Generalised neo-pseudo projective recurrent Finsler space

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.

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.

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.

Chapter VI Finslerian manifolds of constant sectional curvature [4

In this paper, we investigate a class of Finsler spaces, termed generalized Rh-recurrent spaces (denoted by GRh-RFn), where the Cartan's third curvature tensor satisfies a specific condition involving non-null covariant vector fields. We provide several characterizations and properties of these spaces, starting with the equivalence of two distinct forms of the curvature condition, and prove key theorems about the non-vanishing of essential tensors like the Ricci tensor, curvature vector, deviation tensor, and the curvature scalar. Furthermore, we derive important identities associated with the covariant differentiation of tensors in these spaces, such as the h-covariant derivative of the h(v)-torsion tensor and deviation tensor. These identities play a crucial role in understanding the geometric structure of GRh-RFn​, and the results contribute to the broader theory of recurrent spaces in Finsler geometry. The paper concludes by presenting several new identities, which hold in GRh-RFn, thus advancing the theoretical framework of generalized recurrent spaces in differential geometry.

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.

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 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.

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.

Abstract. In the present paper, we introduce a type of Riemann-ian manifolds (namely, W 4 -birecurrent manifold) and study the sev-eral properties of such a manifold on which some geometric condi-tions are imposed. 1. IntroductionIn [7], Lichnerowicz introduced a recurrent manifold of second or-der (called birecurrent manifold) and obtained some interesting results.Later the weak notion of this manifold was studied by many authors in-cluding Thompson [9], Chaki and Roy Chowdhury [2], Ewert-Krzemieni-ewski [3,4], Garai [5]. In [8], Pokhariyal de ned some curvature tensorswith the help of Weyl’s projective curvature tensor and studied theirphysical and geometrical properties. One of the curvature tensors men-tioned in [8] was the W 4 -curvature tensor de ned by W 4 (X;Y;Z;T) = R(X;Y;Z;T) +1(n 1)[g(X;Z)r(Y;T) g(X;Y)r(Z;T)] or in local coordinates,(1.1) W 4ijkl = R ijkl +1(n 1)[g ik r jl g ij r kl ];where Rand rare the Riemannian curvature tensor and the Ricci tensor,respectively. The W 4

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.

A class of Finsler metrics with three parameters is constructed. Moreover, a sufficient and necessary condition for this Finsler metrics to be projectively flat was obtained. AMS subject classifications: 53B40; 53C60; 58B20.

Takano [2] have studied decomposition of curvature tensor in a recurrent space. Sinha and Singh [3] have been studied and defined decomposition of recurrent curvature tensor field in a Finsler space. Singh and Negi studied decomposition of recurrent curvature tensor field in a K¨aehlerian space. Negi and Rawat [6] have studied decomposition of recurrent curvature tensor field in K¨aehlerian space. Rawat and Silswal [11] studied and defined decomposition of recurrent curvature tensor fields in a Tachibana space. Further, Rawat and Kunwar Singh [12] studied the decomposition of curvature tensor field in K¨aehlerian recurrent space of first order. In the present paper, we have studied the decomposition of curvature tensor fields R^h_ijk in terms of two non-zero vectors and a tensor field in EinsteinK¨aehlerian recurrent space of first order and several theorem have been established and proved.

PurposeIn this paper, we give some properties of the α-connections on statistical manifolds and we study the α-conformal equivalence where we develop an expression of curvature R¯ for ∇¯ in relation to those for ∇ and ∇^.Design/methodology/approachIn the first section of this paper, we prove some results about the α-connections of a statistical manifold where we give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds treated in [1, 3], and we construct some examples.FindingsWe give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds, we give the relations between curvature tensors and we construct some examples.Originality/valueWe give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds, we give the relations between curvature tensors and we construct some examples.

Characterization of invariant complex Finsler metrics on the complex Grassmann manifold