International Journal of Geometric Methods in Modern Physics

Four-dimensional conformally flat curvature models of signature $(2,2)$ are considered and complete classification of curvature tensor is obtained. Then, some remarks about Ivanov–Petrova and Walker–Ivanov–Petrova properties are stated.

In this work, we study the evolution of an isotropic universe in an extended theory of gravity obtained geometrically by transforming the normal-gauge Lyra displacement vector field ${\varphi }^{\mu }$ as a complex vectorial function depending on a dynamical scalar field $\phi$. By using the latest observational data, we observe that for ${\omega }^0=-0.98$ the universe starts accelerating at the critical scale factor $a_{\text{ac}}=0.626$ which corresponds to a redshift of $z_{\text{ac}}\approx 0.6$. We also find that the dark energy fluid considered in this model is a generalized fluid with equation of state $p=-\rho +F(\rho ,a)$.

The local Lagrange–Poincaré equations (the reduced Euler–Lagrange equations) for the mechanical system describing the motion of a scalar particle on a finite-dimensional Riemannian manifold with a given free isometric smooth action of a compact semi-simple Lie group are obtained. The equations are written in terms of dependent coordinates which are used to represent the local dynamic given on the orbit space of the principal fiber bundle. The derivation of the equations is performed with the help of the variational principle developed by Poincaré for mechanical systems with symmetry.

Following our earlier work on the perturbative thermodynamic geometry of nonextensive quantum and classical gases [H. Mohammadzadeh, F. Adli and S. Nouri, Phys. Rev. E 94 (2016) 062118], we study $q$-generalized Bose–Einstein, Fermi–Dirac and classical statistics nonperturbatively. We define $q$-generalized polylogarithm functions and evaluate thermodynamics quantities such as internal energy and particle number. We construct the thermodynamic geometry of nonextensive Bose (Fermi) ideal gas and show that the thermodynamic curvature is positive(negative) in full physical range as the same as ordinary statistics. Also, we show that the thermodynamic geometry of nonextensive ideal classical gas is flat, similar to the ordinary one. Therefore, the nonextensive parameter does not change the nature of intrinsic statistical interactions. We argue that the nonextensive boson gas might be more stable than the boson gas due to conjectural interpretation of thermodynamic curvature. In the following, we extract the singular points of thermodynamic curvature of nonextensive Bose gas and relate it to the condensation. We evaluate some thermodynamic quantities such as heat capacities, compressibility and $q$-dependent phase transition temperature. We show that the heat capacity is not differentiable at critical temperature, $T_c^q$ which is reduced by increasing nonextensive parameter $q$. Moreover, the critical temperature and possibility of condensation is investigated for different values of nonextensive parameter in various dimensions.

We propose and investigate the modified Born–Infeld-type gravity model with the function $F(R)=[1-{(1-\beta R/\sigma )}^{\sigma }]/\beta$. At different values of the dimensionless parameter $\sigma$, the action is converted into some models including general relativity ($\sigma =1$), the Starobinsky model ($\sigma =2$), the exponential model of gravity ($\sigma =\infty$), and a model proposed in [S. I. Kruglov, Born–Infeld-like modified gravity, Int. J. Theor. Phys. 52 (2013) 2477–2484; S. I. Kruglov, Notes on Born–Infeld-like modified gravity, Astrophys. Space Sci. 361 (2016) 73] ($\sigma =0.5$). A bound on the parameters, $\beta (1-\sigma )/\sigma \le 2\times 10^{-6}$cm², are due to local tests. The constant curvature solutions are obtained, and we found the potential, and the mass of the scalaron in the Einstein frame. The model describes the inflation of the universe. The de Sitter spacetime is unstable and a Minkowski spacetime is stable. We investigate the cosmological parameters and some critical points of autonomous equations.

The problem of generating solutions of the Einstein field equations with an elastic energy–momentum tensor from the Schwarzschild vacuum solution by means of conformal transformations is analyzed. Applying the formulation of relativistic elasticity, suitable conformal factors are obtained for static and non-static elastic spacetime configurations and particular solutions are presented. This work shows that the technique used here permits generating new elastic matter solutions from a vacuum spacetime.

