International Journal of Geometric Methods in Modern Physics

We discuss a gauge invariant gravity model in a non-Riemannian geometry in which the curvature and the torsion both are zero, the nonmetricity is nonzero. We also argue that only a metric ansatz is enough to start finding solutions to the field equations. As an application, we obtain explicitly a conformally flat solution.

Let $M$ be a smooth manifold and ${𝒮}$ a semi-spray defined on a sub-bundle ${𝒞}$ of the tangent bundle $TM$. In this work, it is proved that the only non-trivial $k$-jet approximation to the exact geodesic deviation equation of ${𝒮}$, linear on the deviation functions and invariant under an specific class of local coordinate transformations, is the Jacobi equation. However, if the linearity property on the dependence in the deviation functions is not imposed, then there are differential equations whose solutions admit $k$-jet approximations and are invariant under arbitrary coordinate transformations. As an example of higher-order geodesic deviation equations, we study the first- and second-order geodesic deviation equations for a Finsler spray.

We study effects of anisotropy (although low) on the ghost and generalized ghost dark energy (DE) models in the framework of fractal cosmology. We obtain the equation of state parameter, ${\omega }_{\mathrm{\text{DE}}}$, the deceleration parameter, and the evolution equation of the ghost and generalized ghost dark energy. We find that, in both models, ${\omega }_{\mathrm{\text{DE}}}$ cannot cross the phantom line and eventually the universe approaches a de-Sitter phase of expansion. We show that the anisotropy effects on ghost and generalized ghost dark energy (GDE) in fractal cosmology correspond to $\mathrm{\Lambda }$CDM limit on the statefinder plane. We evaluate the anisotropy effects on both the linear perturbation and the spherical collapse from the DE models and compare them with the results of the DE of the Friedmann–Robertson–Walker and $\mathrm{\Lambda }$CDM models. We also show that in ghost and generalized ghost cosmologies, the growth factor $g(a)$ rise front the values for an $\mathrm{\Lambda }$CDM universe. Finally, we constrain the model parameters by using the maximum likelihood analysis and a combined dataset of baryon acoustic oscillation (BAO) and OHD.

In our paper, we introduce and study lightlike hypersurfaces of a metallic semi-Riemannian manifold. We examine some geometric properties of invariant lightlike hypersurfaces. We show that the induced structure on an invariant lightlike hypersurface is also metallic. We also define screen semi-invariant lightlike hypersurfaces, investigate integrability conditions for the distributions and give some examples.

Over the $(1,1)$-dimensional supercircle $S^{1|1}$, we investigate the first $𝔬𝔰𝔭(1|2)$-relative cohomology space associated with the embedding of the Lie superalgebra ${𝒦}(1)$ of contact vector fields in the Lie superalgebra ${𝒮}\mathrm{\Psi }{𝒟}{𝒪}(S^{1|1})$ of superpseudodifferential operators with smooth coefficients, where $𝔬𝔰𝔭(1|2)$ is the orthosymplectic Lie superalgebra. Likewise, we study the same problem for the affine Lie superalgebra $𝔞𝔣𝔣(1|1)$ instead of $𝔬𝔰𝔭(1|2)$. We classify $𝔞𝔣𝔣(1|1)$-trivial deformations of the standard embedding of the Lie superalgebra ${𝒦}(1)$ into the Lie superalgebra ${𝒮}\mathrm{\Psi }{𝒟}{𝒪}(S^{1|1})$. This approach leads to the deformations of the central charge induced on ${𝒦}(1)$ by the canonical central extension of ${𝒮}\mathrm{\Psi }{𝒟}{𝒪}(S^{1|1})$.

Over the $(1,2)$-dimensional supercircle $S^{1|2}$, we investigate the first $𝔬𝔰𝔭(2|2)$-relative cohomology space associated with the embedding of the Lie superalgebra ${𝒦}(2)$ of contact vector fields in the Lie superalgebra ${𝒮}\mathrm{\Psi }{𝒟}{𝒪}(S^{1|2})$ of superpseudodifferential operators with smooth coefficients, where $𝔬𝔰𝔭(2|2)$ is the orthosymplectic Lie superalgebra. Likewise, we study the same problem for the affine Lie superalgebra $𝔞𝔣𝔣(2|1)$ instead of $𝔬𝔰𝔭(2|2)$. We classify generic formal $𝔬𝔰𝔭(2|2)$-trivial deformations of the ${𝒦}(2)$-module structure on the superspace of the supercommutative algebra ${𝒮}{𝒫}(2)$ of pseudodifferential symbols on $S^{1|2}$.

