Non-symmetric Connection Research Articles

Proposed Theorems on the Lifts of Kenmotsu Manifolds Admitting a Non-Symmetric Non-Metric Connection (NSNMC) in the Tangent Bundle

The main aim of the proposed paper is to investigate the lifts of Kenmotsu manifolds that admit NSNMC in the tangent bundle. We investigate several properties of the lifts of the curvature tensor, the conformal curvature tensor, and the conharmonic curvature tensor of Kenmotsu manifolds that admit NSNMC in the tangent bundle. We also study and discover that the lift of the Kenmotsu manifold that admit NSNMC is regular in the tangent bundle. Additionally, we find that the data provided by the lift of Ricci soliton on the lift of Ricci semi-symmetric Kenmotsu manifold that admits NSNMC in the tangent bundle are expanding.

Lightlike Hypersurfaces of an Indefinite Kaehler Manifold with an $(\ell,\,m)$-type Metric Connection

Jin introduced a non-symmetric metric connection, called an {\it $(\ell,m)$-type metric connection} \cite{Jin1, Jin2}. There are two examples of $(\ell, m)$-type: a semi-symmetric metric connection when ${\ell}=1$ and $m=0$ and a quater-symmetric connection for ${\ell}=0$ and $m=1$ . Our purpose is to investigate lightlike hypersurfaces of an indefinite (complex) Kaehler manifolds with an $(\ell,m)$-type metric connection under the tangent characteristic vector field on such hypersurfaces.

Concircularly semi-symmetric metric connection

Results on a concircularly semi-symmetric metric connection on a Riemannian manifold are presented. Six linearly independent curvature tensors with respect to this non-symmetric linear connection are studied, and the tensors coincident with the Weyl projective curvature tensor and the concircular curvature tensor of the Levi-Civita connection are determined. Transformations of connections when the Riemannian curvature tensor, Weyl projective curvature tensor and the concircular curvature tensor of the Levi-Civita connection are invariant are also observed. Moreover, new conditions for the Einstein manifold are derived.

Symplectic scalar curvature on the supermanifold of differential forms

In this paper, we study the notion of symplectic scalar curvature on the supermanifold over an ordinary Fedosov manifold whose structural sheaf is that of differential forms. In this purely geometric context, we introduce two families of odd super-Fedosov structures, the first one is very general and uses a graded symmetric connection, leading to a vanishing odd symplectic scalar curvature, while the second one is based on a graded non-symmetric connection and has a nontrivial odd symplectic scalar curvature. As a simple example of the second case, we determine that curvature when the base Fedosov manifold is the torus.

Some Results on α-cosymplectic Manifolds Admitting a Non-symmetric Non-metric Connection

In this study, we study α-cosymplectic manifolds admitting a non-symmetric non-metric connection. Moreover, several results about Ricci semi-symmetric and semi-symmetric α-cosymplectic manifolds admitting the non-symmetric non-metric connection are going to be obtained.

Curvature properties of metric and semi-symmetric linear connections

Considering a non-symmetric linear connection and its dual one appear 6 independent curvature tensors [15]. In the present paper we study properties of curvature tensors of semi-symmetric metric connections on a generalized Riemannian manifold. It is shown that if linear connection preserves the symmetric part of the generalized Riemannian metric with semi-symmetric torsion with an appropriate additional condition, then these six curvature tensors can be expressed by linear combination of Weyl conformal curvature tensor, Weyl projective curvature tensor and the concircular curvature tensor. In the case of the curvature tensor , we have proved that if exists a linear connection preserving the symmetric part of the generalized Riemannian metric with semi-symmetric torsion, whose curvature tensor vanishes, it is necessary and sufficient that the Riemannian metric be projectively flat. For tensor we also get a useful result. If exists a linear connection preserving the symmetric part of the generalized Riemannian metric with semi-symmetric torsion, whose curvature tensor and Ricci tensor vanish, it is necessary and sufficient that the Riemannian metric be conformally flat.

Projective Curvature Tensors of Some Special Manifolds with Non-symmetric Linear Connection

We consider geodesic mappings between some special manifolds with the non-symmetric linear connection. We obtain some tensors that are invariant with respect to geodesic mappings by relaxing the so-called “equitorsion condition”. Derived tensors have algebraic expressions analogous to the Weyl tensor of projective curvature, apart from that the generalized Ricci and curvature tensors appeared instead of the usual Ricci and curvature tensors. Further, we study some properties of mentioned invariant tensors as well as their relations with the Weyl tensor of projective curvature. Finally, we introduce some kinds of recurrent manifolds with the non-symmetric linear connection as well as their generalized Ricci and projective counterparts.

