International Journal of Geometric Methods in Modern Physics

This paper investigates the new interior solution of stellar compact spheres in the framework of metric $f(R)$ gravity. In this connection, we derived the Einstein field equations for static anisotropic spherically symmetric spacetime in the mechanism of Karmakar condition. The obtained results of the field equations have been studied with well-known Starobinskian model $f(R)=R+\zeta R^2$ by using three different compact stars like RXJ1856-37, $\mathrm{\text{Her}}X$-$1$, $\mathrm{\text{Vela}}X$-$12$. Moreover, the constants involved in the solution of metric potentials have been determined through smooth matching conditions between the interior geometry and exterior spacetime. Thereafter, the physical significance of the obtained results is examined in the form of fluid variables, equation of state (EoS) parameters, energy conditions, anisotropic stress and stability analysis by using the graphical plot. The approximated values of the constants and the mass-radius relation have been calculated through different stellar star objects ($\mathrm{\text{RXJ}}1856$-$37$ ($r_b=6\mathrm{\text{Km}}$), $\mathrm{\text{Her}}X$-$1$ ($r_b=6.7\mathrm{\text{Km}}$) and $\mathrm{\text{Vela}}X$-$12$ ($r_b=9.99\mathrm{\text{Km}}$)) shown in Table 1. Finally, we have concluded that our considered compact stellar objects with particular choice of $f(R)$ model in the mechanism of Karmakar condition satisfies all the necessary bounds for potentially stable formation of the stars.

In this paper, we considered the study of Friedmann–Robertson–Walker (FRW) model in the framework of $f(Q,T)$ gravity, an extension of symmetric teleparallel gravity, recently defined by Xu et al. [$f(Q,T)$ gravity, Eur. Phys. J. C 79 (2019) 708]. The nonlinear model $f(Q,T)=-\alpha Q-\beta T^2$, where $\alpha >0$ and $\beta >0$ are constants, is taken into account. The equation of state of perfect fluid is assumed and 31 points of Hubble data are used to constrain the value of model parameter. To explore the evolution of the universe, the numerical solutions of cosmological implications, such as Hubble parameter, deceleration parameter, apparent magnitude and luminosity distance, are determined and the energy conditions are examined. The theoretical results of Hubble parameter are compared with $\mathrm{\Lambda }$CDM model. Further, 57 Supernova data (42 from Supernova cosmology project and 15 from Calán/Tolono supernova survey) are also used to have consistent results of apparent magnitude and luminosity distance.

In this paper, we present the geometry of three-dimensional Heisenberg group $({ℍ}_3,g)$ and its geodesics curves. After studying the Killing magnetic curves and some geodesic Killing magnetic curves, we give the explicit formulas for such curves.

In this paper, we study applications of fractional Heisenberg antiferromagnetic model associated with the magnetic $n$-lines in the normal direction. Evolution equations of magnetic $n$-lines due to inextensible Heisenberg antiferromagnetic flow are computed to construct the soliton surface associated with the inextensible Heisenberg antiferromagnetic flow. Then, their explicit solutions are examined in terms of magnetic and geometric quantities via the conformable fractional derivative method. Finally, we obtain new numerical fractional solutions for nonlinear fractional Schrödinger system with the inextensible Heisenberg antiferromagnetic flow model.

In this paper, we present a study of a fifth-order nonlinear partial differential equation, which was recently introduced in the literature. This equation can be used as a model for bidirectional water waves propagating in a shallow medium. Using elements of an optimal system of one-dimensional subalgebras, we perform similarity reductions culminating in analytic solutions. Rational, hyperbolic, power series and elliptic solutions are obtained. Furthermore, by using the multiple exponential function method we obtain one and two soliton solutions. Finally, local and low-order conserved quantities are derived by enlisting the multiplier approach.

