International Journal of Geometric Methods in Modern Physics

The paper presents Barrow holographic dark energy (infrared cut-off is the Hubble horizon) suggested by Barrow [The area of a rough black hole, Phys. Lett. B 808 (2020) 135643] recently in an anisotropic Bianchi type-I Universe within the framework of $f(Q)$ symmetric teleparallel gravity, where the non-metricity scalar $Q$ is responsible for the gravitational interaction. We consider two cases: Interacting and non-interacting models of pressureless dark matter and Barrow holographic dark energy by solving $f(Q)$ symmetric teleparallel field equations. To find the exact solutions of the field equations, we assume that the time-redshift relation follows a Lambert function distribution as $t(z)=\frac{m{t}_0}{l}g(z)$, where $g(z)=\mathrm{\text{Lambert}}W\left[\right.\frac{l}{m}{e}^{\frac{l-ln(1+z)}{m}}\left]\right.$, $m$ and $l$ are non-negative constants and $t_0$ represents the age of the Universe. Moreover, we discuss several cosmological parameters such as energy density, equation of state (EoS) and skewness parameters, squared sound speed, and $({\omega }_B-{\omega }_B^{{}^{\prime }})$ plane. Finally, we found the values of the deceleration parameter (DP) for the Lambert function distribution as $q_{(z=0)}=-0.45$ and $q_{(z=-1)}=-1$ which are consistent with recent observational data, i.e. DP evolves with cosmic time from initial deceleration to late-time acceleration.

In this study, we analyze mixed quasi-Einstein spacetimes endowed with Gray’s decomposition, as well as generalized Robertson–Walker spacetimes. For mixed quasi-Einstein spacetimes, the shape of the Ricci tensor in all ${𝒪}(n)$-invariant subspaces can be identified using Gray’s decomposition of the gradient of the Ricci tensor. In case one, such a spacetime is found to be static; in three cases, the Ricci tensor is found to be in the form of a  perfect fluid; and in the other three situations, the spacetime becomes a quasi-Einstein spacetime under certain restrictions on the associated vector fields. Finally, it is established that a mixed quasi-Einstein generalized  Robertson–Walker spacetime is a perfect fluid spacetime.

This study investigates a bulk viscous fluid anisotropic cosmological model with $f(Q)$ gravity. Here, $Q$ stands for the nonmetricity factor that drives gravitational interaction. We reconstructed the associated parameters with Tsallis holographic dark energy (THDE). We have solved the modified Einstein’s field equations by considering the bulk viscosity factor $\xi ={\xi }_0+{\xi }_{1}H+{\xi }_2(Ḣ+H^2)$. Under the viscous and nonviscous THDE frameworks, we have obtained the expressions of $f(Q)$ using the power-law form of expansion. We have investigated the nature of various energy conditions for the stability analysis. The positive behavior of DEC and WEC indicates the model’s validation; on the other hand, SEC is violating, indicating the universe’s accelerated expansion. We have also investigated the reconstructed EoS parameter ${\omega }_{\mathrm{\text{rec}},T}$ for bulk viscosity and obtained the one that lies in both quintessence and phantom regions. We also discussed the correspondence of the tachyon scalar field with THDE energy density in $f(Q)$ gravity. This correspondence permits the reconstruction of potentials and dynamics for scalar field models describing accelerated expansion.

In this paper, we have examined the viability of the Bianchi type-V universe in Brans–Dicke (BD) theory of gravitation. We have discussed the interacting and non-interacting scenarios between dark matter (DM) and dark energy (DE) of the derived universe within the framework of BD theory. CCA technique has been applied to constrain the model parameters using 46 values of observational Hubble data (OHD), Pantheon data (the latest compilation of SNIa with 40 binned in the redshift range $0.014\le z\le 1.62)$ and their combined datasets. We establish an exact solution of the field equations to derive the dynamics of the derived universe and the obtained results are found to agree with the observations. We also noted a distinctive change in the sign of the deceleration parameter from positive to negative, as well as the presence of a transition red-shift exists. Using various observational data points, the evolution trajectories for $(r-s)$ diagnostic planes are shown to understand the geometrical behavior of the Bianchi-V model. Some physical properties of the universe are also discussed. It is also worth noting that the conclusions of the cosmological parameter are consistent with modern observational data.

