International Journal of Geometric Methods in Modern Physics

This paper studies the application of the Jacobi–Eisenhart lift, Jacobi metric and Maupertuis transformation to the Kepler system. We start by reviewing fundamentals and the Jacobi metric. Then we study various ways to apply the lift to Kepler-related systems: first as conformal description and Bohlin transformation of Hooke’s oscillator, second in contact geometry and third in Houri’s transformation [T. Houri, Liouville integrability of Hamiltonian systems and spacetime symmetry (2016), www.geocities.jp/football_physician/publication.html], coupled with Milnor’s construction [J. Milnor, On the geometry of the Kepler problem, Am. Math. Mon. 90 (1983) 353–365] with eccentric anomaly.

In this paper, we prove that any bi-harmonic map from a compact orientable Riemannian manifold without boundary $(M,g)$ to Riemannian manifold $(N,h)$ is necessarily constant with $(N,h)$ admitting a strongly convex function $f$ such that ${\mathrm{\text{grad}}}^{N}f$ is a Jacobi-type vector field (or $(N,h)$ admitting a proper homothetic vector field). We also prove that every harmonic map from a complete Riemannian manifold into a Riemannian manifold admitting a proper homothetic vector field, satisfying some condition, is constant. We present an open problem.

Recently, it has been suggested in [S. Chakraborty and N. Dadhich, Brown–York quasilocal energy in Lanczos–Lovelock gravity and black hole horizons, J. High Energ. Phys. 12 (2015) 003.] that the Brown–York mechanism can be used to measure the quasilocal energy in Lovelock gravity. We have used this method in a system of $M0$-branes and show that the Brown–York energy evolves in the process of birth and growth of Lovelock gravity. This can help us to predict phenomenological events which are emerged as due to dynamical structure of Lovelock gravity in our universe. In this model, first, $M0$-branes join each other and form an $M3$-brane and an anti-$M3$-branes connected by an $M2$-brane. This system is named BIon. Universes and anti-universes live on $M3$-branes and $M2$ plays the role of wormhole between them. By passing time, $M2$ dissolves in $M3$’s and nonlinear massive gravities like Lovelock massive gravity emerges and grows. By closing $M3$-branes, BIon evolves and wormhole between branes makes a transition to black hole. During this stage, Brown–York energy increases and shrinks to large values at the colliding points of branes. By approaching $M3$-branes towards each other, the square energy of their system becomes negative and some tachyonic states are produced. To remove these states, $M3$-branes compact, the sign of compacted gravity changes, anti-gravity is created which leads to getting away of branes from each other. Also, the Lovelock gravity disappears and its energy forms a new $M2$ between $M3$-branes. By getting away of branes from each other, Brown–York energy decreases and shrinks to zero.

Based on superconducting qubit coupled to a resonator driven by a strong microwave field, we propose a method to implement two quantum logic gates ($ni$SWAP and NTCP gates) of one qubit simultaneously controlling $n$ qubits selected from $N$ qubits in a circuit QED $(1<n<N)$ by introducing qubit–qubit interaction. The interaction between the qubits and the circuit QED can be achieved by tuning the gate voltage and the external flux. The operation times of the logic gates are much smaller than the decoherence time and dephasing time. Moreover, the numerical simulation under the influence of the gates operations shows that the scheme could be achieved efficiently with presently available techniques.

In this work, we study the equatorial causal geodesics of the Taub–NUT (TN) spacetime in comparison with massless TN spacetime. We emphasized both on the null circular geodesics and time-like circular geodesics. From the effective potential diagram of null and time-like geodesics, we differentiate the geodesics structure between TN spacetime and massless TN spacetime. It has been shown that there is a key role of the NUT parameter to changes the shape of pattern of the potential well in the NUT spacetime in comparison with massless NUT spacetime. We compared the innermost stable circular orbit (ISCO), marginally bound circular orbit (MBCO) and circular photon orbit (CPO) of the said spacetime with graphically in comparison with massless cases. Moreover, we compute the radius of ISCO, MBCO and CPO for extreme TN black hole (BH). Interestingly, we show that these three radii coincides with the Killing horizon, i.e. the null geodesic generators of the horizon. Finally in Appendix A, we compute the center-of-mass (CM) energy for TN BH and massless TN BH. We show that in both cases, the CM energy is finite. For extreme NUT BH, we found that the diverging nature of CM energy. First, we have observed that a non-asymptotic flat, spherically symmetric and stationary extreme BH showing such feature.

