arXiv subjects

Absos Ali Shaikh

Publications and source records attributed to Absos Ali Shaikh.

At least 37 records · Page 2Linked to original sources

arXiv2022

Curvature inheritance symmetry on M-projectively flat spacetimes

The paper aims to investigate curvature inheritance symmetry in M-projectively flat spacetimes. It is shown that the curvature inheritance symmetry in M-projectively flat spacetime is a conformal motion. We have proved that M- projective curvature tensor follows the symmetry inheritance property along a vector field $ξ$, when spacetime admits the conditions of both curvature inheritance symmetry and conformal motion or motion along the vector field $ξ$. Also, we have derived some results for M-projectively flat spacetime with perfect fluid following the Einstein field equations with a cosmological term and admitting the curvature inheritance symmetry along the vector field $ξ$. We have shown that an M-projectively flat perfect fluid spacetime obeying the Einstein field equations with a cosmological term and admitting the curvature inheritance symmetry along a vector field $ξ$ is either a vacuum or satisfies the vacuum-like equation of state. We have also shown that such spacetimes with the energy momentum tensor of an electromagnetic field distribution do not admit any curvature symmetry of general relativity. Finally, an example of M-projectively flat spacetime has been exhibited.

gr-qc↗

arXiv2022

Diameter estimation of $(m,ρ)$-quasi Einstein manifolds

This paper aims to study the $(m,ρ)$-quasi Einstein manifold. This article shows that a complete and connected Riemannian manifold under certain conditions becomes compact. Also, we have determined an upper bound of the diameter for such a manifold. It is also exhibited that the potential function acquiesces to the Hodge-de Rham potential up to a real constant in an $(m,ρ)$-quasi Einstein manifold. Later, some triviality and integral conditions are established for a non-compact complete $(m,ρ)$-quasi Einstein manifold having finite volume. Finally, it is proved that with some certain constraints, a complete Riemannian manifold admits finite fundamental group. Furthermore, some conditions for compactness criteria have also been deduced.

math.DG↗

arXiv2022

Some Characterizations of Quasi Yamabe solitons

In this article, we have proved some results in connection with the potential vector field having finite global norm in quasi Yamabe soliton. We have derived some criteria in particular for the potential vector field on the non-positive Ricci curvature of the quasi Yamabe soliton. Also, a necessary condition for a compact quasi Yamabe soliton has been formulated. We further showed that if the potential vector field has a finite global norm in a complete non-trivial, non-compact quasi Yamabe soliton with finite volume, then the scalar curvature becomes constant and the soliton reduces to a Yamabe soliton.

math.DG↗

arXiv2022

Curvature properties of Bardeen black hole spacetime

The Bardeen solution corresponding to Einstein field equations with a cosmological constant is a regular black hole. The main goal of this manuscript is to investigate the geometric structures in terms of curvature conditions admitted by this spacetime. It is found that this spacetime is pseudosymmetric and possess several kinds of pseudosymmetries. Also, it is a manifold of pseudosymmetry Weyl curvature and the difference tensor C.R-R.C linearly depends on the tensors Q(g;C) and Q(S;C). It is interesting to note that such a spacetime is weakly generalized recurrent manifold and satisfies special recurrent like structure. Further, it is an Einstein manifold of level 2 and Roter type. The energy momentum tensor of this spacetime is pseudosymmetric and finally a worthy comparison between the geometric properties of Bardeen spacetime and Reissner-Nordström spacetime is given.

math.DG↗

arXiv2022

Splitting theorem of Gradient $ρ$-Einstein solitons

In this paper, we have proved a weighted Laplacian comparison of distance function for manifolds with Bakry-Émery curvature bounded from below. Next, we have shown that a gradient $ρ$-Einstein soliton with a bounded integral condition on Ricci curvature splits off a line isometrically. Moreover, using this result, we have established some boundedness conditions on scalar curvature of gradient $ρ$-Einstein soliton.

math.DG↗

arXiv2021

Rigidity and unlikely intersections for stable $p$-adic dynamical systems

Berger asked the question \enquote{To what extent the preperiodic points of a stable $p$-adic power series determines a stable $p$-adic dynamical system} ? In this work we have applied the preperiodic points of a stable $p$-adic power series in order to determine the corresponding stable $p$-adic dynamical system.

math.NT↗

arXiv2021

Diameter estimation of gradient $ρ$-Einstein solitons

Our aim in this article is to give a lower bound of the diameter of a compact gradient $ρ$-Einstein soliton satisfying some given conditions. We have also deduced some conditions of the gradient $ρ$-Einstein soliton with bounded Ricci curvature to become non-shrinking and non-expanding. Further, we have proved that a complete non-compact gradient shrinking or expanding Schouten soliton with non-constant potential and a boundedness condition on scalar curvature must be non-parabolic.

math.DG↗

arXiv2021

$m$-quasi Einstein manifolds with convex potential

The main objective of this paper is to investigate the $m$-quasi Einstein manifold when the potential function becomes convex. In this article, it is proved that an $m$-quasi Einstein manifold satisfying some integral conditions with vanishing Ricci curvature along the direction of potential vector field has constant scalar curvature and hence the manifold turns out to be an Einstein manifold. It is also shown that in an $m$-quasi Einstein manifold the potential function agrees with Hodge-de Rham potential up to a constant. Finally, it is proved that if a complete non-compact and non-expanding $m$-quasi Einstein manifold has bounded scalar curvature and the potential vector field has global finite norm, then the scalar curvature vanishes.

