International Journal of Geometric Methods in Modern Physics

We propose a new unified model that describes dark energy and dark matter in the context of $f(R,\varphi )$ gravity using a massive scalar field in five dimensions. The scalar field is considered in the bulk that surrounds the 3-brane in branworld model. We show that for a specific choice of the $f(R,\varphi )$ function, we can describe the Einstein gravitation in 4-dimensional space-time. We obtain a relationship between the speed of the Universe’s expansion and the speed of the bulk’s expansion. We also propose that the dark matter is represented by the scalar field mass and that the dark energy is a kinetic energy of this field. Finally, we show that, according to conditions, one can obtain the percentages of density of dark matter and the density of ordinary matter.

In this paper, we investigate the charged static spherically symmetric models in $f(R)$ theory of gravity. We consider a linear equation of state (EoS) in the background of anisotropic matter configuration. We formulate the modified field equations and implement Durgapal transformation to examine the gravitational nature of compact stellar objects. For this purpose, we choose a specific gravitational potential and electric charge intensity to analytically solve the set of field equations. We generate three special cases of solutions for specific parametric values of $a,b$ appearing in the expression of gravitational potential. The evolution of physical observables, such as energy density, anisotropy parameter, radial and tangential pressures and electric field intensity, are presented in all cases. Via physical analysis, it is observed that the solution of charged compact spheres satisfies acceptability criteria, models are well behaved, and depict stability and consistency in accordance with $f(R)$ gravity for generated models.

This study deals with the spherically symmetric radiating star (with dissipative perfect fluids) with a central vacuum cavity, evolving under the assumption of expansion-free motion. The analytical model of the such dynamics star is discussed in three regimes — diffusion approximation, geodesic motion and self-similarity — and the solutions of dynamical equations are obtained in its complete generality. The structure scalars, which are related to the fundamental properties of fluid distribution, are also discussed which played a very important role in the dynamics of cavity models. It has been shown that energy density is homogeneous but violates the energy condition under quasi-static diffusion approximation.

We have studied thermal fluctuations from neutral as well as charged and rotating charged cylindrical black holes. In this context, we computed modified expressions for several thermodynamic quantities such as entropy, mass and heat capacity for all these types of cylindrical black holes. It is shown that these quantities change their forms under the influence of thermal fluctuations. We also discuss the phase transitions and thermal stability of these gravitating objects.

Within $F(B^2)$ modified Weyl gravity, we consider a model of a spin-$1/2$ electric charge consisting of interior and exterior regions. The interior region is determined by quantum gravitational effects whose approximate description is carried out using Weyl gravity nonminimally coupled to a massless Dirac spinor field. The interior region is embedded in exterior Minkowski spacetime, and the joining surface is a two-dimensional torus. It is shown that mass, electric charge, and spin of the object suggested may be the same as those for a real electron.

In this paper, we study the thermodynamics of short-range central potentials, namely, the Lee–Wick (LW) potential, and the Plasma potential. In the first part of the paper, we obtain the numerical solution for the orbits equation for these potentials. Posteriorly, we introduce the thermodynamics through the microcanonical and canonical ensembles formalism defined on the phase space of the system. We calculate the density of states associated with the LW and the Plasma potentials. From density of states, we obtain the thermodynamical physical quantities like entropy and temperature as functions of the energy. We also use the Boltzmann–Gibbs formalism to obtain the partition functions, the mean energy and the thermal capacity for these short-range potentials.

This study set out to investigate the effect of anisotropy on the $\mathrm{\Lambda }$CDM model in the framework of Brans−Dicke theory. To this end, astrophysical constraints on this model using current available data including type Ia supernovae (SNIa), the Baryon Acoustic Oscillation (BAO), and the Hubble parameter $H(z)$ data were deployed. Here, we present combined results from these probes, deriving constraints on ${\mathrm{\Omega }}_{{\sigma }_0}$ of $\mathrm{\Lambda }$CDM model and its anisotropy energy density in an anisotropic universe. It is found that ${\mathrm{\Omega }}_{{\sigma }_0}$ can be constrained by the $\mathrm{\text{SNIa}}+H(z)+\mathrm{\text{BAO}}$ data, with the best fitting value ${\mathrm{\Omega }}_{{\sigma }_0}=(-6.52\pm 4.21)\times 10^{-4}$ for the Brans–Dicke cosmology. We extend our study to the case of $\mathrm{\Lambda }$CDM model in an anisotropic universe and Brans–Dicke framework and find out that the equation of state parameter (${\omega }_{\mathrm{\Lambda }}$) cannot cross the phantom line and eventually the universe approaches a quintessence era.

In this paper, we obtain the expressions of the ∗-Ricci operator of a three-dimensional almost Kenmotsu manifold $M^3$ and find that the ∗-Ricci tensor is not symmetric for $M^3$. We obtain a necessary and sufficient condition for the ∗-Ricci tensor to be symmetric and proved that if the ∗-Ricci tensor of a non-Kenmotsu almost Kenmotsu 3-$h$-manifold $M^3$ is symmetric, then $M^3$ is locally isometric to a three-dimensional non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure. Further, it is shown that the ∗-Ricci tensor of a non-Kenmotsu almost Kenmotsu 3-manifold $M^3$ is parallel if and only if $M^3$ is ∗-Ricci flat. In addition, $M^3$ satisfying ${\nabla }_{\xi }h=0$ is locally isometric to ${ℍ}^2(-4)\times ℝ$. Finally, we discuss about $\eta$-parallel ∗-Ricci tensor on almost Kenmotsu 3-manifolds.

