arXiv subjects

Absos Ali Shaikh

Publications and source records attributed to Absos Ali Shaikh.

At least 55 records · Page 3Linked to original sources

arXiv2019

Generalized almost statistical convergence

The objective of this paper is to introduce the notion of generalized almost statistical (briefly, GAS) convergence of bounded real sequences, which generalizes the notion of almost convergence as well as statistical convergence of bounded real sequences. As a special kind of Banach limit functional, we also introduce the concept of Banach statistical limit functional and the notion of GAS convergence mainly depends on the existence of Banach statistical limit functional. We prove the existence of Banach statistical limit functional. Then we have shown the existence of a GAS convergent sequence, which is neither statistical convergent nor almost convergent. Also, some topological properties of the space of all GAS convergent sequences are investigated.

math.FA↗

arXiv2019

On convergence and rational summation of power series in p-adic field

In this paper we have discussed convergence of power series both in p-adic norm as well as real norm. We have investigated rational summability of power series with respect to both p-adic norm and real norm under certain conditions. Then we have studied convergence of specially constructed power series and derived summation formula. Finally, we have studied the adele, idele and some results regarding it with the help of convergent power series.

math.NT↗

arXiv2019

Curvature properties of Melvin magnetic metric

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 $R\cdot R-Q(S,R)=\mathcal L' Q(g,C)$. 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-Einstien, its Ricci tensor is Reimann compatible and Weyl conformal $2$-forms are recurrent. The Maxwell tensor is also pseudosymmetric type.

math.DG↗

arXiv2019

On Curvature properties of Nariai Spacetimes

The charged Nariai spacetimes are the exact solutions of Einstein-Maxwell field equations with positive cosmological constant and such a spacetime is the direct topological product of a $2$-dimentional de-Sitter spacetime with a round $2$-sphere of constant radius. The present paper deals with the investigation of curvature restricted geometric structures admitted by the charged Nariai spacetime metric and it is shown that such a spacetime is locally symmetric, $2$-quasi-Einstein and $Ein(2)$-space. Moreover it realizes the pseudosymmetric type conditions such as Ricci generalized projectively pseudosymmetric and pseudosymmetric due to Weyl conformal curvature tensor. It is interesting to note that the energy momentum tensor can be expressed explicitly with the help of some $1$-forms. We also evaluate the curvature properties of charged Nariai type topological product metric, and it is found that such metric is semisymmetric but not locally symmetric and under certain restrictions belongs to the family of generalized class of recurrent manifolds. Finally we end up our findings with some concluding remarks.

math.DG↗

arXiv2019

Non-existence of Riemannian metric satisfying Yamabe soliton

In this paper we have proved that a compact Riemannian manifold does not admit a metric with positive scalar curvature if there exists a real valued function in this manifold which is strictly positive along a geodesic ray satisfying expanding or steady Yamabe soliton. We have also deduced a relation between scalar curvature and surface area of a geodesic ball in a Riemannian manifold with a pole satisfying steady Yamabe soliton.

math.DG↗

arXiv2019

Conformal image of an osculating curve on a smooth immersed surface

The main intention of the paper is to investigate an osculating curve under the conformal map. We obtain a sufficient condition for the conformal invariance of an osculating curve. We also find an equivalent system of a geodesic curve under the conformal transformation(motion) and show its invariance under isometry and homothetic motion.

math.GM↗

arXiv2019

Rectifying curves under conformal transformation

The main aim of this paper is to investigate the nature of invariancy of rectifying curve under conformal transformation and obtain a sufficient condition for which such a curve remains conformally invariant. It is shown that the normal component and the geodesic curvature of the rectifying curve is homothetic invariant.

math.GM↗

arXiv2019

Some characterizations of Rectifying and osculating curves on a smooth immersed surface

The present paper deals with some characterizations of rectifying and osculating curves on a smooth surface with respect to the reference frame $\{\vec{T},\ \vec{N},\ \vec{T}\times\vec{N}\}$. We have computed the components of position vectors of rectifying and osculating curves along $\vec{T},\ \vec{N},\ \vec{T}\times\vec{N}$ and then investigated their invariancy under isometry of surfaces, and it is shown that they are invariant iff either the normal curvature of the curve is invariant or the position vector of the curve is in the direction of the tangent vector to the curve.

math.GM↗

arXiv2019

Normal curves on a smooth immersed surface

The aim of this paper is to investigate the sufficient condition for the invariance of a normal curve on a smooth immersed surface under isometry. We also find the the deviations of the tangential and normal components of the curve with respect to the given isometry.

