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The primary goal of this paper is to examine spacetimes admitting semiconformal curvature tensor in [Formula: see text] modify gravity. The semiconformal flatness of general spacetime and spacetime in [Formula: see text] gravity with perfect fluid, has been analyzed. For this consideration, we generate the forms of isotropic pressure [Formula: see text] and energy density [Formula: see text]. After that, a few energy conditions are taken into account. Finally, we study the divergence-free semiconformal curvature tensor in [Formula: see text] gravity in presence of perfect fluid. We emphasize that for recurrent or bi-recurrent energy–momentum tensor, Ricci tensor of this spacetime is semi-symmetric and consequently, the resulting spacetimes either accomplish inflation or possess fixed isotropic pressure and energy density.

Sufficient and necessary conditions are provided on warped product manifolds and their base and fiber manifolds for a vector field $$\varphi _{j}$$ to be a $$\varphi \left( \mathrm {Ric}\right) $$ -vector field , that is, $$\nabla _{i}\varphi _{j}=\mu R_{ij}$$ where $$R_{ij}$$ is the Ricci tensor of M and $$ \mu $$ is a scalar. Two warped product space-times admitting $$\varphi \left( \mathrm {Ric}\right) $$ -vector fields are considered. Lorentzian quasi-Einstein manifolds admitting a time-like $$\varphi \left( \mathrm {Ric} \right) $$ -vector field are shown to be either Ricci simple or a perfect fluid GRW space-time. The generators of a Lorentzian generalized quasi-Einstein manifold admitting a time-like $$\varphi \left( \mathrm {Ric} \right) $$ -vector field are eigenvectors of the Ricci tensor with zero eigenvalue.

We prove that an almost pseudo [Formula: see text] symmetric spacetime with Codazzi type of [Formula: see text] tensor is perfect fluid whose velocity vector field [Formula: see text] is parallel. The energy density [Formula: see text] of such perfect fluid spacetime is constant and the state equation is obtained. Also, this spacetime is shown to be a static spacetime. Next, it is shown that such spacetime is a GRW spacetime with Einstein fiber. This kind of spacetime is investigated in f (R) theory of gravity; in this case, we find the forms of the isotropic pressure p and the energy density [Formula: see text] which are constant. Further, some energy conditions are studied in this spacetime. Finally, a concrete example of almost pseudo [Formula: see text] symmetric spacetimes is considered.

In this paper, warped product mixed generalized quasi-Einstein manifolds MG(QE) $$_{n}$$ are studied. It is shown that a Ricci recurrent MG(QE) $$_{n}$$ manifold is a product manifold whose factors are Ricci recurrent. Sufficient conditions for a MG(QE) $$_{n}$$ to be a G(QE) $$_{n}$$ are derived. Then, we classify three types of warped product MG(QE) $$_{n}$$ . For example, it is proved that the fiber manifold is Einstein in the first type, (QE) $$_{n}$$ in the second type and MG(QE) $$_{n}$$ in the third type.

In this paper, we introduce a new type of curvature tensor named H-curvature tensor of type (1, 3) which is a linear combination of conformal and projective curvature tensors. First we deduce some basic geometric properties of H-curvature tensor. It is shown that a H-flat Lorentzian manifold is an almost product manifold. Then we study pseudo H-symmetric manifolds (PHS)n (n > 3) which recovers some known structures on Lorentzian manifolds. Also, we provide several interesting results. Among others, we prove that if an Einstein (PHS)n is a pseudosymmetric (PS)n, then the scalar curvature of the manifold vanishes and conversely. Moreover, we deal with pseudo H-symmetric perfect fluid spacetimes and obtain several interesting results. Also, we present some results of the spacetime satisfying divergence free H-curvature tensor. Finally, we construct a non-trivial Lorentzian metric of (PHS)4.

When minimal length uncertainty emerging from generalized uncertainty principle (GUP) is thoughtfully implemented, it is of great interest to consider its impacts on {\it "gravitational} Einstein field equations (gEFE) and to try to find out whether consequential modifications in metric manifesting properties of quantum geometry due to quantum gravity. GUP takes into account the gravitational impacts on the noncommutation relations of length (distance) and momentum operators or time and energy operators, etc. On the other hand, gEFE relates {\it classical geometry or general relativity gravity} to the energy-momentum tensors, i.e. proposing quantum equations of state. Despite the technical difficulties, we confront GUP to the metric tensor so that the line element and the geodesic equation in flat and curved space are accordingly modified. The latter apparently encompasses acceleration, jerk, and snap (jounce) of a particle in the {\it "quasi-quantized"} gravitational field. Finite higher-orders of acceleration apparently manifest phenomena such as accelerating expansion and transitions between different radii of curvature, etc., Comment: 4 pages, 0 figure, invited talk at 9th int. Workshop on Astronomy and Relativistic Astrophysics (IWARA2020 Video Conference). accepted for publication in Astronomical Note (AN)