The paper focuses on considering some special precessional motions as the spin motions, separating the octonion angular momentum of a proton into six components, elucidating the proton angular momentum in the proton spin puzzle, especially the proton spin, decomposition, quarks and gluons, and polarization and so forth. Maxwell was the first to use the quaternions to study the electromagnetic fields. Subsequently the complex octonions are utilized to depict the electromagnetic field, gravitational field, and quantum mechanics and so forth. In the complex octonion space, the precessional equilibrium equation infers the angular velocity of precession. The external electromagnetic strength may induce a new precessional motion, generating a new term of angular momentum, even if the orbital angular momentum is zero. This new term of angular momentum can be regarded as the spin angular momentum, and its angular velocity of precession is different from the angular velocity of revolution. The study reveals that the angular momentum of the proton must be separated into more components than ever before. In the proton spin puzzle, the orbital angular momentum and magnetic dipole moment are independent of each other, and they should be measured and calculated respectively.

The present paper deals with the study of minimal lightlike submanifolds. We investigate a class of lightlike submanifolds namely, generic lightlike submanifolds under the minimal condition. We give one nontrivial example for minimal generic lightlike submanifolds and derive some characterization theorems for a generic lightlike submanifold to be a minimal lightlike submanifold. We also establish some conditions for the distributions for generic lightlike submanifolds to be minimal. We further derive the expressions for sectional curvature, null sectional curvature and induced Ricci tensor for a minimal lightlike submanifold. Finally, we prove that for a minimal lightlike submanifold, the null sectional curvature vanishes and the induced Ricci tensor is symmetric.

In this paper, we study how to determine the unknown functions for the scalar tensor model $f(R,\varphi )$ where the Ricci scalar is allowed to appear in a nonlinear way. The methods followed to determine these functions are: the matter collineation approach, the Lie group method and the Lagrangian collineation approach. We find several exact analytical solutions for a cosmological model with a FRW metric. We determine that some of the results are also valid for some anisotropic metric (e.g. the self-similar ones).

In this study, we propose to derive circularly symmetric black holes in three-dimensions Maxwell-teleparallel gravity using a non-diagonal traid. Singularities and horizons of these black holes are analyzed. The conserved quantities are calculated using Komar formula and vanishing values are obtained. Therefore, a regularization through relecolization method is applied and finite values of mass and angular momentum are obtained.

This paper is a study of three-dimensional paracontact metric $(\tilde{\kappa },\tilde{\mu },\tilde{\nu })$-manifolds. Three-dimensional paracontact metric manifolds whose Reeb vector field $\xi$ is harmonic are characterized. We focus on some curvature properties by considering the class of paracontact metric $(\tilde{\kappa },\tilde{\mu },\tilde{\nu })$-manifolds under a condition which is given at Definition 3.1. We study properties of such manifolds according to the cases $\tilde{\kappa }>-1,$ $\tilde{\kappa }=-1,\tilde{\kappa }<-1$ and construct new examples of such manifolds for each case. We also show the existence of paracontact metric $(-1,\tilde{\mu }\ne 0,\tilde{\nu }\ne 0)$ spaces with dimension greater than 3, such that ${\tilde{h}}^2=0$ but $\tilde{h}\ne 0.$

We study $f(R)$ gravity models in the language of scalar–tensor (ST) theories. The correspondence between $f(R)$ gravity and ST theories is revisited since $f(R)$ gravity is a subclass of Brans–Dicke models, with a vanishing coupling constant ($\omega =0$). In this treatment, four $f(R)$ toy models are used to analyze the early-universe cosmology, when the scalar field $\varphi$ dominates over standard matter. We have obtained solutions to the Klein–Gordon equation for those models. It is found that for the first model $(f(R)=\beta R^n)$, as time increases the scalar field decreases and decays asymptotically. For the second model $(f(R)=\alpha R+\beta R^n)$, it was found that the function $\varphi (t)$ crosses the $t$-axis at different values for different values of $\beta$. For the third model $\left(\right.f(R)=R-\frac{{\nu }^4}{R}\left)\right.$, when the value of $\nu$ is small, the potential $V(\varphi )$ behaves like the standard inflationary potential. For the fourth model $(f(R)=R-(1-m){\nu }^2\left(\right.\frac{R}{{\nu }^2}{\left)\right.}^m-2\mathrm{\Lambda })$, we show that there is a transition between $1.5<m<1.55$. The behavior of the potentials with $m<1.5$ is totally different from those with $m>1.55$. The slow-roll approximation is applied to each of the four $f(R)$ models and we obtain the respective expressions for the spectral index $n_s$ and the tensor-to-scalar ratio $r$.

