International Journal of Geometric Methods in Modern Physics

In this paper, we investigate the dynamical behavior of the planar polynomial system which describes the evolution of the isotropic star. It is shown that the system has no global $C^{\infty }$ and no global analytic first integrals. It has an invariant algebraic curve with algebraic multiplicity $1$ and an exponential factor that comes from the multiplicity of the infinite invariant straight line. It is proved that the system can be changed into the Liénard system. By using a dominant balance analysis, we prove the system has a general solution which eventuates to a finite-time singularity. Finally, we prove the trajectories of the vector field associated with the planar system of the isotropic star do not create a trajectory manifold which means there is no pseudo-Riemannian metric $g$ in the sense of trajectory metric such that the trajectories of the isotropic star system be geodesic.

The geometric intrinsic approach to Hojman symmetry is developed and use is made of the theory of the Jacobi last multipliers to find the corresponding conserved quantity for non divergence-free vector fields. The particular cases of autonomous Lagrangian and Hamiltonian systems are studied as well as the generalization of these results to normalizer vector fields of the dynamics. The nonautonomous cases, where normalizer vector fields play a relevant role, are also developed.

In this paper, we analyze a few physical characteristics of gravastar with cylindrical geometry in $f(R,T)$ theory, where $R$ is the Ricci scalar and $T$ is the trace of energy-momentum tensor. The gravastar is generally considered to be a substitute of a black hole with three different regions. In this work, we examine the formulation of gravastar-like cylindrical structures in $f(R,T)$ theory. By using Darmois and Israel matching conditions, we formulate a mass function of a thin shell. We calculate the different physical characteristics of gravastar, in particular, entropy within the thin shell, proper length of the intermediate thin shell as well as energy of the shell.

In this paper, we have studied the dynamical aspects of some cosmological models in an isotropic space-time within the framework of an extended gravity theory. Two accelerating cosmological models are presented with quasi-de Sitter (QDS) and linear combination of exponential (LCE) scale factors. The geometrical testing mechanism has been performed and analyzed. While the results of the QDS model show a departure from the concordant $\mathrm{\Lambda }$CDM model, the resulting behavior of the LCE model seems to be compatible with the $\mathrm{\Lambda }$CDM model only at the present epoch and is incompatible in the near past.

The purpose of this paper is to study anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds such that characteristic vector field is vertical or horizontal vector field. We first show that any anti-invariant Riemannian submersions from Sasakian manifold is not a Riemannian submersion with totally umbilical fiber. Then we introduce anti-invariant Riemannian submersions from Sasakian manifolds with totally contact umbilical fibers. We investigate the totally contact geodesicity of fibers of such submersions. Moreover, under this condition, we investigate Ricci curvature of anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds.

The present work is focused on the study of traversable wormholes, proposed by Morris and Thorne [Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity, Am. J. Phys. 56 (1988) 395], using the background of modified gravity. It is performed by using the models: I. $f=R+2\beta T$, II. $f=R+\alpha R^n$ and III. $f=R+\alpha R^n+2\beta T$, where $\alpha$, $\beta$ and $n$ are constants. The Model I belongs to the theory of $f(R,T)$ gravity, Model II belongs to the theory of $f(R)$ gravity and Model III is a combination of Models I and II. These functions have been taken into account for the exploration of wormhole solutions. The shape function, a wormhole metric function, is newly defined which satisfies the flare out condition. Further, the stability condition and energy conditions, namely null, weak and dominant energy conditions, have been examined with respect to each model.

Electron microscopy images of decagonal quasicrystals obtained recently have been shown to be related to cluster coverings with a Hexagon–Bow–Tie decagon as single structural unit. Most decagonal phases show more complex structural orderings than models based on deterministic tilings like the Penrose tiling. We analyze different types of decagonal random tilings and their coverings by a Hexagon–Bow–Tie decagon.

In this paper, we discuss about a set of geometric and physical properties of hyper-generalized quasi-Einstein spacetime. In the beginning, we discuss about pseudosymmetry over a hyper-generalized quasi-Einstein spacetime. Here, we discuss about $W_2$-Ricci pseudosymmetry, $Z$-Ricci pseudosymmetry, Ricci pseudosymmetry and projective Ricci pseudosymmetry over a hyper-generalized quasi-Einstein spacetime. Later on, we take over Ricci symmetric hyper-generalized quasi-Einstein spacetime and derive a set of important geometric and physical theorems over it. Moving further we consider some physical applications of the hyper-generalized quasi-Einstein spacetime. Last, we prove the existence of a hyper-generalized quasi-Einstein spacetime by constructing a non-trivial example.

In this paper, we deal with the Dirac equation and angular momentum, which have an important place in physics in terms of elliptic biquaternions. Thanks to the elliptic biquaternionic representation of angular momentum, we have expressed some useful mathematical and physical results. We obtain the solutions of the Dirac equation with the help of Dirac matrices with elliptic biquaternionic structure. Then, we have expressed the elliptic biquaternionic rotational Dirac equation. This equation could be interpreted as the combination of rotational energy and angular momentum of the particle and anti-particle. Therefore, we also discuss the elliptic biquaternionic form of rotational energy–momentum and of the relativistic mass. Further, we express the spinor wave function by elliptic biquaternions. Accordingly, we also show elliptic biquaternionic rotational Dirac energy–momentum solutions through this function.