Almost Paracontact Almost Paracomplex Riemannian Manifolds with a Pair of Associated Schouten–van Kampen Connections

Two correlated Schouten–van Kampen affine connections on an almost paracontact almost paracomplex Riemannian manifold are introduced and investigated. The considered manifolds are characterized by virtue of the presented non-symmetric connections. Curvature properties of the studied connections are obtained. A family of examples on a Lie group are given in confirmation of the obtained results.

Riccati PDEs That Imply Curvature-Flatness

In this paper the following three goals are addressed. The first goal is to study some strong partial differential equations (PDEs) that imply curvature-flatness, in the cases of both symmetric and non-symmetric connection. Although the curvature-flatness idea is classic for symmetric connection, our main theorems about flatness solutions are completely new, leaving for a while the point of view of differential geometry and entering that of PDEs. The second goal is to introduce and study some strong partial differential relations associated to curvature-flatness. The third goal is to introduce and analyze some vector spaces of exotic objects that change the meaning of a generalized Kronecker delta projection operator, in order to discover new PDEs implying curvature-flatness. Significant examples clarify some ideas.

Ghost-free higher-order theories of gravity with torsion

In this manuscript we will present the theoretical framework of the recently proposed infinite derivative theory of gravity with a non-symmetric connection. We will explicitly derive the field equations at the linear level and obtain new solutions with a non-trivial form of the torsion tensor in the presence of a fermionic source, and show that these solutions are both ghost and singularity-free.

On the Stability of Analog ReLU Networks

Rectified linear unit (ReLU) networks have become widely used in machine learning and automated inference using neural networks. Various forms of hardware accelerators based on ReLU networks have also been under development. In this brief, the stability problem in analog ReLU networks is addressed. Using the Lyapunov stability theory, it is shown that the origin of an unforced, analog ReLU dynamical system is globally asymptotically stable if the induced Euclidean norm of its connectivity matrix is less than one. An example is given to demonstrate that this upper bound is the best that can be achieved. In particular, the stability result holds for the case of a nonsymmetric connectivity matrix as may occur in some mathematical models of neurobiology.

Kenmotsu Manifolds Admitting a Non-Symmetric Non-Metric Connection

The aim of the present paper is to study the properties of Kenmotsu manifolds equipped with a non-symmetric non-metric connection. We also establish some curvature properties of Kenmotsu manifolds. It is proved that a Kenmotsu manifold endowed with a non-symmetric non-metric is irregular.

A note on F-planar mappings of manifolds with non-symmetric linear connection

Following the basic facts of [Formula: see text]-planar mappings introduced by Mikeš and Sinyukov and further developed by Hinterleitner and Mikeš, we consider the notions of an [Formula: see text]-planar curve and an [Formula: see text]-planar mapping, but in case of manifolds endowed with a non-symmetric linear connection and an affinor structure. Consequently, we investigate infinitesimal [Formula: see text]-planar and particularly geodesic transformations of manifolds with non-symmetric linear connection and find necessary and sufficient conditions for the existence of these transformations.

Nonsymmetrical Connection of Two Identical Building Blocks: Constructing Donor-Acceptor Molecules as Deep Blue Emitting Materials for Efficient Organic Emitting Diodes.

We propose a strategy to construct deep blue emission molecules based on the concept of nonsymmetrical connection of two identical π-conjugated groups. It was demonstrated that the nonsymmetrical connection strategy indeed resulted in the separation of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) and the formation of a donor-acceptor (D-A) structure. For D-A molecules constructed by two identical groups, the degree of charge transfer is weaker and deep blue emission is easily achieved. Two D-A molecules (PIpPI and PImPI) were synthesized by employing diphenyl-phenanthroimidazole (PI) as a building block. The nonsymmetric connection of PI groups endows these molecules with a D-A feature that can result in a bipolar transport property. The nondoped organic light-emitting diodes with PIpPI and PImPI as emitter exhibit deep-blue emission and maximum external quantum efficiencies of 8.84% and 6.83%, respectively.

