International Journal of Geometric Methods in Modern Physics

This paper investigates the energy bounds in modified Gauss–Bonnet gravity with anisotropic background. Locally rotationally symmetric Bianchi type $I$ cosmological model in $f(R,G)$ gravity is considered to meet this aim. Primarily, a general $f(R,G)$ model is used to develop the field equations. In this aspect, we investigate the viability of modified gravitational theory by studying the energy conditions. We take in account four $f(R,G)$ gravity models commonly discussed in the literature. We formulate the inequalities obtained by energy conditions and investigate the viability of the above-mentioned models using the Hubble, deceleration, jerk and snap parameters. Graphical analysis shows that for first two $f(R,G)$ gravity models, null energy condition (NEC), weak energy condition (WEC) and strong energy condition (SEC) are satisfied under suitable values of anisotropy and model parameters involved. Moreover, SEC is violated for the third and fourth models which predicts the cosmic expansion.

This paper investigates the invariance properties of the time fractional Benny–Lin equation with Riemann–Liouville and Caputo derivatives. This equation can be reduced to the Kawahara equation, fifth-order Kdv equation, the Kuramoto–Sivashinsky equation and Navier–Stokes equation. By using the Lie group analysis method of fractional differential equations (FDEs), we derive Lie symmetries for the Benny–Lin equation. Conservation laws for this equation are obtained with the aid of the concept of nonlinear self-adjointness and the fractional generalization of the Noether’s operators. Furthermore, by means of the invariant subspace method, exact solutions of the equation are also constructed.

In this paper, we consider a Lagrangian Riemannian submersion from a Hermitian manifold to a Riemannian manifold and establish some basic inequalities to obtain relationships between the intrinsic and extrinsic invariants for such a submersion. Indeed, using these inequalities, we provide necessary and sufficient conditions for which a Lagrangian Riemannian submersion $\pi$ has totally geodesic or totally umbilical fibers. Moreover, we study the harmonicity of Lagrangian Riemannian submersions and obtain a characterization for such submersions to be harmonic.

We show that all of the nonstretching curve motions specified in the Frenet–Serret frame in the literature can be described by the time evolution of an integral curve of a Hamiltonian dynamical system such that the underlying curve is a geodesic curve on a leaf of the foliation determined by the Poisson structure in three dimensions. As an illustrative example, we show that the focusing version of the nonlinear Schrödinger equation and the complex modified Korteweg–de Vries (mKdV) equation are obtained in this way.

We investigate the functional $\mathrm{\Omega }\mapsto {ℰ}(\mathrm{\Omega })$ where $\mathrm{\Omega }$ runs through the set of compact domains of fixed volume $v$ in any Riemannian manifold $(M,g)$ and where ${ℰ}(\mathrm{\Omega })$ is the mean exit time from $\mathrm{\Omega }$ of the Brownian motion. We give an alternative analytical proof of a well-known fact on its critical points proved by McDonald: the critical points of ${ℰ}(\mathrm{\Omega })$ are harmonic domains.

Over the $(1,1)$-dimensional real superspace ${ℝ}^{1|1}$, we classify $𝔞𝔣𝔣(1|1)$-invariant bilinear differential operators acting on the superspaces of weighted densities. We compute the second $𝔞𝔣𝔣(1|1)$-relative cohomology space of ${𝒦}(1)$ with coefficients in the module of $\lambda$-densities ${𝔉}_{\lambda }^1$ on ${ℝ}^{1|1}$, where ${𝒦}(1)$ is the Lie superalgebra of contact vector fields on ${ℝ}^{1|1}$ and $𝔞𝔣𝔣(1|1)$ is the affine Lie superalgebra. This result allows us to compute the second $𝔞𝔣𝔣(1|1)$-relative cohomology space of ${𝒦}(1)$ with coefficients in the Poisson superalgebra ${𝒮}{𝒫}(1)$. We explicitly give 2-cocycles spanning these cohomology spaces.

In this paper, we present three anholonomy densities and three geometric phases for three formulations of curve evolution in Minkowski 3-space and give applications of three formulation for ferromagnetic and antiferromagnetic spin chains.

In this paper, we study the Jacobi stability on the nonlinear double pendulum by the Kosambi–Cartan–Chern (KCC) theory. We assume that the mass and length of rods of two kinds of pendulums are equal, respectively. Moreover, we consider the case that initial angles of the double pendulum are equal. Under these conditions, we obtain the boundary between Jacobi stable and unstable trajectories for initial angles. It is shown that the condition of Jacobi stable or unstable depends only on deflection angles of the nonlinear double pendulum. Then, we discuss relationships between Jacobi stability, physical parameters and other concepts of stability such as Lyapunov stability and chaos. We suggest that the ratio of length of rods and the mass ratio of pendulums of the double pendulum do not affect the Jacobi stability. It is suggested that the equilibrium points in the Jacobi stable region and in the Jacobi unstable region are Lyapunov stable and Lyapunov unstable, respectively, and that the motions in the Jacobi unstable region are related to the onset of chaotic behavior.

Torse-forming vector fields introduced by Yano [On torse forming direction in a Riemannian space, Proc. Imp. Acad. Tokyo 20 (1944) 340–346] are natural extension of concurrent and concircular vector fields. Such vector fields have many nice applications to geometry and mathematical physics. In this paper, we establish a link between rotational hypersurfaces and torse-forming vector fields. More precisely, our main result states that, for a hypersurface $M$ of ${𝔼}^{n+1}$ with $n\ge 3$, the tangential component $x^T$ of the position vector field of $M$ is a proper torse-forming vector field on $M$ if and only if $M$ is contained in a rotational hypersurface whose axis of rotation contains the origin.