We consider the sets of Dirac–Maxwell and Rarita–Schwinger–Maxwell equations in $ℝ\times S^3$ spacetime. Using the Hopf coordinates, we show that these equations allow separation of variables and obtain the corresponding analytic and numerical solutions. It is also demonstrated that the current of the Dirac field is related to the Hopf invariant on the $S^3\to S^2$ fibration.

In this paper, we investigate the topology of a wormhole from the viewpoint of the theory of retracting and var-folding. We deduce the equatorial geodesics on the line element of the wormhole and discuss the minimum deformation retract related to this space. Using the Lagrangian equations we find that there is a type of minimum retraction of a wormhole with associated topology. We also extend the result to the $n$-dimensional wormhole and show that the end limit of var-folding is 0-dimensional wormhole and obtain the relation between limit var-folding and limit retraction. We find a new application in geometric topology and astrophysics.

In this paper, we introduce the notion of Koszul–Vinberg–Nijenhuis (KVN) structures on a left-symmetric algebroid as analogues of Poisson–Nijenhuis structures on a Lie algebroid, and show that a KVN-structure gives rise to a hierarchy of Koszul–Vinberg structures. We introduce the notions of $\mathrm{\text{KV}}\mathrm{\Omega }$-structures, pseudo-Hessian–Nijenhuis structures and complementary symmetric $2$-tensors for Koszul–Vinberg structures on left-symmetric algebroids, which are analogues of $\mathrm{\text{P}}\mathrm{\Omega }$-structures, symplectic-Nijenhuis structures and complementary $2$-forms for Poisson structures. We also study the relationships between these various structures.

We define a functional ${𝒥}(h)$ for the space of Hermitian metrics on an arbitrary Higgs bundle over a compact Kähler manifold, as a natural generalization of the mean curvature energy functional of Kobayashi for holomorphic vector bundles, and study some of its basic properties. We show that ${𝒥}(h)$ is bounded from below by a nonnegative constant depending on invariants of the Higgs bundle and the Kähler manifold, and that when achieved, its absolute minima are Hermite–Yang–Mills metrics. We derive a formula relating ${𝒥}(h)$ and another functional ${ℐ}(h)$, closely related to the Yang–Mills–Higgs functional, which can be thought of as an extension of a formula of Kobayashi for holomorphic vector bundles to the Higgs bundles setting. Finally, using 1-parameter families in the space of Hermitian metrics on a Higgs bundle, we compute the first variation of ${𝒥}(h)$, which is expressed as a certain $L^2$-Hermitian inner product. It follows that a Hermitian metric on a Higgs bundle is a critical point of ${𝒥}(h)$ if and only if the corresponding Hitchin–Simpson mean curvature is parallel with respect to the Hitchin–Simpson connection.

Within the Hamiltonian framework, the propositions about a classical physical system are described in the Borel $\sigma$-algebra of a symplectic manifold (the phase space) where logical connectives are the standard set operations. Considering the geometric formulation of quantum mechanics we give a description of quantum propositions in terms of fuzzy events in a complex projective space equipped with Kähler structure (the quantum phase space) obtaining a quantized version of a fuzzy logic by deformation of the product $t$-norm.

In this paper, initially we solve the Einstein field equations (EFEs) for a static spherically (SS) symmetric perfect fluid space-times in the $f(T,B)$ gravity with the aid of some algebraic techniques. The extracted solutions are then utilized in order to get conformal vector fields (CVFs). It is important to mention that the adopted techniques enable us to obtain various classes of space-times with viable $f(T,B)$ gravity models which already exist in the literature. Excluding all such classes, we find that there exist three cases for which the space-times admit proper CVFs, whereas in rest of the cases, CVFs become KVFs. We have also highlighted some physical implications of our obtained results.

In this work, we have investigated the cosmological bouncing solution in LRS Bianchi-I space-time in framework of $f(R,T)$ gravity. Our study in this paper is based on the modeling of matter bounce scenario in which the universe starts with a matter-dominated contraction phase and transitions into an ekpyrotic phase. Mathematical simulations have been done in the modified general theory of relativity in the form of $f(R,T)$ theory proposed by Harko et al. [f(R, T) gravity, Phys. Rev. D 84 (2011) 024020], whose functional form is as $f(R,T)=R+2\mu T,$ where $R$ is Ricci scalar, $T$ is trace of energy–momentum tensor and $\mu$ is constant. Taking the non-vanishing scale factor in LRS Bianchi-I space-time, the geometrical parameters such as Hubble parameter and deceleration parameter have been derived and their subsequent use in the expression of pressure, density and EoS parameter $\omega$ confirms qualitatively the initial conditions of the universe at the bounce. With the non-vanishing nature of scale factor, initial universe in finite means ruled out the initial singularity problem. The analysis of violation of energy conditions near the bouncing region and stability of the model shows that the matter bounce approach is highly unstable at the bounce but the rapid decay of perturbations away from the bounce supports the stability of the model.

An interest problem arises to determine the surfaces in the Euclidean three space, which admit at least one nontrivial isometry that preserves the principal curvatures. This leads to a class of surface known as a Bonnet surface. The intention of this study is to examine a Bonnet ruled surface in dual space and to calculate the dual geodesic trihedron of the dual curve associated with the Bonnet ruled surface and derivative equations of this trihedron by the dual geodesic curvature. Also, we find that the dual curvature, the dual torsion for the dual curves associated with the Bonnet ruled surface which are different from any dual curves. Moreover, some examples are obtained about the Bonnet ruled surface.