ON W4-BIRECURRENT MANIFOLDS

The purpose of the article is to study the certain results on para-Kenmotsu manifolds equipped with $M$-projective curvature tensor. Here we investigate para-Kenmotsu manifolds satisfying some curvature conditions $\widetilde{M}\cdot R=0,$ $\widetilde{M}\cdot Q=0$ and $Q\cdot \widetilde{M}=0,$ where $R$, $Q$ and $\widetilde{M}$ respectively denote the Riemannian curvature tensor, Ricci operator and $M$-projective curvature tensor.

The ideas of holonomy group fixing an m-dimensional plane in a Finsler space were given by one of the present authors[1]. In that paper the deformation properties of the space admitting such holonomy group were of main consideration and, indeed, the decomposition characteristics of the space were not touched upon. In the present paper we consider the decomposition of the space due to the existence of holonomy group. The geometry is constructed on the decomposed metric of the space. The decomposition properties of various entities such as the connection parameters, the covariant derivatives, the curvature tensors, and the projective curvature tensors have been studied. In all there are six articles in the paper. The first of these is introductory. The next three articles are dealt with the Cartan's approach to Finsler space whereas the fifth one is dealt with Berwald's approach. The last article is devoted to the theory of decomposition in the projective curvature tensors.

This paper investigates the properties and behaviour of generalized βP-recurrent Finsler spaces with an emphasis on the covariant derivatives of Weyl's projective tensors, which include torsion, curvature, and deviation tensors. We define a GβP-recurrent space as a Finsler space where the second curvature tensor satisfies a specific recurrence condition involving non-zero covariant vector fields. Through a series of mathematical derivations, the paper explores the equivalence of different characterizations of the GβP-recurrent space, proving that the torsion tensor, its associate tensor, the P-Ricci tensor, and the curvature vector are non-vanishing. Further, we analyze the relationship between the covariant vector fields showing their dependence or independence on the direction argument. The study also includes the application of Berwald’s covariant derivative to the projective curvature tensor, concluding that these tensors exhibit generalized recurrence under certain conditions. The results are presented in a series of theorems that contribute to the deeper understanding of the geometric structure of - spaces.

We consider -quasi Einstein manifolds satisfying the conditions , , , and where , , and denote the conformal curvature tensor, the quasi-conformal curvature tensor, the projective curvature tensor and the pseudo projective curvature tensor, respectively.

In the present paper, we study the curvature tensors of (k,mu)-paracontact manifold satisfying the conditions P(ξ ,Y)Z=0, P(ξ ,Y)P=0, P(ξ ,Y)S =0 and (ξ,Y)R=0. According these cases, (k,μ)-paracontact manifolds have been characterized. We obtain the necessary and sufficient conditions of a (k,μ)-paracontact manifold to be η-Einsteins.

In this paper, we focus on an almost contact metric manifold admitting a type of semi-symmetric nonmetric connection. We find the expression for the curvature tensor of such a manifold. Furhermore, we study the properties of the curvature tensor and the projective curvature tensor.

In this paper, we concentrate on a new Finsler space that is a combination of two types of Cartan derivatives of third order for Berwald curvature tensor H_jkh^i. We find the condition that Berwald curvature tensor H_jkh^i is special generalized of generalized mixed trirecurrent. Furthermore, we show that the normal projective curvature tensor N_jkh^i is generalized hv- mixed trirecurrent if and only if the tensor ∂ ̇_j (H_hk-H_kh) behaves as mixed trirecurrent. Also, the Ricci tensor N_jk of the normal projective curvature tensor N_jkh^i is non – vanishing if and only if the tensor ((1-n) ∂ ̇_j ∂ ̇_k H+H_jk+H_kj) is mixed trirecurrent.

