International Journal of Geometric Methods in Modern Physics

In this paper, we study bi-warped product submanifolds of the form $M=M_{𝜃}{\times }_{f_1}{M}_T{\times }_{f_2}{M}_{\perp }$ in a Kenmotsu manifold. We obtain a lower bound for the squared norm of the second fundamental form of a bi-warped product submanifold such as $\parallel h{\parallel }^2\ge m_1{csc}^2𝜃(1+{cos}^2𝜃)(\parallel \overrightarrow{\nabla }(ln{f}_1){\parallel }^2-1)+m_2{cot}^2𝜃(\parallel \overrightarrow{\nabla }(ln{f}_2){\parallel }^2-1)$, where $m_1=dim(M_T)$ and $m_2=dim(M_{\perp })$ and $f_1,f_2$ are the warping functions on $M$. The equality case is also considered.

In this paper, global dynamics of forced Lorenz-84 system are discussed, and some new results are presented. First of all, the periodic attractor is analyzed for the almost periodic Lorenz-84 system with almost periodically forcing, including the existence and the boundedness of those almost periodic solutions, and the bifurcation phenomenon in the driven system. Then, the random attractor set and the bifurcation phenomenon for the driven Lorenz-84 system with stochastic forcing are studied, including the globally exponentially attractive set, positive invariant set, random attractor and stochastic bifurcation behavior. Last but not least, some numerical simulations are also presented for verifying obtained theoretical results.

This work deals with holographic dark energy in the context of unimodular $f(T)$ gravity, which is a modification of teleparallel gravity. We develop the general reconstruction procedure of the $f(T)$ form that can yield the holographic feature of the dark energy. We fit the reconstructed model with the $H(z)$ data and our results show a perfect agreement with the WMAP9 cosmological observational data, at least for the range $-1.10\le {\omega }_V\le -1.05$. We investigate the consistency of the reconstructed model by studying its stability against linear gravitational and matter perturbations, fixing ${\omega }_V$ to $-1.05$. The model presents stability for both de Sitter and power-law solutions and we conclude that it is a good candidate as alternative viable model for characterizing holographic dark energy.

The realistic models of neutron stars are considered for simple $R+\alpha R^2$ gravity and equivalent Brance–Dicke theory with dilaton field in Einsein frame. For negative values of $\alpha$ we have no acceptable results from astrophysical viewpoint: the resulting solution for spherical stars doesn’t coincide with Schwarzschild solution on spatial infinity. The mass of star from viewpoint of distant observer tends to very large values. For $\alpha >0$ it is possible to obtain solutions with required asymptotics and well-defined star mass. The mass confined by stellar surface decreases with increasing of $\alpha$ but we have some contribution to mass from gravitational sphere appearing outside the star. The resulting effect is increasing of gravitational mass from viewpoint of distant observer. But another interpretation take place in a case of equivalent Brance–Dicke theory with massless dilaton field in Einstein frame. The mass of star increases due to contribution of dilaton field inside the star. We also considered the possible constraints on $R^2$ gravity from GW 170817 data. According to results of Bauswein et al. the lower limit on threshold mass is $2.74_{-0.01}^{+0.04}$$M_{\odot }$. This allows to exclude some equations of state (EoS) for dense matter. But in $R^2$ gravity the threshold mass increases for given EoS with increasing of $\alpha$. In principle it can helps in future discriminate between General Relativity and square gravity (of course one need to know EoS with more accuracy rather than now).

In this paper, we have presented bulk viscous cosmological model of the universe in the modified gravity theory in which the Lagragian of the gravitational action contains a general function $f(R,T)$, where $R$ and $T$ denote the curvature scalar and the trace of the energy–momentum tensor, respectively, in the framework of a flat Friedmann–Robertson–Walker model with isotropic fluid. We obtain cosmological solution in $f(R,T)$ theory of gravity, specially of particular choice $f(R,T)=R+2\lambda T$, where $\lambda$ is a constant, in the presence of bulk viscosity that are permitted in Eckart theory, Truncated Israel Stewart (TIS) theory and Full Israel Stewart (FIS) theory. The physical and geometrical properties of the models in Eckart, TIS theory and FIS theory are studied in detail. The analysis of the variation of bulk viscous pressure, energy density, scale factor, Hubble parameter and deceleration parameter with cosmic evolution is done in the respective theories. The models are analyzed by comparison with recent observational data. The cosmological models are compatible with observations.

