International Journal of Geometric Methods in Modern Physics

Quaternions are widely used in physics. Quaternions, an extension of complex numbers, are closely related to many fundamental concepts (e.g. Pauli matrices) in physics. The aim of this study is the geometric structure underlying the quaternions used in physics. In this paper, we have investigated a new structure of unit speed associated curves such as spatial quaternionic and quaternionic osculating direction curves. For this, we have assumed that the vector fields

The purpose of this paper is to study the submersions of a generic submanifold $M$ of a nearly Kaehler manifold $\bar{M}$. It is proved that the base space $B$ of such a submersion which is an almost Hermitian manifold, turns out to be nearly Kaehler. It is also shown under certain conditions that base $B$ becomes Kaehler. Further, we give some curvature relations; specifically, we derive the relation between the holomorphic sectional curvature of $\bar{M}$ restricted to the horizontal distribution and that of $B$.

In this paper, a statistical analysis is performed on many thermodynamic properties, that are extracted from an optomechanical oscillator, which is coupled with an environment at a given temperature $T$. The results are interesting for all the obtained thermodynamic parameters as the position and the momentum variances behaviors, the energies behavior and the heat capacity behavior. The current results and their interpretations could contribute to the experimental realization of some materials in solid states with optimal parameters that conserve the defined material properties.

In this paper, we consider the extension of the Hilbert–Einstein action to analyze several interesting features of the theory. More specifically, the Lagrangian $f(R)$ is replaced by $f(R,L_m)$ in action, where $R$ is the Ricci scalar, and $L_m$ is the matter Lagrangian. We derive the motion equations for a test particle in the Friedmann–Lemaître–Robertson–Walker (FLRW) flat and homogeneous spacetime. We also derive the energy conditions in this framework. Then, we use the cosmographic parameter such as Hubble, deceleration, jerk and snap parameters to constraint the model parameters. As a result, we observe that with the constraint range of model parameters, our model shows the current accelerated expansion of the universe.

In this work, we perform a general study of properties of a class of locally symmetric embedded hypersurfaces in spacetimes admitting a $1+1+2$ spacetime decomposition. The hypersurfaces are given by specifying the form of the Ricci tensor with respect to the induced metric. These are slices of constant time in the spacetime. First, the form of the Ricci tensor for general hypersurfaces is obtained and the conditions under which the general case reduces to those of constant time slices are specified. We provide a characterization of these hypersurfaces, with key physical quantities in the spacetime playing a role in specifying the local geometry of these hypersurfaces. Furthermore, we investigate the case where these hypersurfaces admit a Ricci soliton structure. The particular cases where the vector fields associated with the solitons are Killing or conformal Killing vector fields are analyzed. Finally, in the context of spacetimes with local rotational symmetry, it is shown that only spacetimes in this class with vanishing rotation and spatial twist can admit the hypersurface types considered, and that the hypersurfaces are necessarily flat. If such hypersurface does admit a Ricci soliton structure, the soliton is steady, with the components of the soliton field being constants.

The main purpose of this paper is to study and explore some characteristics of static perfect fluid space-time on paracontact metric manifolds. First, we show that if a $K$-paracontact manifold $M^{2n+1}$ is the spatial factor of a static perfect fluid space-time, then $M^{2n+1}$ is of constant scalar curvature $-2n(2n+1)$ and squared norm of the Ricci operator is given by $4n^2(2n+1)$. Next, we prove that if a $(\kappa ,\mu )$-paracontact metric manifold $M^{2n+1}$ with $\kappa >-1$ is a spatial factor of static perfect space-time, then for $n=1$, $M^{2n+1}$ is flat, and for $n>1$, $M^{2n+1}$ is locally isometric to the product of a flat $(n+1)$-dimensional manifold and an $n$-dimensional manifold of constant negative curvature $-4$. Further, we prove that if a paracontact metric 3-manifold $M^3$ with $Q\varphi =\varphi Q$ is a spatial factor of static perfect space-time, then $M^3$ is an Einstein manifold. Finally, a suitable example has been constructed to show the existence of static perfect fluid space-time on paracontact metric manifold.

In this paper, we describe the extension to study the thermodynamics of the structure formation in the large-scale Universe in the nonlocal gravity formalism using standard statistical mechanics. From the derivation of the grand partition function in a modified version of the Yang–Lee theory, we obtained the corresponding thermodynamics properties that can be in consistency with a Bose–Einstein dark matter framework and derive its gravitational phase transition.

In this paper, we prove that the only almost Kenmotsu ${(k,\mu )}^{\prime }$-manifold of dimension three admitting a non-Killing conformal vector field is the product space ${ℍ}^2(-4)\times ℝ$. All conformal vector fields on this product are determined.

