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Absos Ali Shaikh

Publications and source records attributed to Absos Ali Shaikh.

At least 19 recordsLinked to original sources
arXiv2026

Pseudosymmetry, Ricci soliton and Curvature Inheritance symmetries of Friedmann Lema\^itre Robertson Walker spacetime

The Friedmann--Lema\^{i}tre--Robertson--Walker (FLRW) spacetime, which was first proposed by Friedmann (1922--1924) and Lema\^{i}tre (1927) and subsequently developed by Robertson and Walker (1935), is an isotropic and homogeneous cosmological model of the universe. This paper addresses a significant gap in the differential geometry literature by providing a comprehensive examination of the curvature properties of the FLRW spacetime. It is demonstrated that the FLRW spacetime satisfies the curvature condition R \cdot R - Q(S, R)=L_C Q(g, C) alongside several pseudosymmetric-type conditions related to the conformal and conharmonic curvature tensors. Furthermore, the Tachibana tensors Q(g,C) and Q(S, C) are found to exhibit a linear dependence on the tensor $(C \cdot R + R \cdot C)$. Additionally, the spacetime is shown to be a 2-quasi-Einstein manifold, generalized Roter type and Ein(3). The Ricci tensor is shown to be neither cyclic parallel nor of Codazzi type, yet it satisfies several compatibility requirements concerning the R, C, P, K and W curvature tensors. A thorough analysis of Ricci solitons and curvature inheritance properties reveals that the spacetime admits almost Ricci soliton and $\eta$-Ricci Yamabe soliton structures with respect to the non-Killing soliton vector fields $\frac{\partial}{\partial t}$ and $\frac{\partial}{\partial r}$. Moreover, the spacetime admits generalized curvature inheritance symmetry properties for the Riemann curvature tensor, as well as for the Weyl conformal, concircular, and conharmonic curvature tensors with respect to the coordinate vector field $\frac{\partial}{\partial t}$ and the gradient of $t$. Later, a comparison of the FLRW and Lema\^{i}tre--Tolman--Bondi (LTB) spacetimes is provided in terms of various curvature-related geometric properties and physical characteristics. Finally, a noteworthy conclusion of the entire study is presented.

math.GM↗
arXiv2026

On triviality and scalar curvature estimation of gradient h-almost Yamabe solitons

In this study we have explored gradient $h$-almost Yamabe solitons on both compact and complete non-compact Riemannian manifolds. We have established several sufficient conditions for the triviality of such solitons with respect to integral inequalities involving the scalar curvature and the soliton function. In this regard, we have acquired scalar curvature estimation under certain $L^2$-integrability conditions, and defined signal of the function $h$. Our results extend and refine former works on almost and $h$-almost Yamabe solitons, and characterize the geometric structures of generalized Yamabe solitons.

math.DG↗
arXiv2025

Symmetry and Pseudosymmetry properties with Ricci soliton of the Reissner-Nordstr\"{o}m-de Sitter spacetime

The primary objective of the article is to investigate the symmetry and pseudosymmetry properties of the Reissner-Nordstr\"om-de Sitter (briefly, RNdS) spacetime. The secondary aim of the paper is to explore the notion of Ricci solitons in RNdS spacetimes. The study is important due to the conceding of almost Ricci soliton and almost Ricci Yamabe soliton of the RNdS spacetime. The analysis shows that this spacetime satisfies multiple types of symmetric and pseudosymmetric conditions. It is interesting to note that RNdS spacetime reveled pseudosymmetry, conformal pseudosymmetry, Weyl projective pseudosymmetry, conharmonic pseudosymmetry, and concircular pseudosymmetry. Furthermore, in the RNdS spacetime, obtained by second order covariant derivatives $R\cdot R$ is linearly dependent on $Q(S,R)$ and $Q(g,C)$. It is demonstrated that the RNdS spacetime is 2-quasi Einstein and an Ein(2) space with recurrent conformal 2-forms. We derive the general form of the compatible tensors for this spacetime. The energy-momentum tensor of the RNdS spacetime is also shown to be pseudosymmetric and also the energy momentum tensor is pseudosymmetric due to conformal, conharmonic, concircular and projective curvature tensor. The energy momentum tensor is compatible with these curvature. We study a generalized notion of curvature inheritance and find that, with respect to the non-Killing vector fields $\partial/\partial r$ and $\partial/\partial \theta$, the RNdS spacetime does not satisfy these inheritance conditions. However, the RNdS spacetime is shown to admit an almost Ricci soliton and an almost $\eta$-Ricci Yamabe soliton with respect to the non-Killing vector field $\partial/\partial r$ but with respect to the non-Killing vector field $\partial/\partial \theta$ the spacetime does not admit such notions. Finally, we present a comparison between the RNdS and Vaidya-Bonner-de Sitter (VBdS) spacetimes.

