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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.

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... We note that Morris-Thorne spacetime [103] and Gödel spacetime [28] are Ricci simple manifolds, Robertson-Walker spacetime [26] and Siklos spacetime [29] are quasi-Einstein manifolds, Kantowski-Sachs spacetime [84] and Som-Raychaudhuri spacetime [25] are 2-quasi Einstein manifolds and Kaigorodov spacetime [29] is an Einstein manifold. For curvature properties of Robinson-Trautman metric, Melvin magnetic metric and generalized pp-wave metric, etc., we refer the reader to see [30,[104][105][106]. ...

... We note that the Ricci tensor of Gödel spacetime [28] is cyclic parallel and the Ricci tensor of (t − z)-type plane wave metric [104] is of Codazzi type. Let ζ be a (0, 4)-type tensor on M. Then a symmetric (0,2)-type tensor ν corresponding to the endomorphism I ν is said to be ζ-compatible if ...

The objective of the paper is to study the geometric properties of the point-like global monopole (briefly, PGM) spacetime, which is a static and spherically symmetric solution of Einstein field equation. It is shown that PGM spacetime admits various types of pseudosymmetric structures, such as, pseudosymmetry due to Weyl conformal curvature tensor, pseudosymmetry due to concircular curvature tensor, pseudosymmetry due to conharmonic curvature tensor, Ricci generalized conformal pseudosymmetric due to projective curvature tensor, Ricci generalized projective pseudosymmetric etc. Moreover, it is proved that PGM spacetime is 2-quasi Einstein, generalized quasi-Einstein, Einstein manifold of degree 2 and its Weyl conformal curvature 2-forms are recurrent. It is also shown that the stress energy momentum tensor of the PGM spacetime realizes several types of pseudosymmetry, and its Ricci tensor is compatible for Riemann curvature, Weyl conformal curvature, projective curvature, conharmonic curvature and concircular curvature. Further, it is shown that PGM spacetime admits motion, curvature collineation and Ricci collineation. Also, the notion of curvature inheritance (resp., curvature collineation) for the (1,3)-type curvature tensor is not equivalent to the notion of curvature inheritance (resp., curvature collineation) for the (0,4)-type curvature tensor as it is shown that such distinctive properties are possessed by PGM spacetime. Hence the notions of curvature inheritance defined by Duggal [1] and Shaikh and Datta [2] are not equivalent.

... Also, it is noteworthy to mention that Sabina et al. studied various curvature properties of (t − z)-type plane wave spacetime [22], Morris-Thorne wormhole spacetime [21]. ...

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.

... We mention that the Ricci tensor of the (t − z)-type plane wave metric is of Codazzi type [36] and the Ricci tensor is cyclic parallel in Gödel spacetime [33]. ...

The Bardeen solution corresponding to Einstein field equations with a cosmological constant is a regular black hole. The main goal of this manuscript is to investigate the geometric structures in terms of curvature conditions admitted by this spacetime. It is found that this spacetime is pseudosymmetric and possess several kinds of pseudosymmetries. Also, it is a manifold of pseudosymmetry Weyl curvature and the difference tensor C.R-R.C linearly depends on the tensors Q(g;C) and Q(S;C). It is interesting to note that such a spacetime is weakly generalized recurrent manifold and satisfies special recurrent like structure. Further, it is an Einstein manifold of level 2 and Roter type. The energy momentum tensor of this spacetime is pseudosymmetric and finally a worthy comparison between the geometric properties of Bardeen spacetime and Reissner-Nordstr\"om spacetime is given.

... It may be mentioned that Gödel spacetime [26], Som-Raychaudhury spacetime [75], Defrise spacetime [68] are equipped with cyclic parallel Ricci tensor while the (t − z)type plane wave metric [31] has Codazzi Ricci tensor. ...

This article deals with the investigation of geometric properties in terms of curvatures of Lemaître–Tolman–Bondi (briefly, LTB) spacetime (Lemaître 1933; Tolman 1934; Bondi 1947), an inhomogeneous cosmological model of the universe. It is shown that LTB spacetime is an Einstein manifold of level 3, 2-quasi Einstein and generalized Roter type manifold. Also, several curvature conditions of Deszcz pseudosymmetric type are fulfilled by this spacetime. Without considering the mass function we deduce the conditions for which such metric describes a perfect fluid spacetime and a dust solution respectively. The stress energy tensor is Riemann compatible and Weyl compatible as well. As a special case, the curvature properties of Robertson Walker spacetime are obtained and a worthy comparison of geometric properties in terms of curvatures of LTB spacetime and Robertson Walker spacetime is drawn.