In this study, we examine the geometric properties of Riemannian submersions with quarter symmetric non-metric connections. Our work provides a compre-hensive analysis of the Weyl projective curvature tensor, the concircular curvature tensor, and the conharmonic curvature tensor. We also investigate how these curvature tensors interact with the total umbilic fibers, particularly focusing on their behavior when such fibers are present in Riemannian submersions. First we introduce basic properties of quarter symmetric non-metric connections. Subse-quently, we compute the relevant curvature tensors and analyze their interplay with total umbilic fibers. In this context, we elucidate relationships between var-ious curvature tensors and certain geometric properties, as well as the effects of these relationships on Riemannian submersions.

We classify almost C(α)-manifolds, which satisfy the curvature conditions (Z ) (ξ,X)R=0, (Z ) (ξ,X) (Z ) =0, (Z ) (ξ,X)S=0 and (Z ) (ξ,X)P=0, where (Z ) is the concircular curvature tensor, P is the Weyl projective curvature tensor, S is the Ricci tensor and R is Riemannian curvature tensor of manifold.

The primary objective of the article is to investigate the symmetry and pseudosymmetry properties of the Reissner-Nordström-de Sitter (briefly, RNdS) spacetime. The secondary aim of the paper is to explore the notion of Ricci solitons in RNdS spacetimes. The study is important due to the conceding of almost Ricci soliton and almost Ricci Yamabe soliton of the RNdS spacetime. The analysis shows that this spacetime satisfies multiple types of symmetric and pseudosymmetric conditions. It is interesting to note that RNdS spacetime revealed pseudosymmetry, Weyl conformal pseudosymmetry, Weyl projective pseudosymmetry, conharmonic pseudosymmetry, and concircular pseudosymmetry. Also the RNdS spacetime is pseudosymmetric due to conformal, projective, conharmonic curvature tensors. Furthermore, in the RNdS spacetime, the second order commutator tensor R · R is linearly dependent on Q(Ric, R) and Q(g, C), and the tensor R · C does not commute with the tensor C · R. The investigation also shows that RNdS spacetime is a Roter type. It is demonstrated that the RNdS spacetime is 2-quasi Einstein and an Ein(2) space with recurrent conformal 2-forms. Riemann compatibility preserved under geodesic mapping and it has a great geometric insight in determining the nature of Pontryagian forms. We derive the general form of the compatible tensors of the RNdS spacetime. The energy-momentum tensor of the RNdS spacetime is shown to be pseudosymmetric, and also the energy momentum tensor is pseudosymmetric due to conformal, conharmonic, concircular and projective curvature tensor. We note that geodesic maps do not necessarily preserve Weyl compatibility. However such kind of compatibility remains invariant under conformal maps. The presence of a Weyl-compatible vector ensures that the Weyl tensor assumes an algebraically special form, and this condition is both necessary and sufficient for the vanishing of its magnetic component. The energy momentum tensor is compatible with projective, conharmonic, concircular, Riemann and conformal curvature. It is shown that the energy momentum tensor of the RNdS spacetime is Weyl compatible and hence the Weyl tensor of a such a spacetime is algebraically special and hence its magnetic component vanishes. Consequently the Weyl tensor in RNdS spacetime is purely electric. We study a generalized notion of curvature inheritance and find that, with respect to the non-Killing vector fields ∂/∂r and ∂/∂θ, the RNdS spacetime does not satisfy such inheritance condition. However, the RNdS spacetime is shown to admit an almost Ricci soliton and an almost η-Ricci Yamabe soliton with respect to the non-Killing vector field ∂/∂r, but with respect to the non-Killing vector field ∂/∂θ the spacetime does not admit such notions. Finally, we present a comparison between the RNdS and Vaidya-Bonner-de Sitter (VBdS) spacetimes.

Gy. Soos [1] and B. Gupta [2] have discussed the properties of Riemannian spaces Vn (n > 2) in which the first covariant derivative of Weyl's projective curvature tensor is everywhere zero; such spaces they call Protective-Symmetric spaces. In this paper we wish to point out that all Riemannian spaces with this property are symmetric in the sense of Cartan [3]; that is the first covariant derivative of the Riemann curvature tensor of the space vanishes. Further sections are devoted to a discussion of projective-symmetric af fine spaces An with symmetric af fine connexion. Throughout, the geometrical quantities discussed will be as defined by Eisenhart [4] and [5].

