Absolute Parallelism Research Articles

”Let Man Remain Dead:” The Posthuman Ecology of Tale of Tales

Abstract In this essay I analyse Matteo Garrone’s Tale of Tales (2015) within the perspective of embodied cognition. I consider film experience as an affective-conceptual phenomenon based on the viewer’s embodiment of the visual structures. Baruch Spinoza stands at the foundation of my analytical approach since his thought was based on the absolute parallelism between the body and the mind. This paradigm redefines anthropocentrism and rejects dualism; however, the criticism of the rationalist ideal is also one of the main characteristics of the film Tale of Tales: by staging baroque and excessive characters, it allows the viewer to embody a notion of subjectivity that is performative and relational. Therefore, by combining the cognitive analysis of the film with my theoretical framework I will present a radical criticism of abstract rationality and present an ecological idea of the human.

On the stability of flat complex vector bundles over parallelizable manifolds

We investigate the flat holomorphic vector bundles over compact complex parallelizable manifolds G/Γ, where G is a complex connected Lie group and Γ is a cocompact lattice in it. The main result proved here is a structure theorem for flat holomorphic vector bundles Eρ associated with any irreducible representation ρ:Γ⟶GL(r,C). More precisely, we prove that Eρ is holomorphically isomorphic to a vector bundle of the form E⊕n, where E is a stable vector bundle. All the rational Chern classes of E vanish, in particular, its degree is zero.We deduce a stability result for flat holomorphic vector bundles Eρ of rank 2 over G/Γ. If an irreducible representation ρ:Γ⟶GL(2,C) satisfies the condition that the induced homomorphism Γ⟶PGL(2,C) does not extend to a homomorphism from G, then Eρ is proved to be stable.

The Spinning Equations of Motion for Objects in AP-Geometry

Equations of spinning objects are obtained in Absolute Parallelism Geometry [AP], a special class of non-Riemannian geometry admitting an alternative non-vanishing curvature and torsion simultaneously. This new set of equations is the counterpart of the Papapetrou equations in the Riemannian geometry. Applying, the concept of geometerization of physics, it may give rise to describe the spin tensor as parameterized commutation relation between path and path deviation equations in both Riemannian and non-Riemannian geometries.

Factorization homology I: Higher categories

We construct a pairing, which we call factorization homology, between framed manifolds and higher categories. The essential geometric notion is that of a vari-framing of a stratified manifold, which is a framing on each stratum together with a coherent system of compatibilities of framings along links between strata. Our main result constructs labeling systems on disk-stratified vari-framed n-manifolds from (∞,n)-categories. These (∞,n)-categories, in contrast with the literature to date, are not required to have adjoints. This allows the following conceptual definition: the factorization homology∫MC of a framed n-manifold M with coefficients in an (∞,n)-category C is the classifying space of C-labeled disk-stratifications over M. The core calculation underlying our main result is the following: for any disk-stratified manifold, the space of conically smooth diffeomorphisms which preserve a vari-framing is discrete.

Structure formation in f(T) gravity and a solution for H0 tension

We investigate the evolution of scalar perturbations in f(T) teleparallel gravity and its effects on the cosmic microwave background (CMB) anisotropy. The f(T) gravity generalizes the teleparallel gravity which is formulated on the Weitzenböck spacetime, characterized by the vanishing curvature tensor (absolute parallelism) and the non-vanishing torsion tensor. For the first time, we derive the observational constraints on the modified teleparallel gravity using the CMB temperature power spectrum from Planck's estimation, in addition to data from baryonic acoustic oscillations (BAO) and local Hubble constant measurements. We find that a small deviation of the f(T) gravity model from the ΛCDM cosmology is slightly favored. Besides that, the f(T) gravity model does not show tension on the Hubble constant that prevails in the ΛCDM cosmology. It is clear that f(T) gravity is also consistent with the CMB observations, and undoubtedly it can serve as a viable candidate amongst other modified gravity theories.

Explicit Absolute Parallelism for 2-Nondegenerate Real Hypersurfaces $$M^5 \subset \mathbb {C}^3$$ of Constant Levi Rank 1

We study the local equivalence problem under biholomorphisms for five-dimensional real hypersurfaces $$M^5$$ of $$\mathbb {C}^3$$ which are 2-nondegenerate and of constant Levi rank 1. We find exactly two invariants, J and W, which are expressed explicitly in terms of the graphing function F of M, the vanishing of which gives a necessary and sufficient condition for M to be locally biholomorphic to a model hypersurface, the tube over the light cone. If one of the two invariants J or W does not vanish on M, we show that the equivalence problem under biholomorphisms reduces to an equivalence problem between $$\{e \}$$-structures, that is, we construct an absolute parallelism on M.