math.GM↗
arXiv2025

An exploration of the curvature and inheritance properties of the Interior black hole spacetime

In continuation of the study in \cite{SDHK_interior_2020}, the present article explores the geometric and curvature properties of the interior black hole (briefly, IBH) spacetime. It is shown that in an IBH spacetime the operator $R\cdot C$ and $C\cdot R$ does not commute with each other and infact the commutator $C\cdot R-R\cdot C$ is linearly dependent with $Q(g,R)$ and $Q(S,R)$ as well as $Q(g,C)$ and $Q(S,C)$. Also in IBH spacetime $R \cdot R$ is linearly dependent with $Q(S,R)$ and $Q(g,C)$. It is exhibited that IBH spacetime is $2$-quasi Einstein, Ein$(2)$ and generalized quasi Einstein spacetime in the sense of Chaki, and its conformal $2$-forms are recurrent. We have derived the universal form of the compatible tensors in such a spacetime. We have also demonstrated that the nature of energy momentum tensor of IBH spacetime is pseudosymmetric (see, Theorem $4.1$). Again it is exposed that with respect to the non-Killing vector field $\frac{\partial}{\partial t},$ the IBH spacetime obeys the generalized curvature inheritance, generalized Ricci inheritance, special type of generalized conformal, concircular, conharmonic and generalized Weyl projective inheritance. Finally a comparison between IBH spacetime and KIselev Black Hole (KBH) is displayed.

physics.gen-ph↗
arXiv2025

On the existence of various generalizations of semisymmetric and pseudosymmetric type manifolds

The objective of the paper is to investigate a sequential study of different generalizations of semisymmetric and pseudosymmetric manifolds with their proper existence by several spacetimes. In the literature of differential geometry, there are many generalizations of such notions in various directions by involving different curvature tensors. In this paper, we have systematically and consecutively reviewed various generalized notions of semisymmetry, such as, Ricci semisymmetry, conformal semisymmetry, pseudosymmetry, Ricci pseudosymmetry, Ricci generalized pseudosymmetry, conformal pseudosymmetry, Ricci generalized Weyl pseudosymmetry and also many other semisymmetry type conditions. Most importantly, we have exhibited a plenty of suitable examples to examine the proper existence of such geometric structures, and they are physically significant as several spacetimes admit such geometric structures. By considering an immersion of Schwarzschild black hole metric, we have also provided a new example of Ricci pseudosymmetric manifolds. Finally, the interesting character of Weyl projective curvature tensors of type $(1,3)$ and $(0,4)$ are exhibited with their interesting examples.

math.DG↗
arXiv2024

Properties of Breuil-Kisin modules inherited by $p$-divisible groups

In this paper, by assuming a faithful action of a finite flat $\mathbb{Z}_p$-algebra $\mathscr{R}$ on a $p$-divisible group $\mathcal{G}$ defined over the ring of $p$-adic integers $\mathscr{O}_K$, we construct a category of new Breuil-Kisin module $\mathfrak{M}$ defined over the ring $\mathfrak{S}:=W(κ)[\![u]\!]$ and study the freeness and projectiveness properties of such a module.