Theory of gravitation based on a non-Riemannian geometry with dynamical torsion field is geometrically analyzed. To this end, the simplest Lagrangian density is introduced as a measure (reminiscent of a sigma model) and the dynamical equations are derived. Our goal is to rewrite this generalized affine action in a suitable form similar to the standard Born–Infeld (BI) Lagrangian. As soon as the functional action is rewritten in the BI form, the dynamical equations lead the trace-free GR-type equation and the field equations for the torsion, respectively: both equations emerge from the model in a sharp contrast with other attempts where additional assumptions were heuristically introduced. In this theoretical context, the Einstein $\kappa$, Newton $G$ and the analog to the absolute $b$-field into the standard BI theory all arise from the same geometry through geometrical invariant quantities (as from the curvature $R$). They can be clearly identified and correctly interpreted both physical and geometrically. Interesting theoretical and physical aspects of the proposed theory are given as clear examples that show the viability of this approach to explain several problems of actual interest. Some of them are the dynamo effect and geometrical origin of $\alpha \mathrm{\Omega }$ term, origin of primordial magnetic fields and the role of the torsion in the actual symmetry of the standard model. The relation with gauge theories, conserved currents, and other problems of astrophysical character is discussed with some detail.

In this work, we take into consideration a generalization of Gauge Theories based on the analysis of the structural characteristics of Maxwell theory, which can be considered as the prototype of such kind of theories (Maxwell-like). Such class of theories is based on few principles related to different orders of commutators between covariant derivatives. Their physical meaning is very simple, and lies in stating that the local transformations of a suitable substratum (the space-time or a particular phase space) and the imposed constraints define a “compensative mechanism” or the “interaction” we want to characterize. After a mathematical introduction, we apply this approach to a modified theory of gravity, in which the algebra of operators of covariant derivatives leads to an additional term in the equation of motion associated with the non-conservation of the energy–momentum tensor. This offers the possibility to include, without ad hoc physical assumptions and directly from the formalism, new forms of coupling between matter and energy and the expression of the mixing between gravity and torsion.

In this paper, we propose a simple parametrization of the Hubble parameter (HP) $H$ in order to explain the late-time cosmic acceleration. We show that our proposal covers many models obtained in different schemes of parametrization under one umbrella. We demonstrate that a simple modification in the functional form of HP can give rise to interesting cosmological phenomena such as big rip singularity, bounce and others. We have also constrained the model parameters using the latest 28 points of $H(z)$ data for three cases which admit transition from deceleration to acceleration.

We have investigated the Duffin–Kemmer–Petiau (DKP) equation, with equally vector and scalar potentials for the general deformed Morse potential, then we obtained creation and annihilation operators and showed that these operators satisfied the commutation relation of the $\mathrm{\text{SU}}(1,1)$group. The Casimir operator defined for Laguerre polynomials.

In this paper, we introduce the notion of an omni $n$-Lie algebra and show that it is the linearization of the higher analogue of the standard Courant algebroid. We also introduce the notion of a nonabelian omni $n$-Lie algebra and show that it is the linearization of the higher analogue of the Courant algebroid associated to a Nambu–Poisson manifold.

Park and Prasad [Semi-slant submersions, Bull. Korean Math. Soc. 50(3) (2013) 951–962.] defined and studied semi-slant submersions as a generalization of slant submersions, semi-invariant submersions, anti-invariant submersions. As a generalization of semi-slant submersions, we introduce conformal semi-slant submersions and study the new submersions from almost Hermitian manifolds onto Riemannian manifolds. We study the integrability of ditributions and the geometry of leaves of a conformal submersion. Moreover, we show that there are certain product structures on base manifold of a conformal semi-slant submersion. We also obtain totally geodesic conditions for such maps. Finally, we give lots of examples.

Integrable spin systems are an important subclass of integrable (soliton) nonlinear equations. They play important role in physics and mathematics. At present, many integrable spin systems were found and studied. They are related with the motion of three-dimensional curves. In this paper, we consider a model of two moving interacting curves. Next, we find its integrable reduction related with some integrable coupled spin system. Then, we show that this integrable coupled spin system is equivalent to the famous Manakov system.