In order to obtain the geometry of a global monopole without cosmological constant and electric charge in $2+1$ dimensions, we make use of the broken $O(2)$ symmetry. In the absence of an exact solution, we determine the series solutions for both the metric and monopole functions in a consistent manner that satisfies all equations in appropriate powers. The new expansion elements are of the form $\frac{1}{r^n}{(lnr)}^m,$ for the radial distance $r$ and positive integers $m$ and $n$ constrained by $m\le n$. To the lowest order of expansion, we find that in analogy with the negative cosmological constant the geometry of the global monopole acts repulsively, i.e. in the absence of a cosmological constant the global monopole plays at large distances the role of a negative cosmological constant.

This paper is an attempt to revisit the Friedmann–Robertson–Walker (FRW) cosmological models under the new scenario of observational cosmology, which has established that the current universe is expanding with an increasing rate, in contrast to the earlier belief that the rate of expansion is constant or slowing down. This paper represents a model which encompasses both, earlier decelerating and the current accelerating universe, passing through a transition phase. The universe is assumed to be filled with two fluids, barotropic and dark energy. We have considered two cases; first, when these fluids are assumed to be non-interacting and second, when they interact with each other. Some physical, kinematic and geometric properties of the model are also discussed along with the acceptability and stability of the solution. The results found are very compatible with the established results as well as recent observations.

We show that an $n$-dimensional generalized Robertson–Walker (GRW) space-time with divergence-free conformal curvature tensor exhibits a perfect fluid stress–energy tensor for any $f(R)$ gravity model. Furthermore, we prove that a conformally flat GRW space-time is still a perfect fluid in both $f(R)$ and quadratic gravity where other curvature invariants are considered.

We establish duality between real forms of the quantum deformation of the four-dimensional orthogonal group studied by Fioresi et al. [Quantum Klein space and superspace, preprint (2017), arXiv:1705.01755] and the classification work made by Borowiec et al. [Basic quantizations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic ${𝔬}^{\star }(4)$ symmetries, J. High Energy Phys. 1711 (2017) 187]. Classically, these real forms are the isometry groups of ${ℝ}^4$ equipped with Euclidean, Kleinian or Lorentzian metric. A general deformation, named $q$-linked, of each of these spaces is then constructed, together with the coaction of the corresponding isometry group.

We show the geometric consequences that the holographic principle has on the spacetime. Namely, we prove that complete spacelike hypersurfaces in a spacetime that satisfies the holographic principle are non-parabolic. This has important consequences on the behaviour of the Brownian motion in these spacelike hypersurfaces, as well as provides a method for finding examples of universes that lead to a violation of this principle.

The main result of this paper gives a characterization of left-invariant almost $\alpha$-coKähler structures on three-dimensional (3D) semidirect product Lie groups ${ℝ}^2{⋊}_Aℝ$ in terms of the matrix $A$. Then, we study the harmonicity of the Reeb vector field $\xi$ of a simply connected homogeneous almost $\alpha$-coKähler three-manifold, in terms of the Gaussian curvature of the canonical foliation.

The importance of the tomographic approach is that either in quantum mechanics as in classical mechanics the state of a physical system is expressed with marginal probability functions called tomograms. The extension of this procedure to quantum cosmology is straightforward. But in this paper, instead of using the tomographic representation, we use tomograms to analyze the properties of the quantum and classical universes, starting from the wave functions in quantum cosmology and from the phase space distributions in classical cosmology. In this, there is a part where we resume the properties of the tomograms. Then, we apply them to study and discuss the properties of the initial conditions introduced by Hartle and Hawking, Vilenkin and Linde and finally we argue about their classical transition. According to the results of this paper it follows that the decay of the cosmological constant from the Planck scale to the present one could be responsible for the transition of the quantum universe to the classical one.