Initial singularity and pure geometric field theories

In the present article we use a modified version of the geodesic equation, together with a modified version of the Raychaudhuri equation, to study initial singularities. These modified equations are used to account for the effect of the spin-torsion interaction on the existence of initial singularities in cosmological models. Such models are the results of solutions of the field equations of a class of field theories termed pure geometric. The geometric structure used in this study is an absolute parallelism structure satisfying the cosmological principle. It is shown that the existence of initial singularities is subject to some mathematical (geometric) conditions. The scheme suggested for this study can be easily generalized.

New conformal invariants in absolute parallelism geometry

The aim of the present paper is to investigate conformal changes in absolute parallelism geometry. We find out some new conformal invariants in terms of the Weitzenböck connection and the Levi-Civita connection of an absolute parallelism space.

Is Teleparallel Gravity Really Equivalent to General Relativity?

AbstractAn axiomatization of the so‐called Teleparallel Equivalent to General Relativity is presented. A set of formal and semantic postulates are elaborated from where the physical meaning of various key concepts of the theory are clarified. These concepts include those of inertia, Lorentz and diffeomorphism invariance, and reference frame. It is shown that Teleparallel Gravity admits a wider representation of space‐time than General Relativity, allowing to define properties of the gravitational field such as energy and momentum that are usually considered problematic. In this sense, although the dynamical equations of both theories are equivalent, their inequivalence from a physical point of view is demonstrated. Finally, the axiomatic formulation is used to compare Teleparallel Gravity with other theories of gravity based on absolute parallelism such as non‐local and f(T) gravity.

Timelike duality, M\u2032-theory and an exotic form of the Englert solution

Through timelike dualities, one can generate exotic versions of M-theory with different spacetime signatures. These are the M*-theory with signature (9, 2, −), the M′-theory, with signature (6, 5, +) and the theories with reversed signatures (1, 10, −), (2, 9, +) and (5, 6, −). In (s, t, ±), s is the number of space directions, t the number of time directions, and ± refers to the sign of the kinetic term of the 3 form.The only irreducible pseudo-riemannian manifolds admitting absolute parallelism are, besides Lie groups, the seven-sphere S7 ≡ SO(8)/SO(7) and its pseudo-riemannian version S3,4 ≡ SO(4, 4)/SO(3, 4). [There is also the complexification mathrm{SO}left(8,mathrm{mathbb{C}}right)/mathrm{SO}left(7,mathrm{mathbb{C}}right) , but it is of dimension too high for our considerations.] The seven-sphere S7 ≡ S7,0 has been found to play an important role in 11-dimensional supergravity, both through the Freund-Rubin solution and the Englert solution that uses its remarkable parallelizability to turn on non trivial internal fluxes. The spacetime manifold is in both cases AdS4 × S7. We show that S3,4 enjoys a similar role in M ′-theory and construct the exotic form AdS4 × S3,4 of the Englert solution, with non zero internal fluxes turned on. There is no analogous solution in M∗-theory.

Finsler functions consistent with absolute parallelisms

In this note we generalize a result of J.A. Wolf on Riemannian metrics consistent with absolute parallelisms to the Finslerian case. Along the way we obtain necessary and sufficient conditions for a parallelizable manifold to become a Lie group whose left parallelism is induced by the given one. It also turns out that a Finsler function consistent with an absolute parallelism is necessarily of Berwald type.

A pure geometric theory of gravity and a material distribution

A field theory is constructed in the context of parameterized absolute parallelism geometry. The theory is shown to be a pure gravity one. It is capable of describing the gravitational field and a material distribution in terms of the geometric structure of the geometry used (the parallelization vector fields). Three tools are used to attribute physical properties to the geometric objects admitted by the theory. Poisson and Laplace equations are obtained in the linearized version of the theory. The spherically symmetric solution of the theory, in free space, is found to coincide with the Schwarzschild exterior solution of the general theory of relativity. The theory respects the weak equivalence principle in free space only. Gravity and material distribution are not minimally coupled.

Logamediate Inflation in f(T) Teleparallel Gravity

Abstract We study logamediate inflation in the context of f(T) teleparallel gravity. f(T)-gravity is a generalization of the teleparallel gravity which is formulated on the Weitzenbock spacetime, characterized by the vanishing curvature tensor (absolute parallelism) and the non-vanishing torsion tensor. We consider an f(T)-gravity model which is sourced by a canonical scalar field. Assuming a power-law f(T) function in the action, we investigate an inflationary universe with a logamediate scale factor. Our results show that, although logamediate inflation is completely ruled out by observational data in the standard inflationary scenario based on Einstein gravity, it can be compatible with the 68% confidence limit joint region of Planck 2015 TT,TE,EE+lowP data in the framework of f(T)-gravity.

