International Journal of Geometric Methods in Modern Physics

The evolute of a regular curve in the Lorentz–Minkowski plane is given by the locus of centers of osculating pseudo-circle of the base curve. But the case when a curve has singularities is not very clear. In this paper, we use lightcone frame to define the $(n,m)$-cusp mixed-type curves and their evolutes in Lorentz–Minkowski plane. In order to attain this goal, we define the $(n,m)$-cusp non-lightlike curves and their evolutes in Lorentz–Minkowski plane first. Then we study the behaviors of the evolutes of the $(n,m)$-cusp mixed-type curves at the $(n,m)$-cusp.

In this work, we explore the accelerated expansion of the conharmonically flat space in relation to an isotropic and spatially homogeneous Friedmann–Robertson–Walker (FRW) universe through a newly proposed dark energy (DE) model namely Sharma–Mittal holographic DE (SMHDE) by taking Hubble horizon as an IR cut-off and also by considering the deceleration parameter as a linear function of Hubble parameter as $q=\beta H+\alpha$, where $\alpha$ and $\beta$ are arbitrary constants. The analysis of different cosmological parameters, equation of state (EoS) parameter, squared speed of sound, statefinder, ${\omega }_D-{\omega }_D^{\prime }$ pair, and quintessence field model has been calculated and discussed in detail. Analyzing the behavior of such cosmological parameters graphically, it is found that the SMHDE model can lead to the accelerated expansion of the universe at present epoch. We have also reconciled the DE with scalar field potential. For this analysis, we take into account the quintessence field for this reconstruction.

The principal objective of this project is to investigate the gravitational lensing by asymptotically flat black holes in the framework of Horndeski theory in weak field limits. To achieve this objective, we utilize the Gauss–Bonnet theorem to the optical geometry of asymptotically flat black holes and apply the Gibbons–Werner technique to achieve the deflection angle of photons in weak field limits. Subsequently, we manifest the influence of plasma medium on deflection of photons by asymptotically flat black holes in the context of Horndeski theory. We also examine the graphical impact of deflection angle on asymptotically flat black holes in the background of Horndeski theory in plasma medium as well as non-plasma medium.

The characterization of curves plays an important role in both geometry and topology of almost contact manifolds. Olszak found the equation $2\alpha \beta +\xi [\beta ]=0$ on normal almost contact manifolds. The pair $(\alpha ,\beta )$ denotes the type of these manifolds. In this study, we obtained the curvatures of non-geodesic Frenet curves on $3$-dimensional normal almost contact manifolds without neglecting $\alpha$ and $\beta$, and provided the results of their characterization. We exemplified these results with examples.

We analyze the existence of equilibrium points for a particle or dust grain in the framework of unperturbed and perturbed Robe’s motion. This particle is moving in a spherical nebula consisting of a homogeneous incompressible fluid, which is considered as the primary body. The second primary body creates the modified Newtonian potential. The perturbed mean motion and equations of motion are found. The equilibrium points (i.e. collinear, noncollinear and out–of–plane points), along with the required conditions of their existence are also analyzed. We emphasize that this analysis can be used to study the oscillations of the Earth’s core under the attraction of the Moon and it is also applicable to study the motion of underwater vehicles.

We consider a timelike sweeping surface with rotation minimizing frames in Minkowski 3-Space ${𝔼}_1^3$. We introduce a new geometric “invariant”, which demonstrates the geometric properties and local singularities of the surface. Subsequently, we give the sufficient and necessary conditions for this surface to be a developable ruled surface. Finally, the singularities of these ruled surfaces are investigated.

The purpose of the offering exposition is to characterize gradient Yamabe, gradient Einstein and gradient $m$-quasi Einstein solitons within the framework of 3-dimensional para-Kenmotsu manifolds. Finally, we consider an example to prove the result obtained in previous section.

For the spherical unit speed nonlightlike curve in pseudo-hyperbolic space and de Sitter space $\gamma$ and a given point P, we can define naturally the pedal curve of $\gamma$ relative to the pedal point P. When the pseudo-sphere dual curve germs are nonsingular, singularity types of such pedal curves depend only on locations of pedal points. In this paper, we give a complete list of normal forms for singularities and locations of pedal points when the pseudo-sphere dual curve germs are nonsingular. Furthermore, we obtain the extension results in dualities, which has wide influence on the open and closed string field theory and string dynamics in physics, and can be used to better solve the dynamics of trajectory particle condensation process.

In this work, we compute the time evolution equations (TEEs) of the type-1 Bishop frame of the curve. Also, we study the time evolution equations for type-1 Bishop curvatures (TEEBCs) as a system of partial differential equations (PDEs). Through this study, we give a necessary and sufficient condition for the normal and binormal Bishop velocities. Also, we construct new models of normal motions of inextensible curves in ${ℝ}^3$. These models are constructed for curves that move according to the type-1 Bishop frame. Also, we make a comparison study between surfaces obtained by the motions of inextensible curves according to the two frames, the type-1 Bishop frame and the Frenet frame.

