere. As a consequence, we have showed that a four dimensional gradient shrinking Ricci soliton satisfying some conditions is isometric to S4 or RP4 or CP2. We have also deduced a condition for the shrinking Ricci soliton to be compact with quadratic volume growth.

The objective, in this paper, is to obtain the curvature properties of (t−z)-type plane wave metric studied by Bondi et al. (1959). For this a general (t−z)-type wave metric is considered and the condition for which it obeys Einstein’s empty spacetime field equations is obtained. It is found that the rank of the Ricci tensor of (t−z)-type plane wave metric is 1 and is of Codazzi type. Also it is proved that it is not recurrent but Ricci recurrent, conformally recurrent and hyper generalized recurrent. Moreover, it is semisymmetric and satisfies the Ricci generalized pseudosymmetric type condition P⋅P=−13Q(Ric,P). It is interesting to note that, physically, the energy momentum tensor describes a radiation field with parallel rays and geometrically it is a Codazzi tensor and semisymmetric. As special case, the geometric structures of Taub’s plane symmetric spacetime metric are deduced. Comparisons between (t−z)-type plane wave metric and pp-wave metric with respect to their geometric structures are viewed.

In this paper we have investigated the curvature restricted geometric properties of the generalized Kantowski–Sachs (briefly, GK–S) spacetime metric, a warped product of 2-dimensional base and 2-dimensional fibre. It is proved that GK–S metric describes a generalized Roter type, 2-quasi Einstein and Ein(3) manifold. It also has pseudosymmetric Weyl conformal tensor as well as conharmonic tensor and its conformal 2-forms are recurrent. Further, it realizes the curvature condition R⋅R=Q(S,R)+L(t,θ)Q(g,C) (see, Theorem 4.1). We have also determined the curvature properties of Kantowski–Sachs (briefly, K–S), Bianchi type-III and Bianchi type-I metrics which are the special cases of GK–S spacetime metric. The sufficient condition under which GK–S metric represents a perfect fluid spacetime has also been obtained.

This paper aims to investigate the curvature restricted geometric properties admitted by Melvin magnetic spacetime metric, a warped product metric with 1-dimensional fibre. For this, we have considered a Melvin type static, cylindrically symmetric spacetime metric in Weyl form and it is found that such metric, in general, is generalized Roter type, Ein(3) and has pseudosymmetric Weyl conformal tensor satisfying the pseudosymmetric type condition R⋅R−Q(S,R)=L′Q(g,C). The condition for which it satisfies the Roter type condition has been obtained. It is interesting to note that Melvin magnetic metric is pseudosymmetric and pseudosymmetric due to conformal tensor. Moreover such metric is 2-quasi-Einstein, its Ricci tensor is Reimann compatible and Weyl conformal 2-forms are recurrent. The Maxwell tensor is also pseudosymmetric type.

This paper is concerned with the study of the geometry of (charged) Nariai spacetime, a topological product spacetime, by means of covariant derivative(s) of its various curvature tensors. It is found that on this spacetime the condition [Formula: see text] is satisfied and it also admits the pseudosymmetric type curvature conditions [Formula: see text] and [Formula: see text]. Moreover, it is [Formula: see text]-dimensional Roter type, [Formula: see text]-quasi-Einstein and generalized quasi-Einstein spacetime. The energy–momentum tensor is expressed explicitly by some [Formula: see text]-forms. It is worthy to see that a generalization of such topological product spacetime proposes to exist with a class of generalized recurrent type manifolds which is semisymmetric. It is observed that the rank of [Formula: see text], [Formula: see text], of Nariai spacetime (NS) is [Formula: see text] whereas in case of charged Nariai spacetime (CNS) it is [Formula: see text], which exhibits that effects of charge increase the rank of Ricci tensor. Also, due to the presence of charge in CNS, it gives rise to the proper pseudosymmetric type geometric structures.

The objective of this paper is to study the curvature restricted geometric properties of anisotropic nonrelativistic scale invariant metrics, namely, Lifshitz and Schrödinger spacetime metrics. It is found that the Lifshitz spacetime metric admits two important pseudosymmetric type curvature conditions [Formula: see text] and [Formula: see text]. Also, it is [Formula: see text]-quasi Einstein and generalized Roter type manifold. Finally, Lifshitz spacetime is compared with Schrödinger spacetime.

(CS)4-spacetimes with Einstein field equations under some curvature restriction named
generalized weakly Ricci-symmetry have been studied. We have proved that if the characteristic
vector field ξ of a generalized weakly Ricci-symmetric (CS)4-spacetime obeying
Einstein equation is a Killing vector field, then such a spacetime admits (i) curvature
collineation, (ii) conformal collineation, (iii) conharmonic collineation, (iv) concircular
collineation, (v) projective collineation, (vi) m -projective collineation. It is further proved
that each of conformally flat, conharmonically flat, concircularly flat, projectively flat and
m-projectively flat generalized weakly Ricci-symmetric (CS)4 -spacetime is infinitesimally
spatially isotropic relative to the unit timelike vector field ξ .

The aim of this note is to define almost Yamabe solitons as special conformal solutions of the Yamabe flow. Moreover, we shall obtain some rigidity results concerning Yamabe almost solitons. Finally, we shall give some characterizations for homogeneous gradient Yamabe almost solitons.

The object of the present paper is to introduce the notion of generalized φ‐recurrent and study its
(LCS)
n
‐manifolds
various geometric properties with the existence by an interesting example.