Connections in sub-Riemannian geometry of parallelizable distributions

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 [Formula: see text] and [Formula: see text].

Almost complex manifolds with non-degenerate torsion

We show that almost complex manifolds [Formula: see text] of real dimension 4 for which the image of the Nijenhuis tensor forms a non-integrable bundle, called torsion bundle, admit a [Formula: see text]-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 [Formula: see text] 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 [Formula: see text] 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 [Formula: see text].

Parallel multigrid technique: Reduction to independent problems

The unsatisfactory operation of a parallel multigrid algorithm is caused by two reasons: the imbalanced load of processors and the intensive exchanges of data between them. The further development of the parallel universal multigrid technique based on the reduction of a difference initial boundary value problem to a set of independent problems is considered. The universal multigrid technique is a single-grid algorithm, which uses the fundamental multigrid principle to minimize the number of problem-dependent components. The use of the same grid for the calculation of a correction eliminates all the difficulties produced by imbalanced loads and intensive exchanges on coarse grids. It has been shown that it is possible to decrease the volume of stored data and the time of computation and to attain nearly absolute parallelism in some cases. The results of some computational experiments with the difference six-order approximation pattern are presented.

An AP structure with Finslerian Flavor: Path equations

The Bazanski approach for deriving paths is applied to Finsler geometry. The approach is generalized and applied to a new developed geometry called "Absolute parallelism with a Finslerian Flavor" (FAP). A sets of path equations is derived for the FAP. This is the horizontal (h) set. A striking feature appears in this set, that is: the coefficient of torsion term, in the set, jumps by a step of one-half from one equation to the other. This is tempting to believe that the h-set admits some quantum features. Comparisons with the corresponding sets in other geometries are given. Conditions to reduce the set of path equations obtained, to well known path equations in some geometries are summarized in a schematic diagram.

On the Motion of a Free Particle in the de Sitter Manifold

Let \({M = SO(1, 4)/SO(1,3) \simeq S^{3}\times \mathbb{R}}\) (a parallelizable manifold) be a submanifold in the structure \({(\mathring{M}, {\mathring{g}})}\) (hereafter called the bulk) where \({\mathring {M} \simeq \mathbb{R}^{5}}\) and \({{\mathring{g}}}\) is a pseudo Euclidian metric of signature (1,4). Let \({{i}:M\rightarrow\mathbb{R}^{5}}\) be the inclusion map and let \({{g}={i}^{\ast}{\mathring{g}}}\) be the pullback metric on M. It has signature (1,3) Let \({{D}}\) be the Levi-Civita connection of \({{g} }\). We call the structure \({(M,{g})}\) a de Sitter manifold and \({M^{dSL} = (M = {\mathbb{R}} \times S^{3},g,D,\tau_{g},\uparrow)}\) a de Sitter spacetime structure, which is of course orientable by \({\tau_{g} \in {\rm sec} \bigwedge^{4} T^{\ast}M}\) and time orientable (by \({\uparrow}\)). Under these conditions (and without using any General Relativity theory concept) we prove that if the motion of a particle restricted to move on M (without the action of any force as detected by observers living in M) happens with constant bulk angular momentum then its motion in the structure MdSL is a timelike geodesic. Also any geodesic motion in the structure MdSL implies that the particle has constant angular momentum in the bulk.

Prolongation of Tanaka structures: an alternative approach

The classical theory of prolongation of G-structures was generalized by N. Tanaka to a wide class of geometric structures (Tanaka structures), which are defined on a non-holonomic distribution. Examples of Tanaka structures include subriemannian, subconformal, CR-structures, structures associated with second-order differential equations, and structures defined by gradings of Lie algebras (in the framework of parabolic geometries). Tanaka’s prolongation procedure associates with a Tanaka structure of finite order a manifold with an absolute parallelism. It is a very fruitful method for the description of local invariants, investigation of the automorphism group, and equivalence problem. In this paper, we develop an alternative constructive approach for Tanaka’s prolongation procedure, based on the theory of quasi-gradations of filtered vector spaces, G-structures, and their torsion functions.

Affine rigidity of Levi degenerate tube hypersurfaces

Let C(2,1) be the class of connected 5-dimensional CR-hypersurfaces that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In a recent article, we proved that the CR-structures in C(2,1) are reducible to so(3,2)-valued absolute parallelisms. In the present paper, we apply this result to study tube hypersurfaces in C^3 that belong to C(2,1) and whose CR-curvature identically vanishes. Every such hypersurface is shown to be affinely equivalent to an open subset of the tube over the future light cone.