arXiv subjects
Publications and source records attributed to Absos Ali Shaikh.
The object of the present note is to discuss about the defining condition of weakly cyclic Ricci symmetric manifolds and weakly cyclic $Z$-symmetric manifolds \cite{DMS15} and the existence of such notion by proper examples.
In this paper we have generalized the notion of $λ$-radial contraction in complete Riemannian manifold and developed the concept of $p^λ$-convex function. We have also given a counter example proving the fact that in general $λ$-radial contraction of a geodesic is not necessarily a geodesic. We have also deduced some relations between geodesic convex sets and $p^λ$-convex sets and showed that under certain conditions they are equivalent.
The object of the present paper is to study the locally $ϕ$- semisymmetric Kenmotsu manifolds along with the characterization of such notion.
In this article, a Leslie-Gower Holling type III predator-prey model with disease in predator has been developed from both biological and mathematical point of view. The total population is divided into three classes, namely, prey, susceptible predator and infected predator. The local stability, global stability together with sufficient conditions for persistence of the ecosystem near biologically feasible equilibria is thoroughly investigated. Boundedness and existence of the system are established. All the important analytical findings are numerically verified using program software MATLAB and Maple.
As a generalization of geodesic function, in the present paper, we introduce the notion of geodesic $φ$-convex function and deduce some basic properties of $φ$-convex function and geodesic $φ$-convex function. We also introduce the concept of geodesic $φ$-convex set and $φ$-epigraph and investigate a characterization of geodesic $φ$-convex functions in terms of their $φ$-epigraphs.
A spacetime denotes a pure radiation field if its energy momentum tensor represents a situation in which all the energy is transported in one direction with the speed of light. In 1989, Wils and later in 1997 Ludwig and Edgar studied the physical properties of pure radiation metrics, which are conformally related to a vacuum spacetime. In the present paper we investigate the curvature properties of special type of pure radiation metrics presented by Ludwig and Edgar. It is shown that such a pure radiation spacetime is semisymmetric, Ricci simple, $R$-space by Venzi and its Ricci tensor is Riemann compatible. It is also proved that its conformal curvature 2-forms and Ricci 1-forms are recurrent. We also present a pure radiation type metric and evaluate its curvature properties along with the form of its energy momentum tensor. It is interesting to note that such pure radiation type metric is $Ein(3)$ and 3-quasi-Einstein. We also find out the sufficient conditions for which this metric represents a generalized pp-wave, pure radiation and perfect fluid. Finally we made a comparison between the curvature properties of Ludwig and Edgar's pure radiation metric and pp-wave metrics.
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.
Generalizing the notion of local $ϕ$-symmetry of Takahashi, in the present paper, we introduce the notion of local $ϕ$-semisymmetry of a Sasakian manifold along with its proper existence and characterization. We also study the notion of local Ricci (resp., projective, conformal) $ϕ$-semisymmetry of a Sasakian manifold and obtain its characterization. It is shown that the local $ϕ$-semisymmetry, local projective $ϕ$-semisymmetry and local concircular $ϕ$-semisymmetry are equivalent. It is also shown that local conformal $ϕ$-semisymmetry and local conharmonical $ϕ$-semisymmetry are equivalent.
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.
It is well known that among all closed bounded curves in the plane with the given perimeter, the circle encloses the maximum area. There are many proofs in the literature. In this article we have given a new proof using some ideas of Demar.
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.
The present paper deals with the proper existence of a generalized class of recurrent manifolds, namely, hyper-generalized recurrent manifolds. We have established the proper existence of various generalized notions of recurrent manifolds. For this purpose we have presented a metric and computed its curvature properties and finally we have obtained the existence of a new class of semi-Riemannian manifolds which are non-recurrent but hyper-generalized recurrent, Ricci recurrent, conharmonically recurrent, manifold of recurrent curvature 2-forms and semisymmetric; not weakly symmetric but weakly Ricci symmetric, conformally weakly symmetric and conharmonically weakly symmetric; non-Einstein but Ricci simple; do not satisfy $P\cdot P =0$ but fulfills the condition $P \cdot P = -\frac{1}{3}Q(S, P)$, $P$ being the projective curvature tensor.
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 Gödel spacetime and Som-Raychaudhuri spacetime.
In the literature, there are two different notions of pseudosymmetric manifolds, one by Chaki [7] and other by Deszcz [16], 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 [7] (resp. pseudo Ricci symmetric [8]) manifold to be Deszcz pseudosymmetric (resp. Ricci pseudosymmetric). We also study the necessary and sufficient conditions for a weakly symmetric [58] (resp. weakly Ricci symmetric [59]) manifold by Tamássy 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 Tamássy and Binh [59]. 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 ([2], [29]) associated to various curvature tensors.
The object of the present paper is to study locally $ϕ$-symmetric LP-Sasakian manifolds admitting semi-symmetric metric connection and obtain a necessary and sufficient condition for a locally $ϕ$-symmetric LP-Sasakian manifold with respect to semi-symmetric metric connection to be locally $ϕ$-symmetric LP-Sasakian manifold with respect to Levi-Civita connection.
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.