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Uday Chand De

Publications and source records attributed to Uday Chand De.

At least 19 recordsLinked to original sources
arXiv2026

Characterization of generalized quasi-Einstein manifolds and modified gravity

In this work, a detailed examination of a specific case of a generalized quasi-Einstein manifold (GQE)n is provided. It begins by exploring generalized quasi-Einstein spacetimes under certain conditions. The analysis then focuses on cases that admit a parallel time-like vector field. Among the findings, it is demonstrated that such spacetimes can be categorized as generalized Robertson-Walker spacetimes, Robertson-Walker spacetimes, and quasi-constant curvature spacetimes. Additionally, the physical implications of these results are discussed. It is also investigated (GQE)4 spacetimes, which accept F(R)-gravity and feature a parallel unit time-like vector field. Finally, various energy conditions are analyzed based on the results related to F(R)-gravity.

gr-qc↗
arXiv2026

Lyra Almost Ricci-Bourguignon Solitons on Twisted Warped Product Manifolds

This paper defines Lyra almost Ricci-Bourguignon solitons on twisted warped product manifolds and investigates the interaction between the twisting function, the Lyra scale, and the soliton potential field. We derive the horizontal, vertical, and mixed components of the soliton equation and obtain trace, gradient, and factor-inheritance characterizations. The mixed equation yields new rigidity phenomena: a base-dependent Lyra scale preserves the ordinary separability obstruction, whereas a fiber-dependent scale admits a weighted compensation law that allows genuinely nonseparable twisting functions. We also establish Einstein reductions for conformal, Killing, homothetic, concurrent, and gradient potential fields. Exact flat and non-flat examples, together with local and compact nonexistence results, demonstrate the geometric significance of the scale?twisting interaction.

math.DG↗
arXiv2026

First Chen Inequality for CR-Warped Product Submanifolds of a Complex Space Form and Applications

In this paper, the first Chen inequality is proved for CR-warped product submanifolds in complex space forms. This inequality involves intrinsic invariants (a leaf-wise $δ$-invariant and the sectional curvature) controlled by an extrinsic one (the mean curvature vector), which provides an answer to Problem [1]. We carefully distinguish the leaf-wise $δ$-invariant of a factor (used in the bound) from the intrinsic Chen invariant of the same factor, the two being related, on the totally real factor, by the Bishop--O'Neill formula. The bound is sharp and is uniform in the sign of the holomorphic sectional curvature $c$. As a geometric application, we derive necessary conditions for the immersed CR-warped product submanifold to be minimal in a complex space form, providing a partial answer to a well-known problem proposed by S.S. Chern (Problem [2]). For further research directions, we address a couple of open problems (Problem [3]} and Problem [4]).

math.DG↗
arXiv2024

On Pseudo $B$-symmetric spacetimes and $f(\mathcal{R})$ gravity

This article delivers the characterization of a pseudo $B$ symmetric spacetimes and we illustrate that a pseudo $B$ symmetric spacetime admitting Codazzi type of $B$-tensor represents a perfect fluid spacetime and if this spacetime admits the time-like convergence criterion, then the pseudo $B$ symmetric spacetime fulfills cosmic strong energy criterion and contains pure matter. Besides, we find in a pseudo $B$ symmetric spacetime with Codazzi type of $B$-tensor the electric part of the Weyl tensor vanishes and has Riemann and Weyl compatible vector fields. Furthermore, it is established that the chosen spacetime with Codazzi type of $B$-tensor is conformally flat and represents a Robertson-Walker spacetime. Also, we calculate the scale factor $\varPsi (t)$ for these spacetimes in a spatially flat Robertson-Walker spacetime. Finally, we study the impact of this spacetime under $f(R)$ gravity scenario and deduce several energy conditions by considering a new model $f\left(\mathcal{R}\right)= e^{(α\mathcal{R})}-ln(β\mathcal{R})$ in which $α$ and $β$ are positive constants.

gr-qc↗
arXiv2024

Pseudo generalized Ricci-recurrent spacetimes with certain applications to modified gravity