We consider N(k)-quasi Einstein manifolds satisfying the conditions \(R(\xi,X). W_2 = 0, W_2 (\xi,X).S = 0, P(\xi,X). W_2 = 0\), where \(W_2\) and \(P\) denote the \(W_2\)-curvature tensor and projective curvature tensor, respectively.

In the nearly cosymplectic manifold, dened a tensor of type (4,0), it's called a projective curvature tensor. In this article we discuss an interesting question; what the geometric meaning of this tensor when it's act on nearly cosymplectic manifold? The answer of this question leads to get an application on Einstein space. In particular, the necessary and sucient conditions that a projective tensor is vanishes are found.

The metric is an important tensorial object that introduces more structure into a (differential) manifold. The metric coefficients allow for example to determine the length of parameter curves in the manifold and make it possible to relate corresponding co- and contravariant objects defined on the manifold. Thus the inner product of either co- or contravariant quantities and in particular the angle between two vectors or (covectors) may be computed. Adopting the Ricci postulate of vanishing covariant derivative of the metric coefficients results in a decomposition of the fully covariant connection into its Riemann part, that depends exclusively on the metric, and the contortion, that depends exclusively on the torsion. Moreover, in this case the length of vectors and their angle with respect to geodesics are preserved upon parallel transport. Based on the metric coefficients the fully covariant curvature tensor, displaying left and right skew symmetry, is introduced and the corresponding modifications of the Bianchi identities are highlighted. Likewise the metric allows to introduce the mixed-variant Ricci tensor together with corresponding identities. Finally the metric enables to compute the Ricci scalar from the previous curvature tensors. The particular case of a symmetric, metrically connected manifold represents a Riemann geometry with associated Riemann curvature tensor as fundamental for example in the Einstein Theory of General Relativity/Gravitation. A generalization is obtained by allowing for metric manifolds with non-vanishing covariant derivative of its metric coefficients. The presence of the non-metricity is then subsequently reflected by extra terms in the decomposition of the connection and in the explicit representation of the curvature tensor, the Ricci tensor, the Ricci scalar, and the corresponding Bianchi identities for the various curvature quantities.

Objectives: The purpose of this paper is to obtain several results in the field of generalised neo-pseudo projective recurrent Finsler space. Methods: A generalization technique is employed to solve the resulting problem. We provide its application in the study of space-time. Findings: In section 1, we have defined and studied some of the basic and useful results for later work. Section 2 deals for the neo-pseudo projective recurrent curvature tensor. The notion of neo-pseudo projective recurrent space of second order has been delineated in the section 3. In the section 4 we have studied the generalised neo-pseudo projective recurrent space and established several new results. Novelty/Conclusion: In this paper we have studied some recurrent properties of neo-pseudo projective curvature tensor in a Finsler space. We have obtained several new results which are as follows:• If the space Fn admits a neo-pseudo projective curvature tensor Qa bgd then Qa bgd is skew-symmetric with regard to last two indices.• If the neo-pseudo projective deviation tensor Qab and pseudo deviation tensor field Tab coincides to each other for q = 1 then space is W-flat.• If Fn admits the projectively flat Q-recurrent space then the relationÑeQabg +ÑbQage +ÑgQaeb = 0 holds good.• If a Finsler space Fn admits projectively flat Q-birecurrent space then the relation KerQabg +KbrQage +KgrQaeb = 0 holds good.• If the space is Q-birecurrent then the generalised Q-recurrent space is Qsymmetric.• For the projective flat generalised Q-recurrent space the relationÑrÑeQagd +ÑrÑgQade +ÑrÑdQaeg = 0 holds good.AMS Subject Classification: 58B20, 53C20, 53C60.