Weyl Projective Curvature Research Articles

Quarter-symmetric connection on an almost Hermitian manifold and on a Kähler manifold

The paper observes an almost Hermitian manifold as an example of a generalized Riemannian manifold and examines the application of a quarter-symmetric connection on the almost Hermitian manifold. The almost Hermitian manifold with quarter-symmetric connection preserving the generalized Riemannian metric is actually the Kähler manifold. Observing the six linearly independent curvature tensors with respect to the quarter-symmetric connection, we construct tensors that do not depend on the quarter-symmetric connection generator. One of them coincides with the Weyl projective curvature tensor of symmetric metric $g$. Also, we obtain the relations between the Weyl projective curvature tensor and the holomorphically projective curvature tensor. Moreover, we examine the properties of curvature tensors when some tensors are hybrid.

Concircularly semi-symmetric metric connection

Results on a concircularly semi-symmetric metric connection on a Riemannian manifold are presented. Six linearly independent curvature tensors with respect to this non-symmetric linear connection are studied, and the tensors coincident with the Weyl projective curvature tensor and the concircular curvature tensor of the Levi-Civita connection are determined. Transformations of connections when the Riemannian curvature tensor, Weyl projective curvature tensor and the concircular curvature tensor of the Levi-Civita connection are invariant are also observed. Moreover, new conditions for the Einstein manifold are derived.

Quarter-Symmetric Metric Connection on a Cosymplectic Manifold

We study the quarter-symmetric metric A-connection on a cosymplectic manifold. Observing linearly independent curvature tensors with respect to the quarter-symmetric metric A-connection, we construct the Weyl projective curvature tensor on a cosymplectic manifold. In this way, we obtain new conditions for the manifold to be projectively flat. At the end of the paper, we define η-Einstein cosymplectic manifolds of the θ-th kind and prove that they coincide with the η-Einstein cosymplectic manifold.

Curvature properties of metric and semi-symmetric linear connections

Considering a non-symmetric linear connection and its dual one appear 6 independent curvature tensors [15]. In the present paper we study properties of curvature tensors of semi-symmetric metric connections on a generalized Riemannian manifold. It is shown that if linear connection preserves the symmetric part of the generalized Riemannian metric with semi-symmetric torsion with an appropriate additional condition, then these six curvature tensors can be expressed by linear combination of Weyl conformal curvature tensor, Weyl projective curvature tensor and the concircular curvature tensor. In the case of the curvature tensor , we have proved that if exists a linear connection preserving the symmetric part of the generalized Riemannian metric with semi-symmetric torsion, whose curvature tensor vanishes, it is necessary and sufficient that the Riemannian metric be projectively flat. For tensor we also get a useful result. If exists a linear connection preserving the symmetric part of the generalized Riemannian metric with semi-symmetric torsion, whose curvature tensor and Ricci tensor vanish, it is necessary and sufficient that the Riemannian metric be conformally flat.

Recurrence Decompositions in Finsler Space

Finsler geometry is a kind of differential geometry originated by P. Finsler. Indeed, Finsler geometry has several uses in a wide variety and it is playing an important role in differential geometry and applied mathematics of problems in physics relative, manual footprint. It is usually considered as a generalization of Riemannian geometry. In the present paper, we introduced some types of generalized $W^{h}$ -birecurrent Finsler space, generalized $W^{h}$ -birecurrent affinely connected space and we defined a Finsler space $F_{n}$ for Weyl's projective curvature tensor $W_{jkh}^{i}$ satisfies the generalized-birecurrence condition with respect to Cartan's connection parameters $\Gamma ^{\ast i}_{kh}$, such that given by the condition (\ref{2.1}), where $\left\vert m\right. \left\vert n\right. $ is\ h-covariant derivative of second order (Cartan's second kind covariant differential operator) with respect to $x^{m}$ \ and $x^{n}$ ,\ successively, $\lambda _{mn}$ and $\mu _{mn~}$ are\ non-null covariant vectors field and such space is called as a generalized $W^{h}$ -birecurrent\ space and denoted briefly by $GW^{h}$ - $BRF_{n}$ . We have obtained some theorems of generalized $W^{h}$ -birecurrent affinely connected space for the h-covariant derivative of the second order for Wely's projective torsion tensor $~W_{kh}^{i}$ , Wely's projective deviation tensor $~W_{h}^{i}$ in our space. We have obtained the necessary and sufficient condition forsome tensors in our space.