In this article we introduce and characterize a pseudo generalized Ricci-recurrent spacetimes. At first, we produce an example to justify the existence of such a spacetime. Then, it is provided that a pseudo generalized Ricci-recurrent generalized Robertson-Walker spacetime represents a perfect fluid spacetime and a pseudo generalized Ricci-recurrent perfect fluid spacetime represents either a dark energy epoch of the Universe or, the velocity vector field is parallel, conservative, acceleration-free, vorticity-free, and shear-free and becomes a static spacetime. Lastly, we study the impact of this spacetime under $f(\mathcal{R})$ gravity scenario and deduce several energy conditions.

gr-qc↗
arXiv2024

Pseudo generalized Ricci-recurrent spacetimes and modified gravity

In this paper we introduce and characterize a pseudo generalized Ricci-recurrent spacetimes and produce an example to verify the existence of such a spacetime. Then we demonstrate that a conformally flat generalized Ricci-recurrent spacetime with certain condition is a pseudo quasi-Einstein spacetime. Besides, it is proved that a pseudo generalized Ricci-recurrent generalized Robertson-Walker spacetime represents a perfect fluid spacetime. Lastly, we study the impact of this spacetime under $f(\mathcal{R},T^2)$ and $f\left(\mathcal{R}^{\ast}\right)$ gravity scenario and deduce several energy conditions.

gr-qc↗
arXiv2024

Characterizations of a Lorentzian Manifold with a semi-symmetric metric connection

In this article, we characterize a Lorentzian manifold $\mathcal{M}$ with a semi-symmetric metric connection. At first, we consider a semi-symmetric metric connection whose curvature tensor vanishes and establish that if the associated vector field is a unit time-like torse-forming vector field, then $\mathcal{M}$ becomes a perfect fluid spacetime. Moreover, we prove that if $\mathcal{M}$ admits a semi-symmetric metric connection whose Ricci tensor is symmetric and torsion tensor is recurrent, then $\mathcal{M}$ represents a generalized Robertson-Walker spacetime. Also, we show that if the associated vector field of a semi-symmetric metric connection whose curvature tensor vanishes is a $f-$ Ric vector field, then the manifold is Einstein and if the associated vector field is a torqued vector field, then the manifold becomes a perfect fluid spacetime. Finally, we apply this connection to investigate Ricci solitons.

math.DG↗
arXiv2024

Investigations on a Riemannian manifold with a semi-symmetric non-metric connection and gradient solitons

This article carries out the investigation of a three-dimensional Riemannian manifold $N^3$ endowed with a semi-symmetric type non-metric connection. Firstly, we construct a non-trivial example to prove the existence of a semi-symmetric type non-metric connection on $N^{3}$. It is established that a $N^3$ with the semi-symmetric type non-metric connection, whose metric is a gradient Ricci soliton, is a manifold of constant sectional curvature with respect to the semi-symmetric type non-metric connection. Moreover, we prove that if the Riemannian metric of $N^3$ with the semi-symmetric type non-metric connection is a gradient Yamabe soliton, then either $N^{3}$ is a manifold of constant scalar curvature or the gradient Yamabe soliton is trivial with respect to the semi-symmetric type non-metric connection. We also characterize the manifold $N^3$ with a semi-symmetric type non-metric connection whose metrics are Einstein solitons and $m$-quasi Einstein solitons of gradient type, respectively.

math.DG↗
arXiv2024

Characterizations of a spacetime of quasi-constant sectional curvature and $\mathcal{F}(\mathcal{R})$-gravity

The main aim of this article is to investigate a spacetime of quasi-constant sectional curvature. At first, the existence of such a spacetime is established by several examples. We have shown that a spacetime of quasi-constant sectional curvature agrees with the present state of the universe and it represents a Robertson Walker spacetime. Moreover, if the spacetime is Ricci semi-symmetric or Ricci symmetric, then either the spacetime represents a spacetime of constant sectional curvature, or the spacetime represents phantom era. Also, we prove that a Ricci symmetric spacetime of quasi-constant sectional curvature represents a static spacetime and the spacetime under consideration is of Petrov type I, D or O. Finally, we concentrate on a quasi-constant sectional curvature spacetime solution in $\mathcal{F}(\mathcal{R})$-gravity. As a result, various energy conditions are studied and analysed our obtained outcomes in terms of a $\mathcal{F}(\mathcal{R})$-gravity model.