In this study, we have explored an anisotropic stellar configuration by applying the Karmarkar condition in the framework of $F(T,{𝒯})$ [where $T$ and ${𝒯}$ stand for torsion and trace of energy–momentum tensor (EMT)] gravity. We have developed field equations of $F(T,{𝒯})$ gravity for a spherically symmetric line element and considered a viable $F(T,{𝒯})$ model, i.e. $F(T,{𝒯})=\alpha T{(r)}^2+\beta {𝒯}(r)$ (where $\alpha$ and $\beta$ are arbitrary constants) for meaningful results. Following [S. K. Maurya, B. S. Ratanpal and M. Govender, Ann. Phys. 382 (2017) 36], a spherically symmetric solution of embedding class I has been explored which resulted into a generalized compact stellar model. For the evaluation of unknown parameters, we use the matching conditions at the boundary by taking spherical space-time metric as interior and Schwarzschild space-time solution as exterior space-time metric. The viability of our proposed model is checked by exploring the nature of thermodynamical variables ($\rho$, $p_r$, $p_t$), anisotropy, energy condition, Tolman–Oppenheimer–Volkov (TOV) forces, Abreu condition, equation of state parameters and adiabatic index taking the observational data of different compact stars like PSR J1416-2230, 4U 1608-52, Cen X-3, EXO 1785-248 and SMC X-1. Our stellar model is found.

In this paper, we use the Hojman symmetry approach to find new $(2+1)$-dimensional $f(R)$ gravity solutions, in comparison to Noether symmetry approach. In the special case of Hojman symmetry vector $X=R$, we recover $(2+1)$-dimensional Bañados–Teitelboim–Zanelli (BTZ) black hole and generalized $(2+1)$-dimensional BTZ black hole solutions, obtained by Noether symmetry approach, and the interesting point is that the cosmological constant is appeared as the direct manifestation of Hojman symmetry.

In this paper, we explore some new off-shell and on-shell conserved quantities for a scalar field in Minkowski space, using integrability condition. The off-shell conserved tensors are related to the kinematics of the field, while a linear combination of the off-shell and the on-shell conserved tensors ends up with the energy–momentum tensor for the scalar field. In the curved background, using Ricci and Bianchi identities, scalar–tensor theory of gravity, that resembles somewhat with Brans–Dicke-type field equations, emerges without requiring the principle of equivalence. Further, starting from the curvature scalar and using these identities, the field equations for modified gravity (Einstein–Hilbert action in the presence of higher-order terms) follow.

In this paper, the behavior of a Klein–Gordon oscillator under Lorentz symmetry breaking determined by a tensor ${(K_F)}_{\mu \nu \alpha \beta }$ out of the Standard Model Extension (SME) is analyzed. We choose a scenario of Lorentz-violating with an electric field inversely proportional to the radial and/or axial distance, and a constant magnetic field along $z$-direction. In this analysis, we see that the Lorentz-violating terms modified the energy spectrum and the wave function of an oscillator field in comparison to the known standard Landau levels. Furthermore, we note that the oscillation frequency $\omega$ of a scalar particle depends on quantum numbers $\{n,l\}$ as well as on Lorentz-violating parameters in the quantum system which shows a quantum effect.

In the framework of extended theory of gravity, an accelerating cosmological model has been presented in this paper in an anisotropic space-time. The extended gravity is considered as $f(R,T)$ gravity with the matter field as viscous fluid. The role of viscous coefficient has been discussed on the cosmic expansion issue. In the developed formalism, a variable Hubble parameter has been incorporated to express the dynamical parameter with respect to redshift. The model is observed to be compatible with recent observational data.

Inspired by Laughlin’s theory of the fractional quantum Hall effect, we explore the topological nature of the quark–gluon plasma (QGP) and the nucleons in the context of the Clifford algebra. In our model, each quark is transformed into a composite particle via the simultaneous attachment of a spin monopole and an isospin monopole. This is induced by a novel kind of meson endowed with both spin and isospin degrees of freedom. The interactions in the strongly coupled quark–gluon system are governed by the topological winding number of the monopoles, which is an odd integer to ensure that the overall wave function is antisymmetric. The states of the QGP and the nucleons are thus uniquely determined by the combination of the monopole winding number $m$ and the total quark number $N$. The radius squared of the QGP droplet is expected to be proportional to $mN$. We anticipate the observation of such proportionality in the heavy ion collision experiments.

Recently, PNDP-manifolds have been introduced, and these have shown to be useful in applicative aspects especially in the field of cosmology, introducing a new geometric/topological approach to the concept of “emerging space”. However, the applications considered so far concern only trivial and flat PNDP-manifolds. Therefore, we show the existence of non-Ricci-flat solutions for PNDP-manifolds with positive “virtual” dimension which have hyperbolic $\tilde{B}$-manifold. Based on this, we have advanced an applicative hypothesis, accompanied by some open considerations, in the context of $AdS$ cosmology.

Dirac stated that the Cartesian coordinates are uniquely suited for expressing the canonical commutation relations in a simple form. By contrast, expressing these commutation relations in any other coordinate system is more complicated and less obvious. The question that we address in this work is the reason why this is true. We claim that this unique role of the Cartesian coordinates is a result of the existence and uniqueness of the Hofer metric on the space of canonical transformations of the phase space of the system getting quantized.