Sufficient conditions for a Lorentzian generalized quasi-Einstein manifold M,g,f,μ to be a generalized Robertson–Walker spacetime with Einstein fibers are derived. The Ricci tensor in this case gains the perfect fluid form. Likewise, it is proven that a λ,n+m-Einstein manifold M,g,w having harmonic Weyl tensor, ∇jw∇mwCjklm=0 and ∇lw∇lw

This study aims mainly at investigating the effects of concircular flatness and concircular symmetry of a warped product manifold on its fibre and base manifolds. Concircularly flat and concircularly symmetric warped product manifolds are investigated. The divergence free concircular curvature tensor on warped product manifolds is considered. Finally, we apply some of these results to generalized Robertson-Walker and standard static space-times.

The purpose of this note is to study and explore some collineation vector fields on standard static spacetimes Ifâ¯Ãâ¯M(also called fâ associated SSST). Conformal vector fields, Ricci and matter collineations are studied. Many implications for the existence of these collineations on fâassociated SSSTs are obtained. Moreover, Ricci soliton structures on fâ associated SSSTs admitting a potential conformal vector field are considered. Keywords: Conformal vector fields, Ricci collineations, Standard static spacetime, Ricci soliton, MSC: Primary 53C21, Secondary 53C25 53C50

In this note, we study and explore locally symmetric f−associated standard static spacetimes If×M. Necessary and sufficient conditions on f−associated standard static spacetimes to be locally symmetric are derived. Some implications for these conditions are considered.

In this note, we introduce a new type of warped products called as sequential warped products to cover a wider variety of exact solutions to Einstein?s field equation. First, we study the geometry of sequential warped products and obtain covariant derivatives, curvature tensor, Ricci curvature and scalar curvature formulas. Then some important consequences of these formulas are also stated. We provide characterizations of geodesics and two different types of conformal vector fields, namely, Killing vector fields and concircular vector fields on sequential warped product manifolds. Finally, we consider the geometry of two classes of sequential warped product space-time models which are sequential generalized Robertson-Walker space-times and sequential standard static space-times.

We give new necessary and sufficient conditions on the Weyl tensor for generalized Robertson-Walker (GRW) space-times to be perfect-fluid space-times. For GRW space-times, 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 all but one, the Ricci tensor is Einstein or has the form of perfect fluid. We discuss the corresponding equations of state that result from the Einstein equation in dimension 4, where perfect-fluid GRW space-times are Robertson-Walker., Comment: 10 pages, final version

A. Gray presented an interesting $O\left( n\right) $ invariant decomposition of the covariant derivative of the Ricci tensor. Manifolds whose Ricci tensor satisfies the defining property of each orthogonal class are called Einstein-like manifolds. In the present paper, we answered the following question: Under what condition(s), does a factor manifold $M_{i},i=1,2$ of a doubly warped product manifold $M=_{f_{2}}M_{1}\times _{f_{1}}M_{2}$ lie in the same Einstein-like class of $M$? By imposing sufficient and necessary conditions on the warping functions, an inheritance property of each class is proved. As an application, Einstein-like doubly warped product space-times of type $\mathcal{A},$ $\mathcal{B}$ or $\mathcal{P}$ are considered.

This paper deals with the study of perfect fluid spacetimes. It is proven that a perfect fluid spacetime is Ricci recurrent if and only if the velocity vector field of perfect fluid spacetime is parallel and α = β. In addition, in a stiff matter perfect fluid Yang pure space with p + σ ≠ 0, the integral curves generated by the velocity vector field are geodesics. Moreover, it is shown that in a generalized Robertson–Walker perfect fluid spacetime, the Weyl tensor is divergence-free and the gradient of the potential function of the concircular vector field is pointwise collinear with the velocity vector field of perfect fluid spacetime. We also characterize the perfect fluid spacetimes whose Lorentzian metrics are Yamabe and gradient Yamabe solitons, respectively.

