Absolute parallelism for 2-nondegenerate CR structures via bigraded Tanaka prolongation

This article is devoted to the local geometry of everywhere 2-nondegenerate CR manifolds M of hypersurface type. An absolute parallelism for such structures was recently constructed independently by Isaev and Zaitsev, Medori and Spiro, and Pocchiola in the minimal possible dimension ( dim ⁡ M = 5 {\dim M=5} ), and for dim ⁡ M = 7 {\dim M=7} in certain cases by the first author. In the present paper, we develop a bigraded (i.e., ℤ × ℤ {\mathbb{Z}\times\mathbb{Z}} -graded) analog of Tanaka’s prolongation procedure to construct an absolute parallelism for these CR structures in arbitrary (odd) dimension with Levi kernel of arbitrary admissible dimension. We introduce the notion of a bigraded Tanaka symbol – a complex bigraded vector space – containing all essential information about the CR structure. Under the additional regularity assumption that the symbol is a Lie algebra, we define a bigraded analog of the Tanaka universal algebraic prolongation, endowed with an anti-linear involution, and prove that for any CR structure with a given regular symbol there exists a canonical absolute parallelism on a bundle whose dimension is that of the bigraded universal algebraic prolongation. Moreover, we show that for each regular symbol there is a unique (up to local equivalence) such CR structure whose algebra of infinitesimal symmetries has maximal possible dimension, and the latter algebra is isomorphic to the real part of the bigraded universal algebraic prolongation of the symbol. In the case of 1-dimensional Levi kernel we classify all regular symbols and calculate their bigraded universal algebraic prolongations. In this case, the regular symbols can be subdivided into nilpotent, strongly non-nilpotent, and weakly non-nilpotent. The bigraded universal algebraic prolongation of strongly non-nilpotent regular symbols is isomorphic to the complex orthogonal algebra 𝔰 ⁢ 𝔬 ⁢ ( m , ℂ ) {\mathfrak{so}(m,\mathbb{C})} , where m = 1 2 ⁢ ( dim ⁡ M + 5 ) {m=\tfrac{1}{2}(\dim M+5)} . Any real form of this algebra – except 𝔰 ⁢ 𝔬 ⁢ ( m ) {\mathfrak{so}(m)} and 𝔰 ⁢ 𝔬 ⁢ ( m - 1 , 1 ) {\mathfrak{so}(m-1,1)} – corresponds to the real part of the bigraded universal algebraic prolongation of exactly one strongly non-nilpotent regular CR symbol. However, for a fixed dim ⁡ M ≥ 7 {\dim M\geq 7} the dimension of the bigraded universal algebraic prolongations of all possible regular CR symbols achieves its maximum on one of the nilpotent regular symbols, and this maximal dimension is 1 4 ⁢ ( dim ⁡ M - 1 ) 2 + 7 {\frac{1}{4}(\dim M-1)^{2}+7} .

Teleparallel gravity with non-vanishing curvature

Guided by the rules of Einstein’s geometrization philosophy, a pure geometric field theory is constructed. The Lagrangian used to derive the field equations of the theory is a curvature scalar of a version of absolute parallelism (AP) geometry known in the literature as the parameterized absolute parallelism (PAP) geometry. The linear connection of this version has simultaneously non-vanishing curvature and torsion. Analysis of the theory obtained shows clearly that it is a pure gravity theory. The theory is a teleparallel one, since the building blocks of both PAP and AP geometries are the same. It is shown analytically that the theory has a trivial version in the AP-geometry, if gravity is attributed to curvature not to torsion. In the case of spherical symmetry, solutions of the field equations give rise to the Schwarzschild exterior field. The theory depends on two principles: covariance and unification. The weak equivalence principle is satisfied under a certain condition. The work preserves Einstein’s main idea that gravity is just space–time curvature, although it is not a metric theory. It is shown that the theory reduces to vacuum general relativity upon taking the parameter of the geometry b = 0.

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

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.

Parallelizable Manifolds

In this chapter, absolute parallelism is formulated in an equivalent form in terms of a splitting, and the language necessary for this formulation is developed.

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.

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].

Einstein Geometrization Philosophy and Differential Identities in PAP-Geometry

The importance of Einstein’s geometrization philosophy, as an alternative to the least action principle, in constructing general relativity (GR), is illuminated. The role of differential identities in this philosophy is clarified. The use of Bianchi identity to write the field equations of GR is shown. Another similar identity in the absolute parallelism geometry is given. A more general differential identity in the parameterized absolute parallelism geometry is derived. Comparison and interrelationships between the above mentioned identities and their role in constructing field theories are discussed.