We examine the bundle structure of the field of nowhere vanishing null vector fields on a (time-oriented) Lorentzian manifold. Sections of what we refer to as the null tangent are by definition nowhere vanishing null vector fields. It is shown that the set of nowhere vanishing null vector fields comes equipped with a para-associative ternary partial product. Moreover, the null tangent bundle is an example of a non-polynomial graded bundle.

This paper investigates the (2+1)-dimensional variable coefficients Kundu–Mukherjee–Nasker equation which obtained by some transformations. The infinitesimal generators of the equation are obtained by the Lie group method. Next, the representative elements of the one-dimensional subalgebras optimal system are determined with adjoint representation. Based on the optimal system, (1+1)-dimensional partial differential equations can be obtained by similarity reductions. With the help of $(G^{\prime }/G)$-expansion method, several exact solutions including hyperbolic function solutions, trigonometric function solutions and rational function solutions of variable coefficients Kundu–Mukherjee–Nasker equation are obtained. Finally, the conservation laws of the variable coefficients Kundu–Mukherjee–Nasker equation can be obtained by using the conservation theorem proposed by Ibragimov.

The main emphasis of this paper is to find the viable solutions of Einstein Maxwell field equations of compact star in context of modified $f(R)$ theory of gravity. Two different models of modified $f(R)$ gravity are considered. In particular, we choose isotropic matter distribution and Bardeen’s model for compact star to find the boundary conditions as an exterior space-time geometry. We use the conformal Killing geometry to compute the metric potentials. We discuss the behavior of energy density and pressure distribution for both models. Moreover, we analyze different physical properties such as behavior of energy density and pressure, equilibrium conditions, equation of state parameters, causality conditions and adiabatic index. It is noticed that both $f(R)$ gravity models are suitable and provide viable results with Bardeen geometry.

In this paper, we study the Kählerian nature of Taub-NUT and Kerr spaces which are gravitational instanton and black hole solutions in general relativity. We show that Euclidean Taub-NUT metric is hyper-Kähler with respect to the usual almost complex structures by employing an alternative explicit coframe, and Euclidean Kerr metric is globally conformally Kähler. We also show that conformally scaled Euclidean Kerr space admits a Kähler structure by applying a conformal scaling factor stemming from the Lee-form of the original metric or alternatively a factor coming from self-dual part of the Weyl tensor $(W^{+})$.

Relativistic and nonrelativistic quantum mechanics formulated in a noncommutative space-space have recently become the object of renewed interest. In the context of extended relativistic quantum mechanics (ERQM) symmetries with arbitrary spin-orbit coupling quantum number $k$, we approximate to solve the deformed Dirac equation (DDE) for a new suggested new generalized Schiöberg and Manning–Rosen potentials within the generalized (Coulomb and Yukawa)-like tensor interactions (NGSM-GLTs). In the framework of the spin and pseudospin (p-spin) symmetry, we obtain the global new energy eigenvalue which equals the energy eigenvalue in usual relativistic quantum mechanics (RQM) as the main part plus three corrected parts produced from the effect of the spin-orbit interaction, the new modified Zeeman, and the rotational Fermi term, by using the parametric of the well-known Bopp’s shift method and standard perturbation theory using Greene–Aldrich approximation to handle $1/r$, $1/r^2$ and other terms in the effective potential. The new values that we got appeared sensitive to the quantum numbers ($j,k,l/\tilde{l},s/\tilde{s},m/\tilde{m}$), the mixed potential depths ($D_0$,$D_1$,$\sigma$,$\alpha$,$A$), the range of the potential $\delta$, and noncommutativity parameters ($\mathrm{\Theta }$,$\tau$,$\chi$). We recovered several potentials, including the improved Schiöberg and Manning–Rosen potentials within the improved Yukawa-like tensor interaction, the new Schiöberg and Manning–Rosen potentials within the improved Coulomb-like tensor interaction, the new Schiöberg potential within the improved Yukawa-like tensor interaction, the new Manning–Rosen potential within the improved Yukawa-like tensor interaction, and the new Schiöberg and Manning–Rosen potentials potential problems in the context of nonrelativistic extended quantum mechanics symmetries.

