International Journal of Geometric Methods in Modern Physics

Generalized Robertson–Walker spacetimes extend the notion of Robertson–Walker spacetimes, by allowing for spatial non-homogeneity. A survey is presented, with main focus on Chen's characterization in terms of a timelike concircular vector. Together with their most important properties, some new results are presented.

We show that almost complex manifolds $(M^4,J)$ of real dimension 4 for which the image of the Nijenhuis tensor forms a non-integrable bundle, called torsion bundle, admit a ${\mathbb{Z}}_2$-structure locally, that is, a double absolute parallelism. In this way, the problem of equivalence for such almost complex manifolds can be solved; moreover, the classification of locally homogeneous manifold $(M^4,J)$ is explicitly given when the Lie algebra of its infinitesimal automorphisms is non-solvable (indeed reductive). It is also shown that the group of the automorphisms of $(M^4,J)$ is a Lie group of dimension less than or equal to 4, whose isotropy subgroup has at most two elements, and that there are not non-constant holomorphic functions on $(M^4,J)$.

We obtain the expressions for sectional curvature, holomorphic sectional curvature and holomorphic bisectional curvature of a GCR-lightlike submanifold of an indefinite Sasakian manifold and obtain some characterization theorems on holomorphic sectional and holomorphic bisectional curvature.

In this work, we investigate the cosmological application of modified Chaplygin gas (MCG) interacting with pressureless dark matter (DM) in the $f(T)$ modified gravity framework, where $T$ is the torsion scalar in teleparallelism. The interaction term has been chosen proportional to the MCG density with positive coupling constant. In the Einstein general relativity (GR) framework, the interacting MCG has been found to have equation of state (EoS) parameter behaving like quintessence. However, the $f(T)$ gravity reconstructed via the interacting MCG has been found to have EoS crossing the phantom boundary of $-1$. Thus, one can generate a quintom-like EoS from an interacting MCG model in flat universe in the modified gravity cosmology framework. The reconstructed $f(T)$ model has been found to interpolate between dust and $\mathrm{\Lambda }$CDM. Stability of the reconstructed $f(T)$ has been investigated and it has been observed that the model is stable against gravitational perturbation. Cosmological evolution of primordial perturbations has also been investigated and the self-interacting potential has been found to increase with cosmic time and the squared speed of sound has been found to be non-negative.

The aim of this paper is to present a new conceptual approach towards the well-known Schmidt's b-boundary of spacetime. Instead of working in the category of infinitely differentiable manifolds (called also smooth manifolds), we switch to the category of Sikorski differential spaces. Yet, such an approach was developed some time ago, but in this paper, we develop it further. In particular, we mix it with algebraic approach á la Nestruev. It lead us in a natural way towards the new concept of so called generator boundary (in short: gen-boundary). Surprisingly, such a study in the context of cosmology has not been done yet. We provide a through explanation of jointed Nestruev and Sikorski's formalisms. Further, as a result, the concept of gen-boundary emerges. We discuss its properties. Afterwards, having the developed techniques at hand, we re-discuss the Schmidt's b-boundary in this new (more algebraic) language. Interestingly, a classification of spacetime singularities naturally emerges from the proposed formalism.

In this paper, we will take into account the most complete background with torsion and curvature, providing the most exhaustive coupling for the Dirac field, we will discuss the integrability of the interaction of the matter field and the reduction of the matter field equations.

Canonical formulation of higher order theory of gravity requires to fix (in addition to the metric), the scalar curvature, which is acceleration in disguise, at the boundary. On the contrary, for the same purpose, Ostrogradski's or Dirac's technique of constrained analysis, and Horowit'z formalism, tacitly assume velocity (in addition to the co-ordinate) to be fixed at the end points. In the process when applied to gravity, Gibbons–Hawking–York term disappears. To remove such contradiction and to set different higher order theories on the same footing, we propose to fix acceleration at the endpoints/boundary. However, such proposition is not compatible to Ostrogradski's or Dirac's technique. Here, we have modified Horowitz's technique of using an auxiliary variable, to establish a one-to-one correspondence between different higher order theories. Although, the resulting Hamiltonian is related to the others under canonical transformation, we have proved that this is not true in general. We have also demonstrated how higher order terms can regulate the issue of branched Hamiltonian.

The notion of a parallelizable distribution has been introduced and investigated. A non-integrable parallelizable distribution carries a natural sub-Riemannian structure. The geometry of this structure has been studied from the bi-viewpoint of absolute parallelism geometry and sub-Riemannian geometry. Two remarkable linear connections have been constructed on a sub-Riemannian parallelizable distribution, namely, the Weitzenböck connection and the sub-Riemannian connection. The obtained results have been applied to two concrete examples: the spheres $S^3$ and $S^7$.

