International Journal of Geometric Methods in Modern Physics

Cosmography represents an important branch of cosmology which aims to describe the universe without the need of postulating a priori any particular cosmological model. All quantities of interest are expanded as a Taylor series around here and now, providing in principle, a way of directly matching with cosmological data. In this way, cosmography can be regarded a model-independent technique, able to fix cosmic bounds, although several issues limit its use in various model reconstructions. The main purpose of this review is to focus on the key features of cosmography, emphasizing both the strategy for obtaining the observable cosmographic series and pointing out any drawbacks which might plague the standard cosmographic treatment. In doing so, we relate cosmography to the most relevant cosmological quantities and to several dark energy models. We also investigate whether cosmography is able to provide information about the form of the cosmological expansion history, discussing how to reproduce the dark fluid from the cosmographic sound speed. Following this, we discuss limits on cosmographic priors and focus on how to experimentally treat cosmographic expansions. Finally, we present some of the latest developments of the cosmographic method, reviewing the use of rational approximations, based on cosmographic Padé polynomials. Future prospects leading to more accurate cosmographic results, able to better reproduce the expansion history of the universe, are also discussed in detail.

This is a survey of the principal results about the geodesic completeness of nondegenerate hypersurfaces in Lorentzian manifolds from a structural point of view. Some of these results retain their validity in the case of semi-Riemannian submanifolds in semi-Euclidean spaces, as well.

Let $\alpha (s)$ be an arc on a connected oriented surface $S$ in $E^3$, parameterized by arc length $s$, with torsion $\tau$ and length $l$. The total square torsion $H$ of $\alpha$ is defined by $H={\int }_0^l{\tau }^{2}ds$. The arc $\alpha$ is called a relaxed elastic line of second kind if it is an extremal for the variational problem of minimizing the value of $H$ within the family of all arcs of length $l$ on $S$ having the same initial point and initial direction as $\alpha$. In this study, we obtain differential equation and boundary conditions for a relaxed elastic line of second kind on an oriented surface.

First we show that, associated to any Poisson vector field $E$ on a Poisson manifold $(M,\pi )$, there is a canonical Lie algebroid structure on the first jet bundle $J^{1}M$ which, depends only on the cohomology class of $E$. We then introduce the notion of a cosymplectic groupoid and we discuss the integrability of the first jet bundle into a cosymplectic groupoid. Finally, we give applications to Atiyah classes and $L\infty$-algebras.

We construct a tangent bundle exponential map and locally autoparallel coordinates for geometries based on a general connection on the tangent bundle of a manifold. As concrete application we use these new coordinates for Finslerian geometries and obtain Finslerian geodesic coordinates. They generalize normal coordinates known from metric geometry to Finsler geometric manifolds and it turns out that they are identical to the Douglas–Thomas normal coordinates introduced earlier. We expand the Finsler Lagrangian of a Finsler spacetime in these new coordinates and find that it is constant to quadratic order. The quadratic order term comes with the nonlinear curvature of the manifold. From physics these coordinates may be interpreted as the realisation of an Einstein elevator in Finslerian spacetime geometries.

Motivated by the study of the interrelation between functorial and algebraic quantum field theory (AQFT), we point out that on any locally trivial bundle of compact groups, representations up to homotopy are enough to separate points by means of the associated representations in cohomology. Furthermore, we observe that the derived representation category of any compact group is equivalent to the category of ordinary (finite-dimensional) representations of the group.

We provide a generalization of the approach to geometric probability advanced by the great mathematician Gian Carlo Rota, in order to apply it to generalized probabilistic physical theories. In particular, we use this generalization to provide an improvement of the Jaynes’ MaxEnt method. The improvement consists in providing a framework for the introduction of symmetry constraints. This allows us to include group theory within MaxEnt. Some examples are provided.

In this paper, we will present Lagrangian and Hamiltonian $k$-symplectic formalisms, we will recall the notions of symmetry and conservation law and we will define the notion of pseudosymmetry as a natural extension of symmetry. Using symmetries and pseudosymmetries, without the help of a Noether type theorem, we will obtain new kinds of conservation laws for $k$-symplectic Hamiltonian systems and $k$-symplectic Lagrangian systems.

We investigate in this paper the possibilities that the observed cold dark matter density can be generated by decays of a heavy scalar field which dominate the universe at the quantum regime. Indeed, we present two approaches based on an extension of quantum field theory to the case when spacetime topology fluctuates (spacetime foam, at the quantum regime). In this extension the number of bosonic fields becomes a variable and the ground state is characterized by a finite particle number density. In the second approach it is the gauge-group parameters which became dynamical. This is tributary on the Centrally Extended Group and Cohomology.

The symmetric $q$-analysis is used to construct a type of minimum-uncertainty $q$-coherent states in the Fock representation space of the symmetric $q$-oscillator ∗-algebra with $0<q<1$. Then, its corresponding $q$-Hermite polynomials are derived by using the $q$-Bargmann–Fock realization of the symmetric $q$-oscillator algebra.