The Morris-Thorne wormhole is a spherically symmetric solution of Einstein field equations with cosmological constant. The present article aims to investigate the geometric properties in terms of curvatures admitted by this spacetime. It is found that such a spacetime possesses several kinds of symmetries, such as, Ricci generalized pseudosymmetry, Ricci generalized projectively pseudosymmetry, pseudosymmetry due to Weyl conformal curvature, semisymmetry due to conharmonic curvature etc. Also, it is an Einstein manifold of level 2 as well as special quasi-Einstein manifold. The Tachibana tensor due to energy momentum tensor of the wormhole satisfies some conditions of pseudosymmetric type and also the energy momentum tensor is Weyl compatible and Riemann compatible. Finally, this spacetime is compared with the Gödel spacetime with respect to their admitting geometric structures.

The Vaidya–Bonner metric is a non-static generalization of Reissner–Nordström metric and this paper deals with the investigation of the curvature restricted geometric properties of such a metric. The scalar curvature vanishes and several pseudosymmetric-type curvature conditions are fulfilled by this metric. Also, it is a [Formula: see text]-quasi-Einstein, [Formula: see text] and generalized Roter type manifold. As a special case, the curvature properties of Reissner–Nordström metric are obtained. It is noted that Vaidya–Bonner metric admits several generalized geometric structures in comparison to Reissner–Nordström metric and Vaidya metric.

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.

The main objective of the present paper is to investigate the curvature properties of generalized pp-wave metric. It is shown that generalized pp-wave spacetime is Ricci generalized pseudosymmetric, 2-quasi-Einstein and generalized quasi-Einstein in the sense of Chaki. As a special case it is shown that pp-wave spacetime is semisymmetric, semisymmetric due to conformal and projective curvature tensors, $R$-space by Venzi and satisfies the pseudosymmetric type condition $P\cdot P = -\frac{1}{3}Q(S, P)$. Again we investigate the sufficient condition for which a generalized pp-wave spacetime turns into pp-wave spacetime, pure radiation spacetime, locally symmetric and recurrent. Finally, it is shown that the energy-momentum tensor of pp-wave spacetime is parallel if and only if it is cyclic parallel. And the energy momentum tensor is Codazzi type if it is cyclic parallel but the converse is not true as shown by an example. Finally we make a comparison between the curvature properties of the Robinson-Trautman metric and generalized pp-wave metric.

The curvature properties of Robinson-Trautman metric have been investigated. It is shown that Robinson-Trautman metric admits several kinds of pseudosymmetric type structures such as Weyl pseudosymmetric, Ricci pseudosymmetric, pseudosymmetric Weyl conformal curvature tensor etc. Also it is shown that the difference $R\cdot R - Q(S,R)$ is linearly dependent with $Q(g,C)$ but the metric is not Ricci generalized pseudosymmetric. Moreover, it is proved that this metric is Roter type, 2-quasi-Einstein, Ricci tensor is Riemann compatible and its Weyl conformal curvature 2-forms are recurrent. It is also shown that the energy momentum tensor of the metric is pseudosymmetric and the conditions under which such tensor is of Codazzi type and cyclic parallel have been investigated. Finally, we have made a comparison between the curvature properties of Robinson-Trautman metric and Som-Raychaudhuri metric.

The object of the present paper is to study the characterization of warped product manifolds satisfying some pseudosymmetric type conditions, especially, due to projective curvature tensor. For this purpose we consider a warped product manifold satisfying the pseudosymmetric type condition $R\cdot R = L_1 Q(g,R) + L_2 Q(S,R)$ and evaluate its characterization theorem. As special cases of $L_1$ and $L_2$ we find out the necessary and sufficient condition for a warped product manifold to satisfy various pseudosymmetric type, such as pseudosymmetry, Ricci generalized pseudosymmetry, semisymmetry due to projective curvature tensor ($P\cdot R = 0$), pseudosymmetry due to projective curvature tensor ($P\cdot R = L Q(g,R)$) etc. Finally we present some suitable examples of warped product manifolds satisfying such pseudosymmetric type conditions.

