International Journal of Geometric Methods in Modern Physics

In this paper, we obtain the effect of backreaction on the scale factor of the Friedmann–Lemaître–Robertson–Walker (FLRW) and de Sitter spaces. We consider a non-minimally coupled massive scalar field to the curvature scalar. For our purpose, we use the results of vacuum expectation values of energy–momentum tensor, which have been obtained previously. By substituting the quantum energy density into the Friedmann equation, we obtain the linear order perturbation of the scale factor. So, the effect of backreaction leads to the new scale factor.

This paper is concerned with the study of the geometry of (charged) Nariai spacetime, a topological product spacetime, by means of covariant derivative(s) of its various curvature tensors. It is found that on this spacetime the condition $\nabla R=0$ is satisfied and it also admits the pseudosymmetric type curvature conditions $C\cdot R=\frac{(1+L_0)}{6r_0^2}Q(g,R)$ and $P\cdot R=-\frac{1}{3}Q(S,R)$. Moreover, it is 4-dimensional Roter type, $2$-quasi-Einstein and generalized quasi-Einstein spacetime. The energy–momentum tensor is expressed explicitly by some 1-forms. It is worthy to see that a generalization of such topological product spacetime proposes to exist with a class of generalized recurrent type manifolds which is semisymmetric. It is observed that the rank of $(S-\alpha g)$, $\alpha =-1/r_0^2$, of Nariai spacetime (NS) is 0 whereas in case of charged Nariai spacetime (CNS) it is 2, which exhibits that effects of charge increase the rank of Ricci tensor. Also, due to the presence of charge in CNS, it gives rise to the proper pseudosymmetric type geometric structures.

In this paper, we have studied different energy conditions for some metrics arising from the classification of Kantowski–Sachs spacetimes via homothetic vector fields (HVFs). The homothetic equations for Kantowski–Sachs spacetimes are analyzed using a Maple algorithm which transforms these equations to the reduced involutive form (Rif), giving all possible cases in the form of a tree, known as Rif tree, where the branches of the resulting tree give the potential spacetime metrics which may possess proper HVFs. The set of homothetic equations is integrated for all branches of the Rif tree and the explicit form of HVFs is obtained in ease case. For all the obtained metrics, we have found the bounds for different energy conditions. The stability of these metrics is also discussed.

We study the thick brane model within $f(R)\sim R^n$ the modified gravity in $D$-dimensional spacetimes with $D\ge 6$. The system under consideration consists of two branes orthogonal to each other: the four-dimensional Lorentzian brane and $(D-5)$-dimensional Euclidean one. It is numerically shown that, for a given $D$, regular vacuum asymptotically anti-de Sitter solutions exist only in the range $1<n<D/2$. Depending on the values of $n$ and boundary conditions imposed on the Lorentzian brane, the solutions can pass or not pass through a fixed point located on the Lorentzian brane, and also be $Z_2$-symmetric or nonsymmetric. In the large-$D$ limit, we find the approximate analytic solutions. It is also shown that a test scalar field is trapped on the Lorentzian brane at any $D$.

We study the procedure that allows to rewrite the actions of some modified gravity theories like mimetic gravity and Regge–Teitelboim embedding theory as Einstein–Hilbert actions with additional matter contributions. It turns out that this procedure can be applied to brane action, which leads to a polynomial brane action. We also examine the connection between this procedure and Polyakov trick in string theory.

Lorentz symmetry is one of the fundamental symmetries of nature; however, it can be broken by several proposals such as quantum gravity effects, low energy approximations in string theory and dark matter. In this paper, Lorentz symmetry is broken in supersymmetric quantum electrodynamics using aether superspace formalism without breaking any supersymmetry. To break the Lorentz symmetry in three-dimensional quantum electrodynamics, we must use the $N=1$ aether superspace. A new constant vector field is introduced and used to deform the deformed generator of supersymmetry. This formalism is required to fix the unphysical degrees of freedom that arise from the quantum gauge transformation required to quantize this theory. By using Yokoyama’s gaugeon formalism, it is possible to study these gaugeon transformations. As a result of the quantum gauge transformation, the supersymmetric algebra gets modified and the theory is invariant under BRST symmetry. These results could aid in the construction of the Gravity’s Rainbow theory and in the study of superconformal field theory. Furthermore, it is demonstrated that different gauges in this deformed supersymmetric quantum electrodynamics can be related to each other using the gaugeon formalism.

It is shown that for a proper Generalized Cauchy–Riemann ($GCR$)-lightlike submanifold of an indefinite nearly Kaehler manifold such that $D_0$ defines a totally geodesic foliation in $M$, there does not exist any warped product $GCR$-lightlike submanifold of the type $M_{\perp }{\times }_{\lambda }{M}_T$. Then, the existence of $GCR$-lightlike warped product submanifolds of the type $M_T{\times }_{\lambda }{M}_{\perp }$ in indefinite nearly Kaehler manifolds is obtained by establishing a characterization in terms of the shape operator. Further, we prove that for a proper $GCR$-lightlike warped product submanifold of an indefinite nearly Kaehler manifold, the induced connection $\nabla$ can never be a metric connection. Finally, we derive some characterizations in terms of the canonical structures $f$ and $\omega$ on a $GCR$-lightlike submanifold of an indefinite nearly Kaehler manifold enabling it to be a $GCR$-lightlike warped product.