We consider an exponentially expanding, flat, Friedmann–Lemaître–Robertson–Walker (FLRW) Universe filled with a free Schroedinger field. The probability fluid of the latter is used to mimic the cosmological fluid (baryonic plus dark matter), thus providing the matter density and pressure terms in the corresponding Friedmann–Lemaître equations. We first obtain the eigenfunctions of the Laplacian operator on flat FLRW space. A quantum operator qualifying as a cosmological constant is defined to act on the Schroedinger field. We then compute the matrix representing the cosmological constant in the basis of Laplacian eigenfunctions. For an estimate of the orders of magnitude involved it suffices to determine the expectation values of this operator. The expectation value that best fits the experimentally measured value of the cosmological constant allows us to identify the quantum state of the Schroedinger field that best represents the matter contents (baryonic and dark) of the current Universe. Finally, the operator inverse (modulo dimensional factors) to the one representing the cosmological constant provides a measure of the gravitational Boltzmann entropy of the Universe. We compute its matrix in the basis of Laplacian eigenfunctions and verify that the expectation values of this entropy operator comply with the upper bound set by the holographic principle.

The parameter curves $(c_1)$ and $(c_2)$ intersecting under the angle $𝜃$ on a timelike surface and any curve $(c)$ passing through the same point with these curves have six different cases according to their causal characters. In this study, the Darboux trihedrons of these three curves and the Darboux instantaneous rotation vectors of these trihedrons are separately calculated for each of these six cases, and the relationships between these vectors are given. In addition, some theorems and results have been obtained on this surface. Besides, the special cases where the parameter curves $(c_1)$ and $(c_2)$ intersect perpendicularly are examined and these cases are compared with previous studies.

The treatment of (1+3)-covariant perturbation in a multifluid cosmology with the consideration of $f(G)$ gravity, $G$ being the Gauss–Bonnet term, is done in this paper. We define a set of covariant and gauge-invariant variables to describe density, velocity and entropy perturbations for both the total matter and component fluids. We then use different techniques such as scalar decomposition, harmonic decomposition, quasi-static approximation together with the redshift transformation to get simplified perturbation equations for analysis. We then discuss a number of interesting applications like the case where the universe is filled with a mixture of radiation and Gauss–Bonnet fluids as well as dust with Gauss–Bonnet fluids for both short- and long-wavelength limits. Considering polynomial $f(G)$ model, we get numerical solutions of energy density perturbations and show that they decay with increase in redshift. This feature shows that under $f(G)$ gravity, specifically under the considered $f(G)$ model, one expects that the formation of the structure in the late Universe is enhanced.

This study is elaborated to reconstruct some cosmological models in the gravitational framework of an extended teleparallel gravity based on the higher-order derivative terms of torsion scalar. To achieve this goal, we consider spatially flat FRW model with perfect fluid as background matter distribution. In this framework, we find some interesting exact solutions by using the hybrid and intermediate forms of scale factor via a well-known reconstruction scheme. For exploring physical significance of the reconstructed solutions, we analyze the behavior of dark energy EoS parameter, the validity of energy condition bounds as well as the generalized second law of thermodynamics graphically. For investigating the stability of reconstructed solutions, we check the behavior of speed of sound $V_s^2$ in each case. It is seen that in de Sitter case, the reconstruction of full Lagrangian function is not possible as it reduces to the case of simple $f(T)$ gravity which has already been discussed in literature. While for power law form of scale factor, it is found that reconstruction is possible and most of the obtained solutions favor the quintessence era of cosmic evolution. Further, it is concluded that both the energy conditions and generalized second thermodynamical law hold for the reconstructed solutions and these solutions exhibit stable behavior (as $V_s^2\ge 0$). Lastly, in case of intermediate form of scale factor, the resulting differential equations are very complicated and hence the analytical solution is possible only for the simplest case which, further, indicates cosmologically promising behavior.