In this paper, we considered an axion F(R) gravity model and described, with the help of holographic principle, the cosmological models of viscous dark fluid coupled with axion matter in a spatially flat Friedmann–Robertson–Walker (FRW) universe. This description based on generalized infrared-cutoff holographic dark energy was proposed by Nojiri and Odintsov. We explored the Little Rip, the Pseudo Rip, and the power-law bounce cosmological models in terms of the parameters of the inhomogeneous equation of the state of viscous dark fluid and calculated the infrared cutoffs analytically. We represented the energy conservation equation for the dark fluid from a holographic point of view and showed a correspondence between the cosmology of a viscous fluid and holographic cosmology. We analyzed the autonomous dynamic system. In the absence of interaction between fluids, solutions are obtained corresponding to two cases. In the first case, dark energy is missing and the extension describes the component of dark matter. The second case corresponds to cosmological models with an extension due to dark energy. The solutions obtained are investigated for stability. For a cosmological model with the interaction of a special type, the stability of solutions of the dynamic system is also investigated.

In this paper, we discuss thermodynamics for spherically symmetric and static traversable wormholes which include Morris–Thorne wormholes and charged wormholes in the background of $f(R,T)$ gravity. The local coordinates have been used to find trapping horizons of these objects and generalized surface gravity has been worked out on the trapping horizons. The expression for the unified first law has also been derived from the gradient of Misner–Sharp energy with the help of gravitational field equations and from this law the first law of wormhole dynamics has been obtained. We have done this analysis for the simplest case of $f(R,T)$ gravity where $f(R,T)=R+2{\lambda }_{1}T$, $R$ and $T$ being the traces of the Ricci and stress–energy tensors. Also, we have extended these thermodynamic results to non-minimal curvature-matter coupling.

We review the notion of a ${\mathrm{\text{Dyck}}}^m\mathrm{\text{-algebra}}$, an algebraic structure introduced recently by López, Préville-Ratelle and Ronco during their work on the splitting of associativity via $m$-Dyck paths, and we also introduce Rota${}^m$-algebras: both structures can be considered as generalizations of dendriform structures. We obtain examples of Dyck${}^m$-algebras in terms of planar rooted binary trees equipped with a particular type of Rota–Baxter operator, and we present examples of Rota^m-algebras using left averaging morphisms. As an application, we observe that the structures presented here allow us to introduce quite naturally a “non-associative version” of the Kadomtsev–Petviashvili hierarchy.

This paper aims to discuss the gravitationally induced particle creation in the framework of $f({𝒢},T)$ theory, which involves the non-minimal coupling (NMC) between Gauss–Bonnet (GB) invariant, ${𝒢}$ and trace of the energy–momentum tensor (EMT), $T$. Here, NMC between matter and gravitational sector results in non-divergence of EMT. We discuss the generalized conservation equation for the irreversible process of matter creation with the help of generalized second law of thermodynamics (GSLT). Particle creation rate, creation pressure, entropy production rate and temperature are obtained for this theory using flat FRW geometry. We work on three particular $f({𝒢},T)$ models and study cosmological implications of open irreversible thermodynamics. Furthermore, the impact of NMC on cosmological evolution and entropy production is briefly discussed.

In this paper, we propose the generalized description of electromagnetism and linear gravity based on the combined dual numbers and complex quaternion algebra. In this approach, the electromagnetic and gravitational fields can be considered as the components of one combined dual-complex quaternionic field. It is shown that all relations between potentials, field strengths and sources can be formulated in the form of compact quaternionic differential equations. The alternative reformulation of equations of gravitoelectromagnetism based on formalism of $8\times 8$ matrices is also discussed. The results reveal the similarity and isomorphism of distinctive algebraic structures.

In this paper, we study almost coKähler manifolds admitting $k$-almost Yamabe solitons $(g,V,k,\lambda )$. First, we obtain a classification of almost coKähler $(\kappa ,\mu )$-manifolds admitting nontrivial closed $k$-almost Yamabe solitons. Next, we consider an almost $\alpha$-coKähler manifold admitting a nontrivial $k$-almost Yamabe soliton and prove that it is locally the Riemannian product of an almost Kähler manifold with the real line if the potential vector field $V$ is collinear with the Reeb vector field. For the potential vector field $V$ being orthogonal to the Reeb vector field, we also obtain two results.

In this paper, we study almost quasi-Yamabe solitons and gradient almost quasi-Yamabe solitons in $3$-dimensional $f$-Kenmotsu manifolds.

Being motivated by a well-known Nash’s embedding theorem, Chen introduced a method to discover the relationship for the extrinsic invariants controlled by the intrinsic one. In this paper, we extend Chen’s inequality for the intrinsic and extrinsic invariants for pointwise bi-slant warped products in locally conformal Kaehler space forms with quarter-symmetric and semi-symmetric connections. The equality case of the inequality is also investigated. Several applications of the inequality are given. Furthermore, we provide two non-trivial examples of such immersions.

The use of an infinity of fluctuating paths of least time that are compatible with the quantum mechanics indeterminacy provides a new interpretation in geometrical optic of the interference pattern of Young’s double slit experiment, which suggests that the wave behavior of matter and radiation is dictated by the space-time geodesics. Moreover, the association of a wave function to each path of least time as a probability amplitude together with an uncertainty for momentum and position allows to derive the Schrödinger’s equation starting from the geodesic’s characteristics. A new insight is obtained regarding the van der Waals torque as well as Casimir attraction/repulsion mechanism.