math.GM↗

arXiv2019

Certain Results On $N(\Kappa)$-contact Metric Manifolds

In this paper, $N(κ)$-contact metric manifolds satisfying the conditions $\widetilde{C}(ξ,X)\cdot\widetilde{C}=0$, $\widetilde{C}(ξ,X)\cdot R=0$, $\widetilde{C}(ξ,X)\cdot S=0$, $\widetilde{C}(ξ,X)\cdot C=0$, $C\cdot S=0$ and $R\cdot C=f_{C}Q(g,C)$ have been investigated and obtained their classification. Among others it is shown that a Weyl-pseudosymmetric $N(κ)$-contact metric manifold is either locally isometric to the Riemannian product $E^{n+1}(0)\times S^{n}(4)$ or an $η$-Einstein manifold. Finally, an example is given.

math.DG↗

arXiv2019

Curves on a smooth surface with position vectors lie in the tangent plane

The present paper deals with a study of curves on a smooth surface whose position vector always lies in the tangent plane of the surface and it is proved that such curves remain invariant under isometry of surfaces. It is also shown that length of the position vector, tangential component of the position vector and geodesic curvature of a curve on a surface whose position vector always lies in the tangent plane are invariant under isometry of surfaces.

math.GM↗

arXiv2019

Curvature properties of Vaidya metric

As a generalization of the Schwarzschild solution, Vaidya presented a radiating metric to develop a model of the exterior of a star including its radiation field, called Vaidya metric. The present paper deals with the investigation on the curvature properties of Vaidya metric. It is shown that Vaidya metric can be considered as a model of different pseudosymmetric type curvature conditions, namely, $C\cdot C = \frac{m}{r^3} Q(g, C)$, $R\cdot R - Q(S,R) = \frac{m}{r^3} Q(g,C)$ etc. It is also shown that Vaidya metric is Ricci simple, vanishing scalar curvature and its Ricci tensor is Riemann-compatible. As a special case of the main result, we obtain the curvature properties of Schwarzschild metric. Finally, we compare the curvature properties of Vaidya metric with another radiating metric, namely, Ludwig-Edgar pure radiation metric.

math.DG↗

arXiv2018

Some results on $η$-Ricci Soliton and gradient $ρ$-Einstein soliton in a complete Riemannian manifold

The main purpose of the paper is to prove that if a compact Riemannian manifold admits a gradient $ρ$-Einstein soliton such that the gradient Einstein potential is a non-trivial conformal vector field, then the manifold is isometric to the Euclidean sphere. We have showed that a Riemannian manifold satisfying gradient $ρ$-Einstein soliton with convex Einstein potential possesses non-negative scalar curvature. We have also deduced a sufficient condition for a Riemannian manifold to be compact which satisfies almost $η$-Ricci soliton (see Theorem 2).

math.DG↗

arXiv2018

Rectifying curves on a smooth surface immersed in the euclidean space

The main objective of the present paper is to investigate a sufficient condition for which a rectifying curve on a smooth surface remains invariant under isometry of surfaces, and also it is shown that under such an isometry the component of the position vector of a rectifying curve on a smooth surface along the normal to the surface is invariant.

math.GM↗

arXiv2018

Integral Liouville theorem in a complete Riemannian manifold

If the Killing vector field in a Riemannian manifold is the gradient of a smooth real valued function, then it is called Killing potential. In this paper we have deduced a necessary condition for the existence of Killing potential in a complete Riemannian manifold. Yau proved the Liouville theorem of harmonic function in a Riemannian manifold using gradient estimation and after that many authors have generalized this concept and investigated various types of Liouville theorems of harmonic functions. In this article we have also proved a Liouville theorem in integral form of harmonic functions in a complete Riemannian manifold. Finally we have studied the behaviour of harmonic functions in a complete Riemannian manifold that satisfies some gradient Ricci solition and showed that harmonic function is a constant multiple of distance function along some geodesics.

math.DG↗

arXiv2018

Polynomial growth of subharmonic functions in a strongly symmetric Riemannian manifold

In this article we have studied some properties of subharmonic functions in a strongly symmetric Riemannian manifold with a pole. As a generalization of polynomial growth of a function we have introduced the notion of polynomial growth of some degree of a function with respect to a real function and proved that any non-negative twice differentiable subharmonic functions in an $n$-dimensional manifold always admit polynomial growth of degree $1$ with respect to a non-negative real valued subharmonic function on real line. We have also given a lower bound of the integration of a convex function in a geodesic ball.

math.DG↗

arXiv2018

Geodesic Sandwich Theorem with an Application

The main goal of the paper is to prove the sandwich theorem for geodesic convex functions in a complete Riemannian manifold. Then by using this theorem we have proved an inequality in a manifold with bounded sectional curvature. Finally, we have shown that the gradient of a convex function is orthogonal to the tangent vector at some point of any geodesic.

math.DG↗

arXiv2018

Non-existence of certain type of convex functions on a Riemannian manifold with a pole

This paper is devoted to the study of non-existence of certain type of convex functions on a Riemannian manifold with a pole. To this end, we have developed the notion of odd and even function on a Riemannian manifold with a pole and proved the non-existence of non-trivial and non-negative differentiable odd convex function whose gradient is complete. Finally, we have deduced some isoperimetric type inequality related with convex function.

math.DG↗