On the Weyl projective curvature tensor of the projective semi-symmetric connection in an SP-Sasakian manifold

The objective of the present paper is to study the W2-curvature tensor of the projective semi-symmetric connection in an SP-Sasakian manifold. It is shown that an SP-Sasakian manifold satisfying the conditions ܲ\simP ⋅W2\sim ܹ = 0 is an Einstein manifold and ܹW2\sim . ܲP\sim = 0 is a quasi Einstein manifold.

On Anti-Kähler manifolds with complex semi-symmetric metric $F-$connection

In this paper, we construct a complex semi-symmetric metric $F$-connection on an anti-Kähler manifold. First, we present some results concerning the torsion tensor of the complex semi-symmetric metric $F$-connection. Finally, we find expressions of the curvature tensor, the conharmonic curvature tensor and the Weyl projective curvature tensor of such connection, and study their properties.

On certain a generalized \(N_{|m}\) ̶ Recurrent Finsler space

A Finsler space \(F_n\) for which the normal projective curvature tensor \(N_{jkh}^i\) satisfies \(N_{jkh|m}^i = λ_m N_{jkh}^i + μ_m (δ_h^i g_{jk} - δ_k^i g_{jh} ), N_{jkh}^i ≠ 0\), where \(λ_m\) and \(μ_m\) are non-zero covariant vectors field, will be called a generalized \(N_{|m}\) ̶ recurrent space. The curvature vector \(H_k\), the curvature scalarH and Ricci tensor \(N_{jk}\) are non-vanishing. When the generalized \(N_{|m}\) ̶ recurrent space is affinely connected space and under certain conditions, we obtain various results. Also, in generalized \(N_{|m}\) ̶ recurrent space, Weyl's projective curvature tensoris a generalized recurrent tensor.

A Characterisation for Finsler Metrics of Constant Curvature and a Finslerian Version of Beltrami Theorem

We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature. When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classic projective Weyl tensor. In the general case, the Weyl-type curvature tensor differs from the Weyl projective curvature, it is not a projective invariant, and hence Beltrami Theorem does not work in Finsler geometry. We provide the relation between the Weyl-type curvature tensors of two projectively related Finsler metrics. Using this formula we show that a projective deformation preserves the property of having constant flag curvature if and only if the projective factor is a Hamel function. This way we provide a Finslerian version of Beltrami Theorem.

On a Class of P-Kenmotsu Manifolds Admitting Weyl-projective Curvature Tensor of Type (1, 3)

We study a class of para-Kenmotsu manifolds admitting Weyl-projective curvature tensor of type (1, 3). At the end, it is shown that an n-dimensional (n > 2) P-Kenmotsu manifold is Ricci semisymmetric if and only if it is an Einstein manifold.

On almost Kenmotsu manifolds with nullity distributions

The object of this paper is to characterize the curvature conditions $R\cdot P=0$ and $P\cdot S=0$ with its characteristic vector field $\xi$ belonging to the $(k,\mu)'$-nullity distribution and $(k,\mu)$-nullity distribution respectively, where $P$ is the Weyl projective curvature tensor. As a consequence of the main results we obtain several corollaries.

On canonical almost geodesic mappings which preserve the weyl projective tensor

We study a partial case of canonical almost geodesic mappings of the first type of spaces with affine connection that preserve Weyl projective curvature tensor and certain other tensors. Main equations under consideration are reduced to a closed Cauchy system type in covariant derivatives. Therefore a general solution to these equations depends on a finite number of constants. We submit an example of above mappings between flat spaces. We establish that projective Euclidean and equiaffine spaces form closed classes of spaces with respect to these mappings.

On golden semisymmetric metric $F$-connections

In this paper, we construct a golden semisymmetric metric $F$-connection on a locally decomposable golden Riemannian manifold and investigate some properties of its curvature, conharmonic curvature, Weyl projective curvature, and torsion tensors. Moreover, we define the transposed connection of this connection and study its curvature properties.

On Semi-symmetric Para Kenmotsu Manifolds

In this paper we study some remarkable properties of para Kenmotsu (briefly <i>p</i>-Kenmotsu) manifolds satisfying the conditions <i>R(X,Y).R</i>=0, <i>R(X,Y).</i><i>P</i>=0 and <i>P</i><i>(X,Y).R</i>=0, where R(X, Y) is the Riemannian curvature tensor and P(X, Y) is the Weyl projective curvature tensor of the manifold. It is shown that a semi-symmetric <i>p</i>-Kenmotsu manifold (<i>M</i><SUB><i>n</i></SUB>, <i>g</i>) is of constant curvature and hence is an <i>s</i><i>p</i>-Kenmotsu manifold. Also, we obtain the necessary and sufficient condition for a <i>p</i>-Kenmotsu manifold to be Weyl projective semi-symmetric and shown that the Weyl projective semi-symmetric <i>p</i>-Kenmotsu manifold is projectively flat. Finally we prove that if the condition <i>P</i><i>(X,Y).R</i>=0 is satisfied on a <i>p</i>-Kenmotsu manifold then its scalar curvature is constant.