The projective curvature tensor $P$ is invariant under a geodesic preserving transformation on a semi-Riemannian manifold. It is well known that $P$ is not a generalized curvature tensor and hence it possesses different geometric properties than other generalized curvature tensors. The main object of the present paper is to study some semisymmetric type and pseudosymmetric type curvature restricted geometric structures due to projective curvature tensor. The reduced pseudosymmetric type structures for various Walker type conditions are deduced and the existence of Venzi space is ensured. It is shown that the geometric structures formed by imposing projective operator on a (0,4)-tensor is different from that for the corresponding (1,3)-tensor. Characterization of various semisymmetric type and pseudosymmetric type curvature restricted geometric structures due to projective curvature tensor is obtained on a Riemannian and a semi-Riemannian manifold, and it is shown that some of them reduce to Einstein manifold for the Riemann case. Finally to support our theorems four suitable examples are presented.

Som-Raychaudhuri spacetime is a stationary cylindrical symmetric solution of
Einstein field equation corresponding to a charged dust distribution in rigid
rotation. The main object of the present paper is to investigate the curvature
restricted geometric structures admitting by the Som-Raychaudhuri spacetime and
it is shown that such a spacetime is a 2-quasi-Einstein, generalized Roter
type, $Ein(3)$ manifold satisfying $R.R = Q(S,R)$, $C\cdot C = \frac{2a^2}{3}
Q(g,C)$, and its Ricci tensor is cyclic parallel and Riemann compatible.
Finally, we make a comparison between G\"odel spacetime and Som-Raychaudhuri
spacetime.

The object of the present paper is to obtain the characterization of a warped
product semi-Riemannian manifold with a special type of recurrent like
structure, called super generalized recurrent. As consequence of this result we
also find out the necessary and sufficient conditions for a warped product
manifold to satisfy some other recurrent like structures such as weakly
generalized recurrent manifold, hyper generalized recurrent manifold etc.
Finally as a support of the main result, we present an example of warped
product super generalized recurrent manifold.

To generalize the notion of recurrent manifold, there are various recurrent
like conditions in the literature. In this paper we present a recurrent like
structure, namely, \textit{super generalized recurrent manifold}, which
generalizes both the hyper generalized recurrent manifold and weakly
generalized recurrent manifold. The main object of the present paper is to
study the geometric properties of super generalized recurrent manifold. Finally
to ensure the existence of such structure we present a proper example by a
suitable metric.

The present paper deals with the proper existence of a generalized class of
recurrent manifolds, namely, hyper-generalized recurrent. It also deals with
the existence of properness of various generalized curvature restricted
geometric structures. For example, the existence of manifolds which are
non-recurrent but Ricci recurrent, conharmonically recurrent and semisymmetric;
not weakly symmetric but weakly Ricci symmetric, conformally weakly symmetric
and conharmonicly weakly symmetric; non-Einstein but Ricci simple; not satisfy
$P\cdot P =0$ but fulfill the condition $P \cdot P = \frac{1}{3}Q(S, P)$ etc.,
$P$ being the projective curvature tensor. For this purpose we have presented a
metric and computed its curvature properties and finally we have checked
various geometric structures admitting by the metric.

Generalized Roter type manifold is a generalization of conformally flat
manifold as well as Roter type manifold, which gives rise the form of the
curvature tensor in terms of algebraic combinations of the fundamental metric
tensor and Ricci tensors upto level 2. The object of the present paper is to
investigate the characterizations of a warped product manifold to be
generalized Roter-type. We also present an example of a warped product manifold
which is generalized Roter type but not Roter type, and also an example of a
warped product manifold which is Roter type but not conformally flat. These
examples ensure the proper existence of such notions.

The main object of the present paper is to study the geometric properties of
a generalized Roter type semi-Riemannian manifold, which arose in the way of
generalization to find the form of the Riemann-Christoffel curvature tensor
$R$. Again for a particular curvature restriction on $R$ and the Ricci tensor
$S$ there arise two structures, e. g., local symmetry ($\nabla R = 0$) and
Ricci symmetry ($\nabla S = 0$); semisymmetry($R\cdot R =0$) and Ricci
semisymmetry ($R\cdot S =0$) etc. In differential geometry there is a very
natural question about the equivalency of these two structures. In this context
it is shown that generalized Roter type condition is a sufficient condition for
various important second order restrictions. Some generalizations of Einstein
manifolds are also presented here. Finally the proper existence of both type of
manifolds are ensured by some suitable examples.