The object of the present paper is to introduce and investigate the P -curvature tensor that generalizes projective, conharmonic, M -projective and the set of W i curvature tensors introduced by Pokhariyal and Mishra. It is proven that pseudo-Riemannian manifolds admitting a traceless P -curvature tensor are Einstein and those admitting flat P -curvature tensor has a constant curvature. Classification theorems for pseudo-Riemannian manifolds admitting a divergence-free P -curvature tensor are given in each subspace of Gray’s decomposition of the covariant derivative of the Ricci tensor. Space–times having a flat P -curvature tensor or a divergence free P -curvature tensor are scrutinized. Finally, perfect fluid space–times admitting various features of the P -curvature tensor are considered.

Recently, it is proven that generalized Robertson–Walker space-times in all orthogonal subspaces of Gray’s decomposition except one (unrestricted) are perfect fluid space-times. GRW space-times in the unrestricted subspace are identified by having constant scalar curvature. Generalized quasi-Einstein GRW space-times have a constant scalar curvature. It is shown that generalized quasi-Einstein GRW space-times reduce to Einstein space-times or perfect fluid space-times.

A set A in Euclidean n-space En, is called an m-convex set if for every m distinct points of A at least one of the line segments joining two points of them lies in A. In this article we study some geometrical and topological properties of these sets in En. Mathematics Subject Classification: 53C42, 52A05

The purpose of the present article is to study and characterize sev- eral types of symmetries of generalized Robertson-Walker space-times. Con- formal vector fields, curvature and Ricci collineations are studied. Many im- plications for existence of these symmetries on generalied Robertson-Walker spacetimes are obtained. Finally, Ricci solitons on generalized Robertson- Walker space-times admitting conformal vector fields are investigated.

The purpose of this article is to study implications of a Ricci soliton warped product manifold to its base and fiber manifolds. First, it is proved that if a warped product manifold is Ricci soliton then its factors are Ricci soliton. Then we study Ricci soliton on warped product manifolds admitting either a conformal vector field or a concurrent vector field. Finally, we study Ricci soliton on some warped product space-times.

This article aimed to study and explore conformal vector fields on doubly warped product manifolds as well as on doubly warped spacetime. Then we derive sufficient conditions for matter and Ricci collineations on doubly warped product manifolds. A special attention is paid to concurrent vector fields. Finally, Ricci solitons on doubly warped product spacetime admitting conformal vector fields are considered.

In this paper, it is proved that the fiber manifold [Formula: see text] of a warped product manifold [Formula: see text] inherits the Einstein-like class type of [Formula: see text] whereas the base manifold does under some conditions. Some related results are considered.

The present article provides a study of $2-$Killing vector fields on warped product manifolds as well as characterization of this structure on standard static and generalized Robertson-Walker space-times. Some conditions for a $2-$Killing vector field on a warped product manifold to be parallel are obtained. Moreover, some results on the curvature of a warped product manifolds in terms of $2-$Killing vector fields are derived. Finally, we apply our results to describe $2-$Killing vector fields of some well known warped product space-time models., Comment: 13 pages, no figures

Let $\mathcal{W}^{n}$ be the class of $C^{\infty }$ complete simply connected $n-$dimensional manifolds without conjugate points. The hyperbolic space as well as Euclidean space are good examples of such manifolds. Let $% W\in \mathcal{W}^{n}$ and let $A$ be a subset of $W$. This article aims at characterization and building convex and starshaped sets in this class from inside. For example, it is proven that, for a compact starshaped set, the convex kernel is the intersection of stars of extreme points only. Also, if a closed unbounded convex set $A$ does not contain a totally geodesic hypersurface and its boundary has no geodesic ray, then $A$ is the convex hull of its extreme points. This result is a refinement of the well-known Karein-Millman theorem., Comment: arXiv admin note: substantial text overlap with arXiv:1301.0688

The purpose of this paper is to study the [Formula: see text]-curvature tensor on (singly) warped product manifolds as well as on generalized Robertson–Walker and standard static space-times. Some different expressions of the [Formula: see text]-curvature tensor on a warped product manifold in terms of its relation with [Formula: see text]-curvature tensor on the base and fiber manifolds are obtained. Furthermore, we investigate [Formula: see text]-curvature flat warped product manifolds. Many interesting results describing the geometry of the base and fiber manifolds of a [Formula: see text]-curvature flat warped product manifold are derived. Finally, we study the [Formula: see text]-curvature tensor on generalized Robertson–Walker and standard static space-times; we explore the geometry of the fiber of these warped product space-time models that are [Formula: see text]-curvature flat.