We present a mathematical model for physical systems. A large class of functions is built through the functional quantization method and applied to the geometric study of the model. Quantized equations of motion along the Hamiltonian vector field are built up. It is seen that the procedure in higher dimension carries more physical information. The metric tensor appears to induce an electromagnetic field into the system and the dynamical nature of the electromagnetic field in curved space arises naturally. In the end, an explicit formula for the curvature tensor in the quantized space is given.

Sign changeable interactions continue to gain serious attention in modern cosmology. On the other hand, recent interest towards nonlinear interactions gave us a motivation to consider various sign changeable nonlinear interactions. In this paper, we will consider cosmological models involving suggested new sign changeable nonlinear interactions. To describe background dynamics of the large scale universe $f(T)$ gravity is used i.e the universe is governed by torsional modified gravity. In our models, we adopted barotropic EoS and pressureless fluid to model dark energy and dark matter, respectively. In addition to other assumptions about the models, particular form of $f(T)$ is considered to perform phase space analysis.

It is known from [K. Yano and M. Kon, Structures on Manifolds (World Scientific, 1984)] that the integration of the Laplacian of a smooth function defined on a compact orientable Riemannian manifold without boundary vanishes with respect to the volume element. In this paper, we find out the some potential applications of this notion, and study the concept of warped product pointwise semi-slant submanifolds in cosymplectic manifolds as a generalization of contact CR-warped product submanifolds. Then, we prove the existence of warped product pointwise semi-slant submanifolds by their characterizations, and give an example supporting to this idea. Further, we obtain an interesting inequality in terms of the second fundamental form and the scalar curvature using Gauss equation and then, derive some applications of it with considering the equality case. We provide many trivial results for the warped product pointwise semi-slant submanifolds in cosymplectic space forms in various mathematical and physical terms such as Hessian, Hamiltonian and kinetic energy, and generalize the triviality results for contact CR-warped products as well.

The aim of this paper is to explore teleparallel conformal Killing vector fields (CKVFs) of locally rotationally symmetric (LRS) Bianchi type V spacetimes in the context of teleparallel gravity and compare the obtained results with those of general relativity (GR). The general solution of teleparallel conformal Killing's equations is found in terms of some unknown functions of $t$ and $x$, along with a set of integrability conditions. The integrability conditions are solved in some particular cases to get the final form of teleparallel CKVFs. It is observed that the LRS Bianchi type V spacetimes admit proper teleparallel CKVF in only one case, while in remaining cases the teleparallel CKVFs reduce to teleparallel Killing vector fields (KVFs). Moreover, it is shown that the LRS Bianchi type V spacetimes do not admit any proper teleparallel homothetic vector field (HVF).

The Page metric on $ℂ{\mathbb{P}}^2\#\overline{ℂ{\mathbb{P}}^2}$ is a cohomogeneity one Einstein–Riemannian metric, and is the only known cohomogeneity one Einstein–Riemannian metric on compact $4$-manifolds. It has been a long standing problem whether there exists another cohomogeneity one Einstein–Riemannian metric on $4$-manifolds. In this paper, we construct some examples of cohomogeneity one Einstein–Randers metrics on simply connected 4-manifolds. This shows that, although cohomogeneity one Einstein–Riemmian 4-manifolds are rare, non-Riemannian examples may exist at large.

The concept of quasi-generalized CR-lightlike was first introduced by the authors in [Quasi generalized CR-lightlike submanifolds of indefinite nearly Sasakian manifolds, Arab. J. Math. 5 (2016) 87–101]. In this paper, we focus on ascreen and co-screen quasi-generalized CR-lightlike submanifolds of indefinite nearly $\mu$-Sasakian manifold. We prove an existence theorem for minimal ascreen quasi-generalized CR-lightlike submanifolds admitting a metric connection. Classification theorems on nearly parallel and auto-parallel distributions on a co-screen quasi-generalized CR-lightlike submanifold are also given. Several examples are also constructed, where necessary, to illustrate the main ideas.

In this paper, the Lie approximate symmetry analysis is applied to investigate new solutions of the nonlinear Klein–Gordon equation with a small parameter. The nonlinear Klein–Gordon equation is used to model many nonlinear phenomena. The hyperbolic function method and Riccati equation method are employed to solve some of the obtained reduced ordinary differential equations. We construct new analytical solutions with a small parameter which is effectively obtained by the proposed method.

In this paper, we consider a manifold with a dynamical vector field and enquire about the possible tangent bundle structures which would turn the starting vector field into a second-order one. The analysis is restricted to manifolds which are diffeomorphic with affine spaces. In particular, we consider the problem in connection with conformal vector fields of second-order and apply the procedure to vector fields conformally related with the harmonic oscillator ($f$-oscillators). We select one which covers the vector field describing the Kepler problem.