In this paper, we study the geometry of a certain class of compact dynamical horizons with a time-dependent induced metric in locally rotationally symmetric class II spacetimes. We first obtain a compactness condition for embedded $3$-manifolds in these spacetimes, satisfying the weak energy condition, with non-negative isotropic pressure $p$. General conditions for a $3$-manifold to be a dynamical horizon are imposed, as well as certain genericity conditions, which in the case of locally rotationally symmetric class II spacetimes reduces to the statement that “the weak energy condition is strictly satisfied or otherwise violated”. The compactness condition is presented as a spatial first-order partial differential equation in the sheet expansion $\varphi$, in the form $\widehat{\varphi }+(3/4){\varphi }^2-cK=0$, where $K$ is the Gaussian curvature of $2$-surfaces in the spacetime and $c$ is a real number parametrizing the differential equation, where $c$ can take on only two values, $0$ and $2$. Using geometric arguments, it is shown that the case $c=2$ can be ruled out and the ${𝕊}^3$ ($3$-dimensional sphere) geometry of compact dynamical horizons for the case $c=0$ is established. Finally, an invariant characterization of this class of compact dynamical horizons is also presented.

In the frame work of Saturn–Titan system, the resonant orbits of first-order are analyzed for three different families of periodic orbits, namely, interior resonant orbits, exterior resonant orbits and $f$-Family orbits. This analysis is developed by considering Saturn as a spherical and oblate body. The initial position, semi-major axis, eccentricity, orbital period and order of resonant orbits of these families are investigated for different values of Jacobi constant and oblateness parameter.

This paper is fundamentally devoted to the cosmological reconstruction and dynamic studying in homogeneous BIANCHI-I space-time under the $f(T)$ background. Its content is supported by the fact that in the General Relativity description of the standard cosmological paradigm, the evolution from an anisotropic universe into an Friedmann–Lemaitre–Robertson–Walker (FLRW) one can be achieved by a period of inflationary expansion. Nowadays, modified gravity theories like $f(T)$ are widely accepted to provide a real description of some universe evolution phases like inflation era, matter-dominated era, etc. So, we aim to examine here what $f(T)$ gravity model can accommodate with an anisotropic universe, an expanding universe and even the transition between both evolutions. To reach this goal, we use a reconstruction method based on dynamic equations in Bianchi-I space-time by assuming a particular form for the metric anisotropy and by specifying some time functions describing average scale factor. Most of the obtained models are consistent with certain known results in the literature but other add new results in this work. In the second part of this work, the dynamical behaviors of the Bianchi-I space-time are addressed through the reconstruction of an autonomous dynamical system. For an aleatory choice of anisotropic fluid, the numerical analysis of the system shows that the metric anisotropy decreases with expansion. Then, an attractor point is reached and becomes unstable by the end of inflation. Such interesting properties found in this work on Bianchi-I space-time are often interpreted as graceful exit from inflation which doesn’t occur in ordinary FLRW space-time.

We deal with non-autonomous Hamiltonian systems of one degree of freedom. For such differential systems, we compute analytically some of their periodic solutions, together with their type of stability. The tool for proving these results is the averaging theory of dynamical systems. We present some applications of these results.

In this work, we propose a non-interacting model of Barrow holographic dark energy (BHDE) using Barrow entropy in a spatially flat FLRW Universe considering the IR cutoff as the Hubble horizon. We study the evolutionary history of important cosmological parameters, in particular, deceleration parameter, equation of state (EoS) parameter, the BHDE and matter density parameter, and also observe satisfactory behaviors in the BHDE model. The stability of the BHDE model has been examined by squared sound speed $v_s^2$. In addition, to describe the accelerated expansion of the Universe, the correspondence of the BHDE model with the quintessence scalar field has been reconstructed.

Developable surfaces are defined to be locally isometric to a plane. These surfaces can be formed by bending thin flat sheets of material, which makes them an active research topic in computer graphics, computer aided design, computational origami and manufacturing architecture. We obtain condition for developable and minimal ruled surfaces using rotation frame. Also, the validity of the theorems is illustrated with examples.

Although it is not a fundamental question, determining exact and general solutions for a given theory has advantages over a numerical integration in many specific cases. Of course, respecting the peculiarities of the problem. Revisiting the integration of the General Relativity Theory field equations for the Kantowski–Sachs spacetime describes a homogeneous but anisotropic universe whose spatial section has the topology of $R\times S^2$, we integrate the equations for arbitrary curvature parameter and write the solutions considering the process of gravitational collapse. We took the opportunity and made some comments involving some features of the model such as energy density, shear, viscosity and the production of gravitational waves via Petrov classification.