The aim of this work is to study translation surfaces in a Minkowski 3-space ${ℝ}_1^3$ with density. Translation surfaces in ${ℝ}_1^3$ are defined as the two generating curves which lie in orthogonal planes. They have actually three different possible parametrizations according to the intersecting straight line of the two planes. We completely classify all translation surfaces with zero weighted mean curvature in ${ℝ}_1^3$ with density $e^z$ by solving the second-order non-linear ODE with some smooth functions.

In this paper, we have investigated the deflection angle of light by wormholes using a new geometrical method known as Gibbons–Werner method (GW). In particular, we have calculated the deflection angle of light in the weak limit approximation in two wormhole space–time geometries: Ellis wormhole and Janis–Newman–Winnicour (JNW) wormhole. We have employed the famous Gauss–Bonnet theorem (GBT) to the Ellis wormhole optical geometry and JNW wormhole optical geometry, respectively. By using GBT, we computed the deflection angles in leading orders by these wormholes and our results were compared with the ones in the literature.

Over the $(1,2)$-dimensional real superspace, we compute the cohomology space of the affine Lie superalgebra $𝔞𝔣𝔣(2)$ with coefficient in a large class of $𝔞𝔣𝔣(2)$-modules $M$. We apply our results to the module $M={𝔉}_{\lambda }^2$ of weight densities and the module $M={𝔇}_{\lambda ,\mu }^2$ of linear differential operators acting on a superspace of weighted densities. We study nontrivial deformations of the natural action of the Lie superalgebra $𝔞𝔣𝔣(2)$ on the direct sum of the superspaces of weighted densities.

The present paper reports a study on the bouncing behavior of the viscous modified Chaplygin gas (MCG) in Einstein as well as modified gravity framework. For a bouncing scale factor proposed by Cai et al., Class. Quantum Grav. 28 (2011) 215011, we have studied the cosmology of MCG in presence of bulk viscosity. In Einstein gravity framework, we have studied the equation of state parameter and it has been found to cross $-1/3$ indicating the end of the early accelerated expansion and it has also been observed that for flat FRW universe the presence of bulk viscosity induces the crossing of phantom boundary. Role of the model parameters of the MCG has also been investigated before and after the bounce. A Hubble flow dynamics has been carried out and, it was revealed that MCG is capable of realizing inflationary phase as well as an exit from inflation. A $f(T)$ gravitational paradigm has also been considered, where the MCG density has been reconstructed in presence of bulk viscosity. Role of $\sigma$ of the bouncing scale factor, describing how fast the bounce takes place, has also been studied in this framework.

Variational symmetries and conservation laws of a modified de Sitter metric are classified. The resulting Noether symmetries in each case are compared with conservation laws given by Killing vectors.

The universe modeled with van der Waals fluid approximation, where the van der Waals fluid equation of state contains a single parameter ${\omega }_v$. Analytical solutions to the Einstein’s field equations are obtained by assuming the mean scale factor of the metric follows volumetric exponential and power-law expansions. The model describes a rapid expansion where the acceleration grows in an exponential way and the van der Waals fluid behaves like an inflation for an initial epoch of the universe. Also, the model describes that when time goes away the acceleration is positive, but it decreases to zero and the van der Waals fluid approximation behaves like a present accelerated phase of the universe. Finally, it is observed that the model contains a type-III future singularity for volumetric power-law expansion.

The order horismos induces the Zeeman $Z$ topology, which is coarser than the Fine Zeeman Topology $F$. The causal curves in a spacetime under $Z$ are piecewise null. $F$ is considered to be the most physical topology in a spacetime manifold $M$, as the group of homeomorphisms of $M$ is isomorphic to the group of homothetic transformations of $M$. $Z$ was used in the a ambient boundary-ambient space cosmological model, in order to show that there is no possibility of formation of spacetime singularities. In this paper, we question this result, by reviewing the corresponding papers, and we propose new questions toward the improvement of this model.

The inflationary expansion of our early-time universe is considered in terms of the van der Waals equation, as equation of state for the cosmic fluid, where a bulk viscosity contribution is assumed to be present. The corresponding gravitational equations for the energy density in a homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker universe are solved, and an analytic expression for the scale factor is obtained. Attention is paid, specifically, to the role of the viscosity term in the accelerated expansion; the values of the slow-roll parameters, the spectral index, and the tensor-to-scalar ratio for the van der Waals model are calculated and compared with the most recent astronomical data from the Planck satellite. By imposing reasonable restrictions on the parameters of the van der Waals equation, in the presence of viscosity, it is shown to be possible for this model to comply quite precisely with the observational data. One can therefore conclude that the inclusion of viscosity in the theory of the inflationary epoch may definitely improve the cosmological models.

The emergence of early cosmological anisotropies and late accelerated expansion can be achieved considering Bianchi cosmological models equipped with torsion. In this paper, we present a Bianchi IX cosmology with torsion where both the previous issues are addressed by the exact solutions derived from the model. The standard Friedman–Robertson–Walker universe is recovered as soon as torsion is switched off and metric becomes isotropic.