The present paper deals with the existence of a new class of semi-Riemannian manifolds which are weakly generalized recurrent, pseudo quasi-Einstein and fulfill the condition R·R=Q(S,R). For this purpose, we present a metric, compute its curvature properties, and finally check various geometric structures arising from the different curvatures by means of their covariant derivatives of first and second order.

Derdziński and Shen's theorem on the restrictions on the Riemann tensor imposed by existence of a Codazzi tensor holds more generally when a Riemann compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity for a new “Codazzi deviation tensor”, with a geometric significance. The above general properties are studied, with their implications on Pontryagin forms. Examples are given of manifolds with Riemann compatible tensors, in particular those generated by geodesic mappings. Compatibility is extended to generalized curvature tensors, with an application to Weyl's tensor and general relativity.

Abstract: We extend a remarkable theorem of Derdziński and Shen, on the restrictions imposed on the Riemann tensor by the existence of a nontrivial Codazzi tensor. We show that the Codazzi equation can be replaced by a more general algebraic condition. The resulting extension applies both to the Riemann tensor and to generalized curvature tensors.

In this paper, we introduce the notion of recurrent conformal 2-forms on a pseudo-Riemannian manifold of arbitrary signature. Some theorems already proved for the same differential structure on a Riemannian manifold are proven to hold in this more general contest. Moreover other interesting results are pointed out; it is proven that if the associated covector is closed, then the Ricci tensor is Riemann compatible or equivalently, Weyl compatible: these notions were recently introduced and investigated by one of the present authors. Further some new results about the vanishing of some Weyl scalars on a pseudo-Riemannian manifold are given: it turns out that they are consequence of the generalized Derdziński–Shen theorem. Topological properties involving the vanishing of Pontryagin forms and recurrent conformal 2-forms are then stated. Finally, we study the properties of recurrent conformal 2-forms on Lorentzian manifolds (space-times). Previous theorems stated on a pseudo-Riemannian manifold of arbitrary signature are then interpreted in the light of the classification of space-times in four or in higher dimensions
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We introduce the new algebraic property of Weyl compatibility for symmetric tensors and vectors. It is strictly related to Riemann compatibility, which generalizes the Codazzi condition while preserving much of its geometric implications. In particular, it is shown that the existence of a Weyl compatible vector implies that the Weyl tensor is algebraically special, and it is a necessary and sufficient condition for the magnetic part to vanish. Some theorems (Derdziński and Shen [11], Hall [15]) are extended to the broader hypothesis of Weyl or Riemann compatibility. Weyl compatibility includes conditions that were investigated in the literature of general relativity (as in McIntosh et al. [16, 17]). A simple example of Weyl compatible tensor is the Ricci tensor of an hypersurface in a manifold with constant curvature

In this paper we recall the closedness properties of generalized curvature 2-forms,
which are said to be Riemannian, Conformal, Projective, Concircular and Conhar-
monic curvature 2-forms, given in [?]. Moreover, we extend the concept of recurrent
generalized curvature tensor to the associated curvature 2-forms while generalizing
some known results.
In particular, we introduce the recurrence of the Conformal curvature 2-form and
give some interesting theorems. In the �nal section we focus on the closedness of the
associated 2-forms for curvature-like tensors.

The object of the present paper is to study weakly projective symmetric manifolds and its decomposability with several non-trivial exam-ples. Among others it is shown that in a decomposable weakly projective symmetric manifold both the decompositions are weakly Ricci symmetric.

The notion of quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations as well as during considerations of quasi-umbilical hypersurfaces. For instance, the Robertson-Walker spacetimes are quasi-Einstein manifolds. The object of the present paper is to study Lorentzian quasi-Einstein manifolds. Some basic geometric properties of such a manifold are obtained. The applications of Lorentzian quasi-Einstein manifolds to the general relativity and cosmology are investigated. Theories of gravitational collapse and models of Supernova explosions [5] are based on a relativistic fluid model for the star. In the theories of galaxy formation, relativistic fluid models have been used in order to describe the evolution of perturbations of the baryon and radiation components of the cosmic medium [32]. Theories of the structure and stability of neutron stars assume that the medium can be treated as a relativistic perfectly conducting magneto fluid. Theories of relativistic stars (which would be models for supermassive stars) are also based on relativistic fluid models. The problem of accretion onto a neutron star or a black hole is usually set in the framework of relativistic fluid models. Among others it is shown that a quasi-Einstein spacetime represents perfect fluid spacetime model in cosmology and consequently such a spacetime determines the final phase in the evolution of the universe. Finally the existence of such manifolds is ensured by several examples constructed from various well known geometric structures.