On an Almost C(α)-Manifold Satisfying Certain Conditions on the Concircular Curvature Tensor

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.

On projective N-curvature inheritance

The concept of a Lie recurrence was introduced by the first author [6]. It is an infinitesimal transformation \(\overline{x}^{i}={x}^{i}+\varepsilon {v}^{i}({x}^{j})\) with respect to which the Lie derivative of a curvature tensor is proportional to itself. Apart from other results related to a Lie recurrence, it was established that the Weyl projective curvature tensor is Lie recurrent with respect to a Lie recurrence but its converse is not necessarily true. However, an infinitesimal transformation with respect to which the Weyl projective curvature tensor and the Ricci tensor are Lie recurrent, is necessarily a Lie recurrence. Singh [12] studied an infinitesimal transformation with respect to which the Lie derivative of the curvature tensor is proportional to itself and called such transformation as curvature inheritance. Obviously, a curvature inheritance is nothing but a Lie recurrence. Singh [13] also considered a curvature inheritance which is a projective motion and called it a projective curvature inheritance. Gatoto and Singh [1,2] studied \(\widetilde{K}\)-curvature inheritance and projective \(\widetilde{K}\)-curvature inheritance. Pandey and Pandey [9] studied \(\widetilde{K}\) projective Lie recurrence. Mishra and Yadav [3] studied projective curvature inheritance in an NP-Fn. In the present paper we have established that an infinitesimal transformation in a Finsler space is Lie recurrence if and only if the normal projective curvature tensor is Lie recurrent. A part from this result we have generalized almost all theorems of Mishra and Yadav [3].

On Equitorsion Geodesic Mappings of General Affine Connection Spaces

In the papers [19], [20] several Ricci type identities are obtained by using non-symmetric affine connection. In these identities appear 12 curvature tensors, 5 of which being independent [21], while the rest can be expressed as linear combinations of the others. In the general case of a geodesic mapping f of two non-symmetric affine connection spaces GAN and GAN it is impossible to obtain a generalization of the Weyl projective curvature tensor. In the present paper we study the case when GAN and GAN have the same torsion in corresponding points. Such a mapping we name ”equitorsion mapping”. With respect to each of mentioned above curvature tensors we have obtained quantities E θ i jmn (θ = 1, · · · , 5), that are generalizations of the Weyl tensor, i.e. they are invariants based on f . Among E θ only E 5 is a tensor. All these quantities are interesting in constructions of new mathematical and physical structures. AMS Subj. Class.: 53B05.

A classification of Kaehlerian manifolds satisfying some special conditions

where Rkjfh, Rj~ and R denote the Riemannian curvature tensor, the Ricci tensor and the scalar curvature respectively, and Sji=Fj"R,~. "Ibis tensor corresponds to the Weyl's projective curvature tensor of a Riemannian space and is invariant under any holomorphically projective correspondence. On the other hand, as a complex analogue to the Weyl's conformal curvature tensor, Bochner [10] (see also Yano and Bochner [6] introduced in a Kaehlerian manifold the curvature tensor

Remarks on the Brans-Dicke theory ofN dimensions

A study of the Brans-Dicke theory in vacuum shows that one of the Brans-Dicke equations is represented in terms of the Riemann curvature tensor, the Weyl projective curvature tensor and its covariant derivative. Studying the case that the covariant derivative of the Weyl projective curvature tensor vanishes, we make a comment on a relation between the Brans-Dicke space-time and a space-time of constant curvature.

Relativistic segnificance of curvature tensors

In thi paper new curvature tensors have been defined on the lines of Weyl′s projective curvature tensor and it has been shown that the “distribution” (order in which the vectors in question are arranged before being acted upon by the tensor in question) of vector field over the metric potentials and matter tensors plays an important role in shaping the various physical and geometrical properties of a tensor viz the formulation of gravitational waves, reduction of electromagnetic field to a purely electric field, vanishing of the contracted tensor in an Einstein Space and the cyclic property.