math.DG↗

arXiv2021

Existence of finite global norm of potential vector field in a Ricci soliton

In this article, we investigate global norm of potential vector field in Ricci soliton. In particular, we have deduced certain conditions so that the potential vector field has finite global norm in expanding Ricci soliton. We have also showed that if the potential vector field has finite global norm in complete non-compact Ricci soliton having finite volume, then the scalar curvature becomes constant.

math.GM↗

arXiv2020

Some characterizations of $ρ$-Einstein solitons

In this article we have showed that a gradient $ρ$-Einstein soliton with a vector field of bounded norm and satisfying some other conditions is isometric to the Euclidean sphere. Later, we have proved that a non-trivial complete gradient $ρ$-Einstein soliton with finite weighted Dirichlet integral and certain restriction on Ricci curvature must be of constant scalar curvature and steady Ricci flat. Finally, we have proved that a non-shrinking or non-expanding gradient traceless Ricci soliton possessing some conditions must be steady.

math.DG↗

arXiv2020

Extension property of continuous functions in a Riemannain manifold with a pole

The Brouwer fixed point theorem says that any continuous function from disc to itself has a fixed point. By using simple geometrical technique we have generalized the result in manifold and proved that any continuous function on the boundary of a bounded convex domain of a $2$-dimensional Riemannian manifold with a pole having at least one fixed point can be extended to the convex domain without any interior fixed point.

math.DG↗

arXiv2020

Isometry theorem of gradient Shrinking Ricci solitons

In this paper, we have proved that if a complete conformally flat gradient shrinking Ricci soliton has linear volume growth or the scalar curvature is finitely integrable and also the reciprocal of the potential function is subharmonic, then the manifold is isometric to the Euclidean sphere. As a consequence, we have showed that a four dimensional gradient shrinking Ricci soliton satisfying some conditions is isometric to $\mathbb{S}^4$ or $\mathbb{RP}^4$ or $\mathbb{CP}^2$. We have also deduced a condition for the shrinking Ricci soliton to be compact with quadratic volume growth.

math.DG↗

arXiv2020

On Ricci solitons whose potential is convex

In this paper we consider the Ricci curvature of a Ricci soliton. In particular, we have showed that a complete gradient Ricci soliton with non-negative Ricci curvature possessing a non-constant convex potential function having finite weighted Dirichlet integral satisfying an integral condition is Ricci flat and also it isometrically splits a line. We have also proved that a gradient Ricci soliton with non-constant concave potential function and bounded Ricci curvature is non-shrinking and hence the scalar curvature has at most one critical point.

math.DG↗

arXiv2020

Some characterizations of gradient Yamabe solitons

In this article we have proved that a gradient Yamabe soliton satisfying some additional conditions must be of constant scalar curvature. Later, we have showed that in a gradient expanding or steady Yamabe soliton with non-negative Ricci curvature if the potential function satisfies some integral condition then it is subharmonic, in particular, for steady case the potential function becomes harmonic. Also we have proved that, in a compact gradient Yamabe soliton, the potential function agrees with the Hodge-de Rham potential upto a constant.

math.DG↗

arXiv2020

Compact gradient $ρ$-Einstein soliton is isometric to the Euclidean sphere

In this paper we have investigated some aspects of gradient $ρ$-Einstein Ricci soliton in a complete Riemannian manifold. First, we have proved that the compact gradient $ρ$-Einstein soliton is isometric to the Euclidean sphere by showing that the scalar curvature becomes constant. Second, we have showed that in a non-compact gradient $ρ$-Einstein soliton satisfying some integral condition, the scalar curvature vanishes.

math.DG↗

arXiv2020

On the freeness and projectiveness of Breuil-Kisin module

In the work we have considered Breuil-Kisin module over the ring of witt vectors $W(κ)$ over the residue field $κ$ of characteristic $p$ and a finite flat $\mathbb{Z}_p$-algebra $R$. Then considered Breuil-Kisin modules $M$ over the ring $W(κ)$ and taking the action of $R$ on $W(κ)$, we get again a Breuil-Kisin module $M$ over the ring $R \otimes_{\mathbb{Z}_p} W(κ)$. We have studied freeness and projectiveness of this module.

math.NT↗

arXiv2020

Isometry theorem of Cartan-Hadamard manifold

Cartan-Hadamard manifold is a simply connected Riemannian manifold with non-positive sectional curvature. In this article, we have proved that a Cartan-Hadamard manifold satisfying steady gradient Ricci soliton with the integral condition of potential function is isometric to the Euclidean space. Next we have proved a compactness theorem for gradient shrinking Ricci soliton satisfying some scalar curvature condition. Finally, we have showed that a gradient expanding Ricci soliton with linear volume growth and positive potential function is an Einstein manifold.

math.DG↗

arXiv2019

On Rational Invariant Summation of p-Adic Power Series with Binomial Coefficient

In the work we have considered p-adic functional series with binomial coefficients and discussed its p-adic convergence. Then we have derived a recurrence relation following with a summation formula which is invariant for rational argument. More precisely, we have investigated a sufficient condition under which the p-adic power series converges to a rational invariant sum 0. Finally we have shown application of the invariant summation formula to get some intersting relations involving Bernoulli numbers and Bernoulli polynomials.

math.NT↗