We construct a two-parameter bicovariant differential calculus on ${{ℛ}}_q^{1/2}$ with the help of the covariance point of view using the Hopf algebra structure of ${{ℛ}}_q^{1/2}$. To achieve this, we first use the consistency of calculus and the approach of $R$-matrix which satisfies both ungraded and graded Yang–Baxter equations. In particular, based on this differential calculus, we investigate Cartan–Maurer forms for this $q$-superspace. Finally, we obtain the quantum Lie superalgebra corresponding the Cartan–Maurer forms.

The aim of this paper is to investigate teleparallel conformal Killing vector fields (CKVFs) in plane symmetric non-static spacetimes. Ten teleparallel conformal Killing’s equations are obtained which are linear in the components of the teleparallel CKVF $X$. A general solution of these equations comprising the components of CKVF and conformal factor is presented, which subject to some integrability conditions. For seven particular choices of the metric functions, the integrability conditions are completely solved to get the final form of teleparallel CKVFs and conformal factor. In four different cases we get proper CKVFs, while in the remaining three cases it is shown that teleparallel CKVFs reduce to teleparallel homothetic or teleparallel Killing vector fields.

A global vector field $v$ on a “spacetime” differentiable manifold $\mathrm{\text{V}}$, of dimension $N+1$, defines a congruence of world lines: the maximal integral curves of $v$, or orbits. The associated global space ${\mathrm{\text{N}}}_v$ is the set of these orbits. A “$v$-adapted” chart on $\mathrm{\text{V}}$ is one for which the ${ℝ}^N$ vector $x\equiv (x^j)(j=1,\dots ,N)$ of the “spatial” coordinates remains constant on any orbit $l$. We consider non-vanishing vector fields $v$ that have non-periodic orbits, each of which is a closed set. We prove transversality theorems relevant to such vector fields. Due to these results, it can be considered plausible that, for such a vector field, there exists in the neighborhood of any point $X\in \mathrm{\text{V}}$ a chart $\chi$ that is $v$-adapted and “nice”, i.e. such that the mapping $\bar{\chi }:l\mapsto x$ is injective — unless $v$ has some “pathological” character. This leads us to define a notion of “normal” vector field. For any such vector field, the mappings $\bar{\chi }$ build an atlas of charts, thus providing ${\mathrm{\text{N}}}_v$ with a canonical structure of differentiable manifold (when the topology defined on ${\mathrm{\text{N}}}_v$ is Hausdorff, for which we give a sufficient condition met in important physical situations). Previously, a local space manifold ${\mathrm{\text{M}}}_{\mathrm{\text{F}}}$ had been associated with any “reference frame” $\mathrm{\text{F}}$, defined as an equivalence class of charts. We show that, if $\mathrm{\text{F}}$ is made of nice $v$-adapted charts, ${\mathrm{\text{M}}}_{\mathrm{\text{F}}}$ is naturally identified with an open subset of the global space manifold ${\mathrm{\text{N}}}_v$.

In this paper, we consider Yang–Mills theory on manifolds $ℝ\times X$ with a $d$-dimensional Riemannian manifold $X$ of special holonomy admitting gauge instanton equations. Instantons are considered as particle-like solutions in $d+1$ dimensions whose static configurations are concentrated on $X$. We study how they evolve in time when considered as solutions of the Yang–Mills equations on $ℝ\times X$ with moduli depending on time $t\in ℝ$. It is shown that in the adiabatic limit, when the metric in the $X$ direction is scaled down, the classical dynamics of slowly moving instantons corresponds to a geodesic motion in the moduli space ${ℳ}$ of gauge instantons on $X$. Similar results about geodesic motion in the moduli space of monopoles and vortices in higher dimensions are briefly discussed.

We investigate how the gravitational baryogenesis mechanism can potentially constrain the form of a Type IV singularity. Specifically, we study two different models with interesting phenomenology, that realize two distinct Type IV singularities, one occurring at the end of inflation and one during the radiation domination era or during the matter domination era. As we demonstrate, the Type IV singularities occurring at the matter domination era or during the radiation domination era are constrained by the gravitational baryogenesis, in such a way so that these do not render the baryon to entropy ratio singular. Both the cosmological models we study cannot be realized in the context of ordinary Einstein–Hilbert gravity, and hence our work can only be realized in the context of $F(R)$ gravity and more generally in the context of modified gravity only.

Further gravitational massive modes emerge by extending the geometrical sector of Hilbert–Einstein theory in the most general theory including curvature invariants. Besides massless spin-2, also spin-0 and spin-2 massive and ghost fields have to be considered. We investigate the possible detectability of such additional modes by the Large Hadron Collider and calculate the detectable energy density of the spectrum.

The paper is devoted to the algebraic background of Weil contact elements: in particular, we investigate an eventual absence of orientation reversing maps by the description of Weil algebras having their group of automorphisms with only one connected component.