This paper systematically investigates the Lie symmetry analysis of the time-fractional Buckmaster equation in the sense of Riemann–Liouville fractional derivative. With the aid of infinitesimal symmetries, this equation is transformed into a nonlinear ordinary differential equation of fractional order (FODE), where the fractional derivatives are in Erdelyi–Kober sense. The reduced FODE is solved with the explicit power series method and some figures for the obtained power series solutions are also depicted. Finally, Ibragimov’s method and Noether’s theorem have been employed to conclude the conservation laws of this equation.

Here, using the projectively invariant pseudo-distance and Schwarzian derivative, it is shown that every connected complete Finsler space of the constant negative Ricci scalar is reversible. In particular, every complete Randers metric of constant negative Ricci (or flag) curvature is Riemannian.

We examine Hamiltonian analysis of three-dimensional advection flow $ẋ=v(x)$ of incompressible nature $\nabla \cdot v=0$ assuming that the dynamics is generated by the curl of a vector potential $v=\nabla \times A$. More concretely, we elaborate Nambu–Hamiltonian and bi-Hamiltonian characters of such systems under the light of vanishing or non-vanishing of the quantity $A\cdot \nabla \times A$. We present an example (satisfying $A\cdot \nabla \times A\ne 0$) which can be written as in the form of Nambu–Hamiltonian and bi-Hamiltonian formulations. We present another example (satisfying $A\cdot \nabla \times A=0$) which we cannot able to write it in the form of a Nambu–Hamiltonian or bi-Hamiltonian system while it can be manifested in terms of Hamiltonian one-form and yields generalized or vector Hamiltonian equations ${ẋ}_i=-{𝜖}_{ijk}\partial {\eta }_j/\partial x_k$.

In this paper, we have analyzed the complexity factor for the most general axially symmetric static anisotropic fluid distributions in context of $f(R)$ theory of gravity. For this purpose, we have studied three distinct complexity factors that are organized in terms of three scalar variables (structure scalars) comes from the orthogonal splitting of the curvature tensor. The vanishing of all complexity factors condition for what we choose the simplest fluid distribution that in which system having energy density is homogeneous with isotropic pressure. Although, it has been found that the complexity factors condition can also vanish when inhomogeneous energy density and anisotropy of the pressure cancel each other. Next, we express a class of exact solutions and their graphical analysis as compatible to our models that satisfies the vanishing condition of complexity factors. Finally, it is worth mentioning that these results can reproduce the results of General theory of Relativity under some constraints.

Nearly Kähler- and Kähler–Codazzi-type manifolds are defined in a very similar way. We prove that nearly Kähler-type manifolds make sense only in Hermitian and para-Hermitian contexts, and that Kähler–Codazzi-type manifolds reduce to Kähler-type manifolds in all the four Hermitian, para-Hermitian, Norden and product Riemannian geometries. Kähler–Codazzi condition is also studied on almost complex golden manifolds.

In this paper, we find the $q$-deformed algebra with the finite- and infinite-dimensional Fock space and both the fermionic limit and the bosonic limit. Using the cardinality of set theory, we propose the Hamiltonian interpolating bosonic case and fermionic case, which enables us to construct the proper partition function and internal energy. As examples, we discuss the specific heat of free $p=3$ parafermion gas model and $p=3$ parafermion star.

Using the gauged pseudo-Hermitian fuzzy Ginsparg–Wilson algebra, pseudo fuzzy Dirac and chirality operators on the fuzzy $EAd{S}_2$ have been studied. Also, the spectrum of the gauged pseudo fuzzy Dirac operator in the instanton sector has been studied.

Inspired by the low wave-length limit of topological M-theory, which re-constructs the theory of 3 + 1D gravity in the self-dual variables’ formulation, and by the realization that in Loop Quantum Gravity (LQG) the holonomy of a flat connection can be non-trivial if and only if a non-trivial (space-like) line defect is localized inside the loop, we argue that non-trivial gravitational holonomies can be put in correspondence with space-like M-branes. This suggests the existence of a new duality, which we call $H$ duality, interconnecting topological M-theory with LQG. We spell some arguments to show that fundamental S-strings are serious candidates to be considered in order to instantiate this correspondence to classes of LQG states. In particular, we consider the case of the holonomy flowers in LQG, and show that for this type of states the action of the Hamiltonian constraint, from the M-theory side, corresponds to a linear combination of appearance and disappearance of a SNS1-strings. Consequently, these processes can be reinterpreted, respectively, as enucleations or decays into open or closed strings.

We present investigation devoted to the dynamical study of relativistic hydrodynamics with some thermodynamical characteristics in $f(R,G)$ gravity towards spatially homogeneous isotropic cosmological model filled with isotropic fluid. We govern the features of the derived cosmological model by considering the power-law inflation for the average scale factor. The temperature and entropy density of the proposed model are positive definite. We also discuss the energy conditions to our solutions. The strong energy condition violated, which indicates the accelerated expansion of the proposed model.