This paper is concerned with the study of $LCS$-manifolds and Ricci solitons. It is shown that in a $CS$-spacetime, the fluid has vanishing vorticity and vanishing shear. It is found that in an $LCS$-manifold, $\mathrm{\text{grad}}\alpha$ is an irrotational vector field, where $\alpha$ is a non-zero smooth scalar function. It is proved that in a $CS$-spacetime with generator vector field $\xi$ obeying Einstein equation, $T(\xi ,\xi )>0$ or $<0$ according to $\rho >{\alpha }^2$ or $\rho <{\alpha }^2$, where $\rho$ is a scalar function and $T$ is the energy momentum tensor. Also, it is shown that if $X$ is a non-null spacelike (respectively, timelike) vector field on a $CS$-spacetime with scalar curvature $r$ and cosmological constant $\mathrm{\Lambda }$, then $T(X,X)>0$ if and only if $r>2\mathrm{\Lambda }$ (respectively, $r<2\mathrm{\Lambda }$), and $T(X,X)<0$ if and only if $r<2\mathrm{\Lambda }$ (respectively, $r>2\mathrm{\Lambda }$), and further $T(X,X)=0$ if and only if $r=2\mathrm{\Lambda }$. The nature of the scalar curvature of an $LCS$-manifold admitting Yamabe soliton is obtained. Also, it is proved that an $LCS$-manifold admitting $\eta$-Ricci soliton is $\eta$-Einstein and its scalar curvature is constant if and only if $\alpha$ is constant. Further, it is shown that if $\beta$ is a scalar function with $\beta =-(\xi \rho )$ and $2\alpha \rho -\beta$ vanishes, then the gradients of $\alpha$, $\beta$, $\rho$ are co-directional with the generator $\xi$. In a perfect fluid $CS$-spacetime admitting $\eta$-Ricci soliton, it is proved that the pressure density $p$ and energy density $\sigma$ are constants, and if it agrees Einstein field equation, then we obtain a necessary and sufficient condition for the scalar curvature to be constant. If such a spacetime possesses Ricci collineation, then it must admit an almost $\eta$-Yamabe soliton and the converse holds when the Ricci operator is of constant norm. Also, in a perfect fluid $CS$-spacetime satisfying Einstein equation, it is shown that if Ricci collineation is admitted with respect to the generator $\xi$, then the matter content cannot be perfect fluid, and further $\kappa (p-\sigma )\ne 2\mathrm{\Lambda }$ with gravitational constant $\kappa$ implies that $\xi$ is a Killing vector field. Finally, in an $LCS$-manifold, it is proved that if the $CL$-curvature tensor is conservative, then scalar potential and the generator vector field are co-directional, and if the manifold possesses pseudosymmetry due to the $CL$-curvature tensor, then it is an $\eta$-Einstein manifold.

Recently, a novel equation of state (EoS) parameter for dark energy has been introduced which deals with a special mathematical function, known as the Lambert $W$ function. In this paper, we study the effect on the growth of perturbations for the Lambert $W$ dark energy model. We perform the analysis for two different approaches. In the first case, we consider the universe to be filled with two different fluid components, namely, the baryonic matter component and the Lambert $W$ dark energy component, while in the second case, we consider that there is a single fluid component in the universe whose equation of state parameter is described by the Lambert $W$ function. We then compare the growth rates of Lambert $W$ model with that for a standard $\mathrm{\Lambda }$CDM model as well as the CPL model. Our results indicate that the presence of Lambert $W$ dynamical dark energy sector changes the growth rate and affects the matter fluctuations in the universe to a great extent.

In this paper, we find the necessary and sufficient conditions for a multiply warped product to be a gradient Ricci-harmonic soliton. We also investigate the necessary and sufficient conditions for a multiply generalized Robertson–Walker space-time and a generalized Reissner–Nordström space-time to be a gradient Ricci-harmonic soliton.

The hybrid expansion law (HEL) for average scale factor that yields power-law and exponential-law cosmologies is considered in spatially homogenous and anisotropic Bianchi type-I model in the context of Brans–Dicke (BD) Theory of gravitation. The solutions of the field equations have been calculated by assuming the power-law expression between the average scale factor $(a(t))$ and scalar field ($\varphi$). We studied both interacting and non-interacting forms of dark energy and dark matter and obtained respective solutions. The energy density ${\mathrm{\Omega }}^m$ decreases with time while energy densities ${\mathrm{\Omega }}^{\mathrm{\text{de}}}$ increases with time. In both the cases, the physical acceptability and stability of the models are also studied. The coincidence problem in $\mathrm{\Lambda }$CDM model can be ruled out with proper choice of coupling between dark matter (DM) and dark energy (DE). We also discussed the physical behaviors of the derived models with the current observations applied to late-time acceleration and beginning of the universe. In this model, it is observed that our HEL Bianchi type I universe is highly anisotropic in the beginning of universe and becomes isotropic and overlaps with flat $\mathrm{\Lambda }$CDM model at late times.

In the contest of open quantum systems, we study a class of Kraus operators whose definition relies on the defining representation of the symmetric groups. We analyze the induced orbits as well as the limit set and the degenerate cases.