This paper is concerned with some results on weakly symmetric and
weakly Ricci symmetric warped product manifolds. We prove the necessary and suffici-
ent condition for a warped product manifold to be weakly symmetric and weakly Ricci
symmetric. On the basis of these results two proper examples of warped product weakly
symmetric and weakly Ricci symmetric manifolds are presented.

In the literature, there are two different notions of pseudosymmetric
manifolds, one by Chaki [9] and other by Deszcz [30], and there are many papers
related to these notions. The object of the present paper is to deduce
necessary and sufficient conditions for a Chaki pseudosymmetric [9] (resp.
pseudo Ricci symmetric [10]) manifold to be Deszcz pseudosymmetric (resp. Ricci
pseudosymmetric). We also study the necessary and sufficient conditions for a
weakly symmetric [121] (resp. weakly Ricci symmetric [122]) manifold by Tamassy
and Binh to be Deszcz pseudosymmetric (resp. Ricci pseudosymmetric). We also
obtain the reduced form of the defining condition of weakly Ricci symmetric
manifolds by Tamassy and Binh [122]. Finally we give some examples to show the
independent existence of such types of pseudosymmetry which also ensure the
existence of Roter type and generalized Roter type manifolds and the manifolds
with recurrent curvature 2-form ([3], [64]) associated to various curvature
tensors.

We investigate hypersurfaces in space forms satisfying particular curvature conditions which are strongly related to pseudosymmetry. Expressing certain products of curvature tensors as linear combinations of Tachibana tensors we deduce several pseudosymmetry-type results.

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.

Siklos spacetime represents exact gravitational waves propa�gating on the anti-de-Sitter universe with negative cosmological constant
and it is conformally related to pp-wave spacetime. The object of this
paper is to investigate the curvature restricted geometric structures ad�mitting by the Siklos spacetime and it is shown that such spacetime is
Ein(2), quasi-Einstein and its conformal 2-forms are recurrent. It is also
shown that this spacetime satisfies various pseudosymmetric type curva�ture conditions such as pseudosymmetry, semisymmetry due to conformal
curvature tensor and Ricci generalized conformally pseudosymmetry. The
curvature properties of Siklos spacetime in vacuum has also been inves�tigated. As special case, we have evaluated the curvature properties of
Kaigorodov spacetime and Defrise’s spacetime. Finally, we make compar�ison between the curvature properties of Siklos spacetime and pp-wave
spacetime.
�

In the present paper we investigate some geometric properties of two classes of Riemannian manifolds introduced by A. Gray (1978). The existence of such classes are also ensured by several non-trivial examples. We also study these classes of Rie- mannian manifolds with quasi-Einstein structure introduced by M. C. Chaki and R. K. Maity (2000) and obtained various interesting geometric properties.

In this paper we present some new results about n(≥ 4)-dimensional pseudo-Z symmetric space-times. First we show that if the tensor Z satisfies the Codazzi condition then its rank is one, the space-time is a quasi-Einstein manifold, and the associated 1-form results to be null and recurrent. In the case in which such covector can be rescaled to a covariantly constant we obtain a Brinkmann-wave. Anyway the metric results to be a subclass of the Kundt metric. Next we investigate pseudo-Z symmetric space-times with harmonic conformal curvature tensor: a complete classification of such spaces is obtained. They are necessarily quasi-Einstein and represent a perfect fluid space-time in the case of time-like associated covector; in the case of null associated covector they represent a pure radiation field. Further if the associated covector is locally a gradient we get a Brinkmann-wave space-time for n > 4 and a pp-wave space-time in n = 4. In all cases an algebraic classification for the Weyl tensor is provided for n = 4 and higher dimensions. Then conformally flat pseudo-Z symmetric space-times are investigated. In the case of null associated covector the space-time reduces to a plane wave and results to be generalized quasi-Einstein. In the case of time-like associated covector we show that under the condition of divergence-free Weyl tensor the space-time admits a proper concircular vector that can be rescaled to a time like vector of concurrent form and is a conformal Killing vector. A recent result then shows that the metric is necessarily a generalized Robertson–Walker space-time. In particular we show that a conformally flat (PZS)n, n ≥ 4, space-time is conformal to the Robertson–Walker space-time.