This paper is concerned with the Jacobi stability of the Shimizu–Morioka model by using the KCC-theory. First, by associating the nonlinear connection and Berwald connection, five geometrical invariants of the dynamical model are obtained. Furthermore, the Jacobi stability of the Shimizu–Morioka model at equilibrium is studied in terms of the eigenvalues of the deviation curvature tensor. It shows that the three equilibria are always Jacobi unstable. Finally, the dynamical behavior of the components of the deviation vector is discussed, which geometrically characterizes the chaotic behavior of studied model near the origin. It proved the onset of chaos in the Shimizu–Morioka model.

In this paper, we aim to derive new soliton solutions of (1+1)- and (2+1)-dimensional generalized Sasa–Satsuma equations via the new Kudryashov method. In optical fiber transmission systems, the Sasa–Satsuma equation describes the effects of third-order dispersion, self-steepening and stimulated Raman scattering in the propagation of ultrafast pulses. The considered equations are encountered in various physical applications such as ultra-short and femto-second pulse propagation in optical fibers and dynamics of deep water waves. So, investigation of the novel solutions of the equations is one of the important topics. We have successfully extracted some soliton solutions for the considered equation. The various graphs of the obtained solutions have been depicted in the figures by selecting appropriate parameters. The singular and bright soliton solutions have been revealed in the figures. All acquired solutions have been confirmed to satisfy the considered equations. The results show that the approach may be used to find exact solutions to various nonlinear evolution equations. The new solutions and the paper results may enrich the understanding of the wave propagation in the optical fibers and may shed light on new studies.

The pent–heptagonal carbon nanosheets (CNSs) admit significant industrial applications. We shall compute the distance-based Hosoya polynomial of CNSs including $V{C}_5{C}_7$ and $H{C}_5{C}_7$. The first-, second- and third-order derivatives of Hosoya polynomial, respectively, lead to the Wiener, hyper-Wiener and Tratch–Stankevitch–Zefirov (TSZ) indices which are the distance-based topological indices. These indices have many physical applications and can be correlated to physio-chemical properties of chemical structures. Results exhibit that despite the fact that both sheets have the same chemical formula, they have different Hosoya polynomials and related distance-based topological indices, resulting in distinct quantitative structure–property relation (QSPR) and quantitative structure–activity relation (QSAR) due to configuration differences.

In this study, the Kaniadakis holographic dark energy (KHDE) model is investigated within the framework of a flat FLRW universe by using an infrared cut-off as Hubble horizon. For the different values of the Kaniadakis parameter $\kappa$ and $C^2$, we perform an analysis on the deceleration, the equation of state and the energy density parameters. The current phase shift of the Cosmos may be rationalized by the change in sign of the deceleration parameter in the KHDE model, which went from $+ve$ to $-ve$, signifying a move from a phase of deceleration to one of acceleration. In order to get an understanding of the geometrical behavior shown by the KHDE model, we compute and show the statefinder evolutionary trajectory for the pairs $(r,s)$ and $(q,r)$. The evolutionary trajectory of ${\omega }_D-{\omega }_D^{\prime }$ reveals that the ${\omega }_D$ crosses the phantom divide line ${\omega }_D=-1$ at a late period in the evolutionary process. It has also been determined, by examination of the KHDE models stability analysis, that the model has behaved in an unstable manner during the whole course of the history of the universe. In addition to this, we optimized the model parameters by making use of the distance modulus measurement that was taken from the most current Union 2.1 dataset of supernovae.

The paper aims to investigate curvature inheritance (CI) symmetry in M-projectively flat spacetimes. It is shown that the CI 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 $\xi$, when spacetime admits the conditions of both CI symmetry and conformal motion or motion along the vector field $\xi$. Also, we have derived some results for M-projectively flat spacetime with perfect fluid following the Einstein field equations (EFEs) with a cosmological term and admitting the CI symmetry along the vector field $\xi$. We have shown that an M-projectively flat perfect fluid spacetime obeying the EFEs with a cosmological term and admitting the CI symmetry along a vector field $\xi$ 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.