Beginning with a review of the existing literature on the computation of spectral distances on noncommutative spaces like Moyal plane and fuzzy sphere, adaptable to Hilbert–Schmidt operatorial formulation, we carry out a correction, revision and extension of the algorithm provided in [1] i.e. [F. G. Scholtz and B. Chakraborty, J. Phys. A, Math. Theor. 46 (2013) 085204] to compute the finite Connes’ distance between normal states. The revised expression, which we provide here, involves the computation of the infimum of an expression which involves the “transverse” $(\mathrm{\Delta }{\rho }_{\perp })$ component of the algebra element in addition to the “longitudinal” component $\mathrm{\Delta }\rho$ of [1], identified with the difference of density matrices representing the states, whereas the expression given in [1] involves only $\mathrm{\Delta }\rho$ and corresponds to the lower bound of the distance. This renders the revised formula less user-friendly, as the determination of the exact transverse component for which the infimum is reached remains a nontrivial task, but under rather generic conditions it turns out that the Connes’ distance is proportional to the Hilbert-Schmidt norm of $\mathrm{\Delta }\rho$, leading to considerable simplification. In addition, we can determine an upper bound of the distance by emulating and adapting the approach of [P. Martinetti and L. Tomassini, Commun. Math. Phys. 323 (2013) 107–141]. We then look for an optimal element for which the upper bound is reached. We are able to find one for the Moyal plane through the limit of a sequence obtained by finite-dimensional projections of the representative of an element belonging to a multiplier algebra, onto the subspaces of the total Hilbert space, occurring in the spectral triple and spanned by the eigen-spinors of the respective Dirac operator. This is in contrast with the fuzzy sphere, where the upper bound, which is given by the geodesic of a commutative sphere, is never reached for any finite $n$-representation of $SU(2)$. Indeed, for the case of maximal noncommutativity ($n=1/2$), the finite distance is shown to coincide exactly with the above-mentioned lower bound, with the transverse component playing no role. This, however, starts changing from $n=1$ onwards and we try to improve the estimate of the finite distance and provide an almost exact result, using our revised algorithm. The contrasting features of these types of noncommutative spaces becomes quite transparent through the analysis, carried out in the eigen-spinor bases of the respective Dirac operators.

IK-normal complex contact metric manifolds have some important properties. There are several applications of this kind of contact manifolds in theoretical physics. In this paper, we studied on $H$-curvature tensors for IK-normal complex contact metric manifolds. We have shown that there is no IK-normal complex contact metric manifold with constant sectional curvature and an IK-normal complex contact metric manifold is not Ricci semi-symmetric.

Given an ascending sequence of weak symplectic Banach manifolds on which the Darboux Theorem is true, we can ask about conditions under which the Darboux Theorem is also true on the direct limit. We will show that, in general, without very strong conditions, the answer is negative. In particular, we give an example of an ascending symplectic Banach manifolds on which the Darboux Theorem is true but not on the direct limit. In the second part, we illustrate this discussion in the context of an ascending sequence of Sobolev manifolds of loops in symplectic finite-dimensional manifolds. This context gives rise to an example of direct limit of weak symplectic Banach manifolds on which the Darboux Theorem is true around any point.

We define the notion of tensor product of supersymmetric $N=(1,1)$ spectral data in the context of supersymmetric quantum theory and noncommutative geometry. We explain in which sense our definition is canonical and also establish its compatibility with the tensor product of $N=1$ spectral data defined earlier by Connes. As an application, we show that the unitary connections on the individual $N=(1,1)$ spectral data give rise to a unitary connection on the product $N=(1,1)$ spectral data.

In this paper, we present the modified Fermi–Walker parallel transport, the generalized Fermi–Walker parallel transport and related phases according to Frenet and Darboux frame in Minkowski 3-space.

In this paper, we explore the equivalence between two theories, namely $f(R)$ and scalar–tensor theories of gravity. We use this equivalence to explore several $f(R)$ toy models focusing on the inflation epoch of the early universe. The study is done based on the definition of the scalar field in terms of the first derivative of $f(R)$ model. We have applied the slow-roll approximations during inflationary parameters consideration. The comparison of the numerically computed inflationary parameters with the observations is done. We have inspected that some of the $f(R)$ models produce numerical values of $n_s$ that are in the same range as the suggested values from observations. But for the case of the tensor-to-scalar ratio $r$, we realized that some of the considered $f(R)$ models suffer to produce a value which is in agreement with the observed values for different considered space parameter.

Taking into account various energy density models in the framework of torsion-based modified theory, some wormhole solutions are derived and studied. We use a non-minimal coupling between matter and torsion in order to construct the wormhole geometry for constant, exponential and Lorentzian energy density models. Two sets of torsion dependent models, including (i) teleparalell case with linear forms of torsion scalar and (ii) a ${ℱ}(T)$ gravity with quadratic form of torsion scalar, have been considered throughout this paper. Our study shows that additional terms to the Einstein field equations, due to consider modified gravity, give us the opportunity of obtaining wormhole solutions which may be supported by baryonic sources. We also check the equilibrium phase of the wormhole solutions using anisotropic and hydrostatic forces. The details of the obtained results depend on the numerical values of the free parameters that the function ${ℱ}(T)$ holds.

We develop the concept of a double (more generally $n$-tuple) principal bundle departing from a compatibility condition for a principal action of a Lie group on a groupoid.

In this paper we will study the cosmological dynamical system of an $f(R)$ gravity in the presence of a canonical scalar field $\varphi$ with an exponential potential by constructing the dynamical system in a way that it is rendered autonomous. This feature is controlled by a single variable $m$, which when it is constant, the dynamical system is autonomous. We focus on the $m=0$ case which, as we demonstrate by using a numerical analysis approach, leads to an unstable de Sitter attractor, which occurs after $N\sim 60$ $e$-foldings. This instability can be viewed as a graceful exit from inflation, which is inherent to the dynamics of de Sitter attractors.