In [1] an example of a non-semisymmetric Ricci-symmetric quasi-Einstein austere hypersurface M isometrically immersed in an Euclidean space was constructed. In this paper we state that, at every point of the hypersurface M, the following generalized Einstein metric curvature condition is satisfied: (*) the difference tensor R·C −C ·R and the Tachibana tensor Q(S,C) are linearly dependent. Precisely, (n−2) (R·C −C ·R) = Q(S,C) on M. We also prove that non-conformally flat and non-Einstein hypersurfaces with vanishing scalar curvature having at every point two dictinct principal curvatures, as well as some hypersurfaces having at every point three distinct principal curvatures, satisfy (*). We present examples of hypersurfaces satisfying (*).

Warped product manifolds with p-dimensional base, p=1,2, satisfy some curvature conditions of pseudosymmetry type. These conditions are formed from the metric tensor g, the Riemann-Christoffel curvature tensor R, the Ricci tensor S and the Weyl conformal curvature C of the considered manifolds. The main result of the paper states that if p=2 and the fibre is a semi-Riemannian space of constant curvature, if n is greater or equal to 4, then the (0,6)-tensors R.R-Q(S,R) and C.C of such warped products are proportional to the (0,6)-tensor Q(g,C) and the tensor C is expressed by a linear combination of some Kulkarni-Nomizu products formed from the tensors g and S. Thus these curvature conditions satisfy non-conformally flat non-Einstein warped product spacetimes (p=2, n=4). We also investigate curvature properties of pseudosymmetry type of quasi-Einstein manifolds. In particular, we obtain some curvature property of the Goedel spacetime. 1

We determine curvature properties of pseudosymmetry type of some class of minimal 2-quasiumbilical
hypersurfaces in Euclidean spaces En+1, n � 4. We present examples of such hypersurfaces. The obtained results are used to determine curvature properties of biharmonic hypersurfaces with three distinct principal curvatures in E5. Those hypersurfaces were recently investigated by Y. Fu in [38].

It is known that the difference tensor \(R \cdot C - C \cdot R\) and the Tachibana tensor \(Q(S,C)\) of any semi-Riemannian Einstein manifold \((M,g)\) of dimension \(n \ge 4\) are linearly dependent at every point of \(M\) . More precisely \(R \cdot C - C \cdot R = (1/(n-1))\, Q(S,C)\) holds on \(M\) . In the paper we show that there are quasi-Einstein, as well as non-quasi-Einstein semi-Riemannian manifolds for which the above mentioned tensors are linearly dependent. For instance, we prove that every non-locally symmetric and non-conformally flat manifold with parallel Weyl tensor (essentially conformally symmetric manifold) satisfies \(R \cdot C = C \cdot R = Q(S,C) = 0\) . Manifolds with parallel Weyl tensor having Ricci tensor of rank two form a subclass of the class of Roter type manifolds. Therefore we also investigate Roter type manifolds for which the tensors \(R \cdot C - C \cdot R\) and \(Q(S,C)\) are linearly dependent. We determine necessary and sufficient conditions for a Roter type manifold to be a manifold having that property.

The notions of a generalized quasi Einstein manifold and a manifold of generalized quasi constant curvature are introduced and some properties of such manifolds are obtained.

We investigate semi-Riemannian manifolds satisfying some curvature conditions. Those conditions are strongly related to pseudosymmetry.

The notion of quasi-Einstein manifolds arose during the study of exact solutions of the
Einstein field equations as well as during considerations of quasi-umbilical
hypersurfaces. For instance, the Robertson-Walker spacetimes are quasi-Einstein manifolds.
The object of the present paper is to study quasi-Einstein spacetimes. Some basic
geometric properties of such a spacetime are obtained. The applications of quasi-Einstein
spacetimes in general relativity and cosmology are investigated. Finally, the existence of
such spacetimes are ensured by several interesting examples.