The magnetosphere structure of a magnetar is considered in the context of a theory of gravity with dynamical torsion field beyond the standard General Relativity (GR). To this end, the axially symmetric version of the Grad-Shafranov equation (GSE) is obtained in this theoretical framework. The resulting GSE solution in the case of the magnetosphere corresponds to a stream function containing also a pseudoscalar part. This function solution under axisymmetry presents a complex character that (as in the quantum field theoretical case) could be associated with an axidilaton field. Magnetar–pulsar mechanism is suggested and the conjecture about the origin of the excess energy due the GSE describing the magnetosphere dynamics is claimed. We also show that two main parameters of the electrodynamic processes (as described in GR framework by Goldreich and Julian (GJ) [Astrophys. J. 157 (1969) 869]) are modified but the electron-positron pair rate $Ṅ$ remains invariant. The possible application of our generalized equation (defined in a non-Riemannian geometry) to astrophysical scenarios involving emission of energy by gravitational waves, as described in the context of GR in [S. Capozziello, M. De Laurentis, I. De Martino, M. Formisano and D. Vernieri, Astrophys. Space Sci. 333 (2011) 29–35], is briefly discussed.

Recently, some authors have proposed a new model for the BIon which can be applied in both nanosystems and cosmology. A Bion is a configuration which has been constructed from two branes and a wormhole which connects them. In this paper, we consider the process of formation of a BIon by joining hexagonal accretion disks. These disks exchange gravitational waves with each other and produce some curvature. By increasing the number of connected disks and joining these waves, a wormhole emerges.

Riemann’s principle “force equals geometry” provided the basis for Einstein’s General Relativity — the geometric theory of gravitation. In this paper, we follow this principle to derive the dynamics for any static, conservative force. The geometry of spacetime of a moving object is described by a metric obtained from the potential of the force field acting on it. We introduce a generalization of Newton’s First Law — the Generalized Principle of Inertia stating that: An inanimate object moves inertially, that is, with constant velocity, in its own spacetime whose geometry is determined by the forces affecting it. Classical Newtonian dynamics is treated within this framework, using a properly defined Newtonian metric with respect to an inertial lab frame. We reveal a physical deficiency of this metric (responsible for the inability of Newtonian dynamics to account for relativistic behavior), and remove it. The dynamics defined by the corrected Newtonian metric leads to a new Relativistic Newtonian Dynamics for both massive objects and massless particles moving in any static, conservative force field, not necessarily gravitational. This dynamics reduces in the weak field, low velocity limit to classical Newtonian dynamics and also exactly reproduces the classical tests of General Relativity, as well as the post-Keplerian precession of binaries.

Generalized Robertson–Walker (GRW) spacetime is the generalization of the Robertson–Walker (RW) spacetime and a further generalization of GRW spacetime is the twisted spacetime. In this paper, we generalize the results of the paper [C. A. Mantica, Y. J. Suh and U. C. De, A note on generalized Robertson–Walker spacetimes, Int. J. Geom. Methods Mod. Phys. 13 (2016), Article Id: 1650079, 9 pp., doi:101142/s0219887816500791]. We prove that a Ricci simple Lorentzian manifold with vanishing quasi-conformal curvature tensor is a RW spacetime. Further, we prove that a Ricci simple Lorentzian manifold with harmonic quasi-conformal curvature tensor is a GRW spacetime. As a consequence, we obtain several corollaries. Finally, we have cited some examples of the obtained results.

In [M. K. Karacan, D. W. Yoon and B. Bukcu, Translation surfaces in the three-dimensional simply isotropic space ${𝕀}_3^1$, Int. J. Geom. Methods Mod. Phys. 13(7) (2016) 1650088], there is a mistake in Theorem 5 that appeared in the paper. We here provide a correct theorem.