Classical Electromagnetic Potential as a Part of Gravitational Connection: Ideas and History

We consider a natural form of unified theory of gravity and electromagnetism which was somehow missed at the time of intense search for such an unification, and was noticed only in 1978 but remained quite unknown. The basic idea of this unification is to use the metric and non-symmetric connection as independent variables, which generalizes the so-called Palatini formalism. The certain components of connection in the appearing theory can be naturally identified with electromagnetic potential, and with the proper choice of action the Einstein-Maxwell equations are reproducing. In this paper we compare such an approach with the known ideas of unification. Also we propose the more consistent way of including matter (in the form of classical particles) in the theory and briefly discuss the perspectives of further development of this approach.

Generalized Kähler Spaces in Eisenhart’s Sense Admitting a Holomorphically Projective Mapping

We relax the conditions $$\underset{1}{\nabla }F=0$$ and $$\underset{2}{\nabla }F=0$$ with respect to the non-symmetric linear connections $$\underset{1}{\nabla }$$ and $$\underset{2}{\nabla }$$ and an almost complex structure F of a generalized Kahler space. In such a way we introduce a wider class of generalized Kahler spaces which admits a holomorphically projective mapping. These generalized Kahler spaces are named generalized Kahler spaces in Eisenhart’s sense, since they are defined as a particular case of Eisenhart’s generalized Riemannian spaces. Curvature tensors of generalized Kahler spaces in Eisenhart’s sense have some interesting properties that have been pointed out in the present paper. Also, we consider equitorsion holomorphically projective mappings and examine some invariant geometric objects with respect to these mappings. Our results are deduced without restrictive conditions that were given in some of the previous papers.

On almost geodesic mappings of the second type between manifolds with non-symmetric linear connection

We derive two mixed systems of Cauchy type in covariant derivatives of the first and second kind that ensures the existence of almost geodesic mappings of the second type between manifolds with non-symmetric linear connection. Also, we consider a particular class of these mappings determined by the condition ?F = 0, where ? is the symmetric part of non-symmetric linear connection ?1 and F is the affinor structure. The same special class of almost geodesic mappings of the second type between generalized Riemannian spaces was recently considered in the paper (M.Z. Petrovic, Special almost geodesic mappings of the second type between generalized Riemannian spaces, Bull. Malays. Math. Sci. Soc. (2), DOI :10.1007/s40840-017-0509-5)

Special Almost Geodesic Mappings of the Second Type Between Generalized Riemannian Spaces

We deal with almost geodesic lines of manifolds with non-symmetric linear connection. Also, we consider special almost geodesic mappings of the second type between Eisenhart’s generalized Riemannian spaces as well as between generalized classical (elliptic) and hyperbolic Kahler spaces. These mappings are generalizations of holomorphically projective mappings between generalized classical and hyperbolic Kahler spaces. We prove some existence theorems for special almost geodesic mappings of the second type between generalized Riemannian spaces as well as between generalized classical and hyperbolic Kahler spaces. Finally, we find some invariant geometric objects with respect to these mappings.

Description of gravity in the model with independent nonsymmetric connection

A generalization of General Relativity is studied. The standard Einstein-Hilbert action is considered in the Palatini formalism, where the connection and the metric are independent variables, and the connection is not symmetric. As a result of variation with respect to the metric Einstein equations are obtained. A variation with respect to the connection leads to an arbitrariness in the determination of connection, i.e. the presence of gauge invariance. Then a matter in a form of point particle which interacts with field of connection is introduced. Also the action is complemented by a kinetic term for field of the connection to avoid incompatible equation of motion. Thus after the variation procedures we obtain the Einstein equations, the geodesic equation and the Maxwell`s equations for electromagnetism, where some components of the connection play the role of the electromagnetic potential. Thereby the electromagnetic potential is obtained from the geometry of space-time.

Riemannian surfaces with torsion as homogenization limits of locally Euclidean surfaces with dislocation-type singularities

We reconcile two classical models of edge dislocations in solids. The first, from the early 1900s, models isolated edge dislocations as line singularities in locally Euclidean manifolds. The second, from the 1950s, models continuously distributed edge dislocations as smooth manifolds endowed with non-symmetric affine connections (equivalently, endowed with torsion fields). In both models, the solid is modelled as a Weitzenböck manifold. We prove, using a weak notion of convergence, that the second model can be obtained rigorously as a homogenization limit of the first model as the density of singular edge dislocation tends to infinity.