math.NT↗
arXiv2024

Symmetry and pseudosymmetry properties of Vaidya-Bonner-de Sitter spacetime

The primary focus of the current study is to explore the geometrical properties of the Vaidya-Bonner-de Sitter (briefly, VBdS) spacetime, which is a generalization of Vaidya-Bonner spacetime, Vaidya spacetime and Schwarzschild spacetime. In this study we have shown that the VBdS spacetime describes various types of pseudosymmetric structures, including pseudosymmetry due to conformal curvature, conharmonic curvature and other curvatures. Additionally, it is shown that such a spacetime is 2-quasi-Einstein, Einstein manifold of level 3, generalized Roter type, and that conformal 2-forms are recurrent. The geometric features of the Vaidya-Bonner spacetime, Vaidya spacetime, and Schwarzschild spacetime are obtained as a particular instance of the main determination. It is further established that the VBdS spacetime admits almost Ricci soliton and almost η-Yamabe soliton with respect to non-Killing vector fields. Also, it is proved that such a spacetime possesses generalized conharmonic curvature inheritance. It is interesting to note that in the VBdS spacetime the tensors Q(T,R), Q(S,R) and Q(g,R) are linearly dependent. Finally, this spacetime is compared with the Vaidya-Bonner spacetime with respect to their admitting geometric structures, viz., various kinds of symmetry and pseudosymmetry properties.

math.DG↗
arXiv2023

Construction of Fuchsian Schottky group with conformal boundary at infinity

In this article, we have constructed an interesting type of generalized Schottky group, named as Fuchsian Schottky group of arbitrary finite rank, in the context of the classical Schottky group (i.e., Schottky curves which are Euclidean circles). After that, we initiated the construction of the generalized Fuchsian Schottky group of any finite rank by including orientation-reversing isometries of the hyperbolic plane as side-pairing transformations. Further, we have investigated the hyperbolic ends for any arbitrary finite rank Fuchsian Schottky groups from the point of view of the Euler characteristic in the hyperbolic surface. Finally, we have shown that the compact core of the conformally compact Riemann surface can be decomposed into non-tight pairs of pants by using suitable twist parameters with some fixed Bers' constant. The Fenchel-Nielsen coordinates for Teichmüller space corresponding to any finite rank Fuchsian Schottky groups are also obtained.

math.DG↗
arXiv2023

Ricci solitons and curvature inheritance on Robinson-Trautman spacetimes

The purpose of the article is to investigate the existence of Ricci solitons and the nature of curvature inheritance as well as collineations on the Robinson-Trautman (briefly, RT) spacetime. It is shown that under certain conditions RT spacetime admits almost Ricci soliton, almost $η$-Ricci soliton, almost gradient $η$-Ricci soliton. As a generalization of curvature inheritance \cite{Duggal1992} and curvature collineation \cite{KLD1969}, in this paper, we introduce the notion of \textit{generalized curvature inheritance} and examine if RT spacetime admits such a notion. It is shown that RT spacetime also realizes the generalized curvature (resp. Ricci, Weyl conformal, concircular, conharmonic, Weyl projective) inheritance. Finally, several conditions are obtained, under which RT spacetime possesses curvature (resp. Ricci, conharmonic, Weyl projective) inheritance as well as curvature (resp. Ricci, Weyl conformal, concircular, conharmonic, Weyl projective) collineation, and we have also introduced the concept of generalized Lie inheritance and showed that RT spacetime realizes such a notion.

math.DG↗
arXiv2023

Non-classical generating sets in Fuchsian Schottky groups

The goal of this article is to initiate the study of estimates of the non-classical Schottky structure in the discrete subgroups of the projective special linear group over the real numbers degree $2$. In fact, in this paper, we have investigated the non-classical generating sets in the Fuchsian Schottky groups on the hyperbolic plane with boundary. A Schottky group is usually considered non-classical if the curves used in the Schottky construction are Jordan curves (except the Euclidean circles). More precisely, in this manuscript, we have provided a structure of the rank $2$ Fuchsian Schottky groups with non-classical generating sets by utilizing two suitable hyperbolic M\"obius transformations on the upper-half plane model. In particular, we have derived two non-trivial examples of Fuchsian Schottky groups with non-classical generating sets in the upper-half plane with the circle at infinity as the boundary.

math.DG↗
arXiv2023

Triviality Results and Conjugate Radius Estimation of Ricci Solitons

The investigation of Ricci solitons is the focus of this work. We have proved triviality results for compact gradient Ricci soliton under certain restriction. Later, a rigidity result is derived for a compact gradient shrinking Ricci soliton. Also, we have estimated the conjugate radius for non-compact gradient shrinking Ricci solitons with superharmonic potential. Moreover, an upper bound for the conjugate radius of Ricci soliton with concircular potential vector field is determined. Finally, it is proved that a non-compact gradient Ricci soliton with a pole and non-negative Ricci curvature is non-shrinking.