In this paper, evolution of a Friedmann–Robertson–Walker universe is studied in a higher derivative theory of gravity. The relativistic solutions admitting hybrid expansion law of the universe are explored here. Hybrid expansion law is a general form of scale factor from which one can recover both the power-law expansion and exponential expansion as a special case. The hybrid expansion law is interesting as it addresses the early deceleration phase and presents accelerating phase satisfactorily. It is found that an inflationary scenario with hybrid expansion law is permitted in the $R^2$ gravity fairly well. We consider universe filled with cosmic fluid that describes by an equation of state (EoS) parameter which varies with time. Consequently, we analyze the time variation of energy density parameter, cosmic pressure, equation of state parameter, deceleration parameter and jerk parameter in the cosmological model. The constraints of the model parameters imposed by the cosmological observational data set are determined. The present value of the deceleration parameter $(q)$, EoS parameter and the epoch at which the transition of decelerated phase to accelerated phase are estimated. In the higher derivative theory, we obtain some new and interesting cosmological solutions relevant for building cosmological models.

In this paper, traversable wormholes have been studied in $f(R)=R+\alpha R^m+\beta R^{-n}$ gravity, where $\alpha$, $\beta$, $m>0$ and $n>0$ are constant. A simplest form of shape function and a logarithmic form of redshift function is considered to construct wormhole solutions. The range of parameters providing the wormhole solutions free from the matter violating the energy conditions is explored. Further, the effect of charge is analyzed on wormhole solutions.

This paper aims at additional symmetries of the unextended and extended, commutative and noncommutative dispersionless Gelfand–Dickey (dGD) hierarchies. Being similar to the Lax formalism of the Gelfand–Dickey (GD) hierarchy, we construct the function ${ℒ}$ and Orlov–Schulman function ${ℳ}$ of the hierarchies. Meanwhile, the additional symmetry will be studied with the infinite flows of $v_i$ and $\tau$ function of the dGD hierarchy and one can find that only a part of additional flows can survive under the GD constraints with the corresponding string equation. Furthermore, we pay attention to the additional symmetries of the dispersionless extended Gelfand–Dickey (dEGD) hierarchy which has a quantum torus algebraic structure and show the flows in detail. The additional symmetry of dispersionless noncommutative Gelfand–Dickey (dNCGD) hierarchy and dispersionless extended noncommutative Gelfand–Dickey (dENCGD) hierarchy are studied.

In this paper, we construct metallic Kähler and nearly metallic Kähler structures on Riemannian manifolds. For such manifolds with these structures, we study curvature properties. Also, we describe linear connections on the manifold which preserve the associated fundamental $2$-form and satisfy some additional conditions and present some results concerning them.

In this paper, an algebraic classification of the Roter type spacetimes is given. It follows that the Roter type curvature condition is essentially equivalent with the pseudosymmetry condition on 4-dimensional Lorentzian manifolds.

In the work reported here we have considered Barrow holographic dark energy (BHDE) proposed in E. N. Saridakis, Barrow holographic dark energy, Phys. Rev. D 102 (2020) 123525 as a special case of more generalized version of Nojiri–Odintsov holographic dark energy (NOHDE) proposed in S. I. Nojiri and S. D. Odintsov, Unifying phantom inflation with late-time acceleration: Scalar phantom–non-phantom transition model and generalized holographic dark energy, Gen. Relativ. Gravit. 38 (2006) 1285–1304 and also reconstructed $f(R)$ gravity with BHDE as the form of background evolution. It has been observed that in the case of BHDE reconstructed $f(R)$ gravity the equation of state can have a transition from quintessence to phantom. In a particular case of reconstruction, possibility of Little Rip singularity has been observed. Finally, the generalized second law of thermodynamics has been found to be valid under this reconstruction scheme.

We explore the cosmological models of the late-time universe based on the holographic principle, taking into account the properties of the viscosity of the dark fluid. We use the mathematical formalism of generalized infrared cutoff holographic dark energy, as presented by Nojiri and Odintsov [Covariant generalized holographic dark energy and accelerating universe, Eur. Phys. J. C 77 (2017) 528]. We consider the Little Rip, the Pseudo Rip, and a bounce exponential model, with two interacting fluids, namely dark energy and dark matter in a spatially-flat Friedmann–Robertson–Walker universe. Within these models, analytical expressions are obtained for infrared cutoffs in terms of the particle horizons. The law of conservation of energy is presented, from a holographic point of view.