math.DG↗
arXiv2024

Impact of projective curvature tensor in $f\left(R,G\right)$, $f\left(R,T\right)$ and $f\left(R,L_{m}\right)$-gravity

This article concerns with the characterization of a spacetime and modified gravity, such as $f\left(R,G\right)$, $f\left(R,T\right)$ and $f\left(R,L_{m}\right)$-gravity equipped with the projective curvature tensor. We establish that a projectively flat perfect fluid spacetime represents dark energy era. Also, we prove that a projectively flat perfect fluid spacetime is either locally isometric to Minkowski spacetime or a de-Sitter spacetime. Furthermore, it is shown that a perfect fluid spacetime permitting harmonic projective curvature tensor becomes a generalized Robertson-Walker spacetime and is of Petrov type $I$, $D$ or $O$. Lastly, we investigate the effect of projectively flat perfect fluid spacetime solutions in $f\left(R,G\right)$, $f\left(R,T\right)$ and $f\left(R,L_{m}\right)$-gravity, respectively. We also investigate the spacetime as a $f\left(R,G\right)$-gravity solution of and use the flat Friedmann-Robertson-Walker metric to establish a relation among jerk, snap, and deceleration parameters. Numerous energy conditions are studied in terms of Ricci scalar with the model $f\left(R,G\right)=\exp(R)+α\left(6G\right)^β$. For this model, the strong energy condition is violated but the weak, dominant and null energy conditions are fulfilled, which is in excellent accordance with current observational investigations that show the universe is now accelerating.

gr-qc↗
arXiv2024

Perfect fluid spacetimes and $k$-almost yamabe solitons

In this article, we presumed that a perfect fluid is the source of the gravitational field while analyzing the solutions to the Einstein field equations. With this new and creative approach, here we study $k$-almost yamabe solitons and gradient $k$-almost yamabe solitons. First, two examples are constructed to ensure the existence of gradient $k$-almost Yamabe solitons. Then we show that if a perfect fluid spacetime admits a $k$-almost yamabe soliton, then its potential vector field is Killing if and only if the divergence of the potential vector field vanishes. Besides, we prove that if a perfect fluid spacetime permit a $k$-almost yamabe soliton ($g,k,ρ,λ$), then the integral curves of the vector field $ρ$ are geodesics, the spacetime becomes stationary and the isotopic pressure and energy density remain invariant under the velocity vector field $ρ$. Also, we establish that if the potential vector field is pointwise collinear with the velocity vector field and $ρ(a)=0$ where a is a scalar, then either the perfect fluid spacetime represents phantom era, or the potential function $Φ$ is invariant under the velocity vector field $ρ$. Finally, we prove that if a perfect fluid spacetime permits a gradient $k$-almost yamabe soliton ($g,k,DΦ,λ$) and $R, λ, k$ are invariant under $ρ$, then the vorticity of the fluid vanishes.

math.DG↗
arXiv2024

Characterizations of generalized Robertson-Walker spacetimes concerning gradient solitons

In this article, we examine gradient type Ricci solitons and $(m,τ)$-quasi Einstein solitons in generalized Robertson-Walker ($GRW$) spacetimes. Besides, we demonstrate that in this scenario the $GRW$ spacetime presents the Robertson-Walker ($RW$) spacetime and the perfect fluid ($PF$) spacetime presents the phantom era. Consequently, we show that if a $GRW$ spacetime permits a gradient $τ$- Einstein solitons, then it also represents a $PF$ spacetime under certain condition.