math.DG↗
arXiv2023

Geometrical Properties of a Point-like Global Monopole Spacetime

The aim of this paper is to study the geometric properties of the point-like global monopole (briefly, PGM) spacetime, which is a static and spherically symmetric solution of the Einstein's field equations. It has shown that PGM spacetime admits various types of pseudosymmetry structures, such as pseudosymmetry due to Weyl conformal curvature tensor, pseudosymmetry due to concircular curvature tensor, pseudosymmetry due to conharmonic curvature tensor, Ricci generalized conformal pseudo-symmetric due to projective curvature tensor, Ricci generalized projective pseudo-symmetric. Moreover, it has proved that PGM spacetime is $2$-quasi Einstein, generalized quasi-Einstein, Einstein manifold of degree $2$, and its Weyl conformal curvature $2$-forms are recurrent. The energy-momentum tensor of the PGM spacetime realizes several types of pseudosymmetry, and its Ricci tensor is compatible with Riemann curvature, Weyl conformal curvature, projective curvature, and conharmonic curvature and concircular curvature. Further, it has shown that PGM spacetime admits motion, curvature collineation, and Ricci collineation. Also, the notion of curvature inheritance (resp., curvature collineation) for the (1,3)-type curvature tensor is not equivalent to the notion of curvature inheritance (resp., curvature collineation) for the (0,4)-type curvature tensor as it has shown that such distinctive properties were possessed by PGM spacetime. Hence the notions of curvature inheritance defined by Duggal \cite{Duggal1992} and Shaikh and Datta \cite{ShaikhDatta2022} are not equivalent.

gr-qc↗
arXiv2023

Unlikely intersection in higher-dimensional formal groups

In this article, we identify a class of higher-dimensional formal groups over the ring of $p$-adic integers that are uniquely determined by their $p$-power torsion points. More precisely, we prove that if two simple finite-height formal groups share infinitely many torsion points, then they are equal. This extends a rigidity theorem of Berger \cite{LB1} from the one-dimensional setting to a higher-dimensional family of simple formal groups.

math.NT↗
arXiv2023

On curvature related geometric properties of Hayward black hole spacetime

This paper is devoted to the study of curvature properties of Hayward black hole (briefly, HBH) spacetime, which is a solution of Einstein field equations (briefly, EFE) having non-vanishing cosmological constant. We have proved that the HBH spacetime is an Einstein manifold of level $2$, $2$-quasi Einstein, generalized quasi-Einstein and Roter type manifold. Also, it is shown that the nature of the HBH spacetime is pseudosymmetric and it obeys several types of pseudosymmetries, such as, pseudosymmetry due to concircular, conformal and conharmonic curvature (i.e., $F\cdot F=\mathcal{L}Q(g,F)$ for $F=W,C, K$ with a smooth scalar function $ \mathcal{L} $), and it also possesses the relation $R\cdot R-\mathcal{L} Q(g,C)=Q(S,R)$. It is engrossing to mention that the nature of energy momentum tensor of the HBH spacetime is pseudosymmetric. On the basis of curvature related properties, we have made a comparison among Reissner-Nordström spacetime, interior black hole spacetime and HBH spacetime. Also, it is shown that the HBH spacetime admits an almost $η$-Ricci soliton as well as an almost $η$-Ricci-Yamabe soliton. Finally, an elegant comparative study is delineated between the HBH spacetime and the point-like global monopole spacetime with respect to different kinds of symmetry, such as, motion, curvature collineation, curvature inheritance etc.

math.DG↗
arXiv2023

Scalar curvature estimation of Generalized Ricci-Yamabe solitons

This paper is concerned with the study of generalized gradient Ricci-Yamabe solitons. We characterize the compact generalized gradient Ricci-Yamabe soliton and find certain conditions under which the scalar curvature becomes constant. The estimation of Ricci curvature is deduced and also an isometry theorem is found in gradient Ricci-Yamabe soliton satisfying a finite weighted Dirichlet integral. Further, it is proved that a Ricci-Yamabe soliton reduces to an Einstein manifold when the potential vector field becomes concircular. Moreover, the eigenvalue and the corresponding eigenspace of the Ricci operator are also discussed in case of a Ricci-Yamabe soliton with concircular potential vector field.

math.DG↗
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↗