The class of biwarped product manifolds is a generalized class of product manifolds and a special case of multiply warped product manifolds. In this paper, biwarped product submanifolds of the type $N_T{\times }_{{\psi }_1}{N}_{\perp }{\times }_{{\psi }_2}{N}_{𝜃}$ embedded in the complex space forms are studied. Some characterizing inequalities for the existence of such type of submanifolds are derived. Moreover, we also estimate the squared norm of the second fundamental form in terms of the warping function and the slant function. This inequality generalizes the result obtained by Chen in [B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds I, Monatsh. Math. 133 (2001) 177–195]. By the application of derived inequality, we compute the Dirichlet energies of the warping functions involved. A nontrivial example of these warped product submanifolds is provided.

We find a characterization of a sphere using a compact gradient almost Ricci soliton and the lower bound on the integral of Ricci curvature in the direction of potential field. Also, we use Poisson equation on a compact gradient almost Ricci soliton to find a characterization of the unit sphere.

A study of the universal thermodynamics has been done for the inhomogeneous Friedmann–Lemaitre–Robertson–Walker type (FLRW-type) space-time model. Validity of the generalized second law of thermodynamics has been examined across an arbitrary horizon and integrability condition for the generalized Gibbs equation has been determined. Also, the conditions for thermodynamical equilibrium have been presented in a tabular form. Finally, using unified first law, the corrected Bekenstein entropy has been evaluated for both apparent and event horizons.

Fractional oscillator type equations are well-known model equations to describe several phenomenon in mathematical physics, engineering and biology. In this paper, a new method incorporated by the ultraspherical wavelet operational matrix of general order integration and block-pulse functions are adopted to investigate the solution of fractional oscillator type equations. To facilitate this, the ultraspherical wavelets are first presented and the corresponding operational matrix of fractional-order integration is derived by virtue of block pulse functions. The properties of ultraspherical wavelets and block pulse functions are used to transform the underlying problem to a system of algebraic equations which can be easily solved by any of the usual numerical methods. The efficiency and accuracy of the proposed method is demonstrated by presenting several benchmark test problems. Moreover, special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.

In this paper, we study harmonic evolutes of $B$-scrolls, that is, of ruled surfaces in Lorentz–Minkowski space having no Euclidean counterparts. Contrary to Euclidean space where harmonic evolutes of surfaces are surfaces again, harmonic evolutes of $B$-scrolls turn out to be curves. In particular, we show that the harmonic evolute of a $B$-scroll of constant mean curvature together with its base curve forms a null Bertrand pair. This allows us to characterize $B$-scrolls of constant mean curvature and reconstruct them from a given null curve which is their harmonic evolute.

Classical metrics (Eguchi–Hanson, Taub–NUT, Fubini–Study) satisfy a certain system of ODE that we introduced. By studying this system we found new Ricci-flat incomplete Riemannian metric in explicit formula and described its behavior at infinity and in the singular point.

Following the basic facts of $F$-planar mappings introduced by Mikeš and Sinyukov and further developed by Hinterleitner and Mikeš, we consider the notions of an $F$-planar curve and an $F$-planar mapping, but in case of manifolds endowed with a non-symmetric linear connection and an affinor structure. Consequently, we investigate infinitesimal $F$-planar and particularly geodesic transformations of manifolds with non-symmetric linear connection and find necessary and sufficient conditions for the existence of these transformations.

We find a background solution of the string theory which has a special property distinguished from the usual background solutions. This background solution does not produce the NS-NS two-form fields under T-duality and therefore the background vacua described by this solution essentially do not involve NS-NS type branes in their configurations, unlike the case of the ordinary Calabi–Yau ansatz. As a result the non-linear $\sigma$-models, whose target space metrics are given by these T-dual partners, can both be torsion-free.

An orthogonality space is a set endowed with a symmetric, irreflexive binary relation. By means of the usual orthogonality relation, each anisotropic quadratic space gives rise to such a structure. We investigate in this paper the question to which extent this strong abstraction suffices to characterize complex Hilbert spaces, which play a central role in quantum physics. To this end, we consider postulates concerning the nature and existence of symmetries. Together with a further postulate excluding the existence of nontrivial quotients, we establish a representation theorem for finite-dimensional orthomodular spaces over a dense subfield of $ℂ$.