math.DG↗
arXiv2024

Conformal vector fields on almost Kenmotsu manifolds

In this paper, first we consider that the conformal vector field $\mathbf{X}$ is identical with the Reeb vector field $ς$ and next, assume that $\mathbf{X}$ is pointwise collinear with %the Reeb vector field $ς$, in both cases it is shown that the manifold $\mathbf{N}^{2m+1}$ becomes a Kenmotsu manifold and $\mathbf{N}^{2m+1}$ is locally a warped product $\mathbf{N}' \times_{f} \mathbf{M}^{2m}$, where $\mathbf{M}^{2m}$ is an almost Kähler manifold, $\mathbf{N}'$ is an open interval with coordinate t, and $f = ce^{t}$ for some positive constant c. Beside these, we prove that if a $(\verb"k",\boldsymbolμ)'$-almost Kenmotsu manifold admits a Killing vector field $\mathbf{X}$, then either it is locally a warped product of an almost Kähler manifold and an open interval or $\mathbf{X}$ is a strict infinitesimal contact transformation. Furthermore, we also investigate $\boldsymbolη$-Ricci-Yamabe soliton with conformal vector fields on $(\verb"k",\boldsymbolμ)'$-almost Kenmotsu manifolds and finally, we construct an example.

math.DG↗
arXiv2024

Almost co-Kähler manifolds and $(m,ρ)$-quasi-Einstein solitons

The present paper aims to investigate $(m,ρ)$-quasi-Einstein metrices on almost co-Kähler manifolds $\mathcal{M}$. It is proven that if a $(κ,μ)$-almost co-Kähler manifold with $κ<0$ is $(m,ρ)$-quasi-Einstein manifold, then $\mathcal{M}$ represents a $N(κ)$-almost co-Kähler manifold and the manifold is locally isomorphic to a solvable non-nilpotent Lie group. Next, we study the three dimensional case and get the above mentioned result along with the manifold $\mathcal{M}^3$ becoming an $η$-Einstein manifold. We also show that there does not exist $(m,ρ)$-quasi-Einstein structure on a compact $(κ,μ)$-almost co-Kähler manifold of dimension greater than three with $κ<0$. Further, we prove that an almost co-Kähler manifold satisfying $η$-Einstein condition with constant coefficients reduces to a $K$-almost co-Kähler manifold, provided $ma_{1} \neq (2n-1)b_{1}$ and $m \neq 1$. We also characterize perfect fluid spacetime whose Lorentzian metric is equipped with $(m, ρ)$-quasi Einstein solitons and acquired that the perfect fluid spacetime has vanishing vorticity, or it represents dark energy era under certain restriction on the potential function. Finally, we construct an example of an almost co-Kähler manifold with $(m,ρ)$-quasi-Einstein solitons.

math.DG↗
arXiv2023

Ricci solitons on singly warped product manifolds and applications

The purpose of this article is to study implications of a Ricci soliton warped product manifold to its base and fiber manifolds. First, it is proved that if a warped product manifold is Ricci soliton then its factors are Ricci soliton. Then we study Ricci soliton on warped product manifolds admitting either a conformal vector field or a concurrent vector field. Finally, we study Ricci soliton on some warped product space-times.

math.DG↗
arXiv2022

Characterizations of Perfect fluid spacetimes obeying $f(\mathcal{R})$-gravity equipped with different gradient solitons

The prime object of this article is to study the perfect fluid spacetimes obeying $f(\mathcal{R})$-gravity, when $η$-Ricci solitons, gradient $η$-Ricci solitons, gradient Einstein Solitons and gradient $m$-quasi Einstein solitons are its metrics. At first, the existence of the $η$-Ricci solitons is proved by a non-trivial example. We establish conditions for which the $η$-Ricci solitons are expanding, steady or shrinking. Besides, in the perfect fluid spacetimes obeying $f(\mathcal{R})$-gravity, when the potential vector field of $η$-Ricci soliton is of gradient type, we acquire a Poisson equation. Moreover, we investigate gradient $η$-Ricci solitons, gradient Einstein Solitons and gradient $m$-quasi Einstein solitons in $f(\mathcal{R})$-gravity, respectively. As a result, we establish some significant theorems about dark matter era.

math.DG↗
arXiv2022

Semi-invariant Conformal submersions with horizontal Reeb vector field

The present paper deals with the characterization of a new submersion named semi-invariant conformal $ζ^{\perp }$-Riemannian submersion from almost contact metric manifolds onto Riemannian manifolds which is the generalization of some known submersions on Riemannian manifolds. We give important and adequate conditions for such submersions to be totally geodesic and harmonic. Also, few examples are examined for such submersions endowed with horizontal Reeb vector field.

math.DG↗