Curvature Tensor Field Research Articles

The Ricci decomposition of the inertia tensor for a rigid body in arbitrary spatial dimensions

The rotations of rigid bodies in Euclidean space are characterized by their instantaneous angular velocity and angular momentum. In an arbitrary number of spatial dimensions, these quantities are represented by bivectors (antisymmetric rank-2 tensors), and they are related by a rank-4 inertia tensor. Remarkably, this inertia tensor belongs to a well-studied class of algebraic curvature tensors that have the same index symmetries as the Riemann curvature tensor field used in general relativity. Any algebraic curvature tensor can be decomposed into irreducible representations of the orthogonal group via the Ricci decomposition. We calculate the Ricci decomposition of the inertia tensor for a rigid body in any number of dimensions, and we find that (unlike for the Riemann curvature tensor field) its traceless Weyl tensor is always zero, so the inertia tensor is completely characterized by its (rank-2) Ricci contraction. So unlike in general relativity, the traceless Weyl tensor does not cause any qualitatively new phenomenology for rigid-body dynamics in n≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 4$$\\end{document} dimensions.

A study of recurrent Finsler spaces of higher order with Cartan’s Curvature Tensor

In the present communication, we have derived Bianchi and Veblen identities along with a few more related results in a recurrent and generalized nth-recurrent Finsler space with Cartan’s curvature tensor field. A Finsler space \(F_n\) whose Cartan's third curvature tensor \(R_{jkh}^i\) satisfies the condition \(R_{jkh|m_1|m_2|…|m_n}^{i} = λ_{m_1m_2…m_n} R_{jkh}^{i} + μ_{m_1m_2…m_n} (δ_{h}^{i} g_{jk} - δ_{k}^{i} g_{jh})\), where \(R_{jkh}^i\)≠0 and \(|m_1 |m_2 |…|m_n\) are h-covariant differentiation (Cartan's second kind covariant differential operator) with respect to x^m to nth order, \(λ_{m_1 m_2…m_n }\) and \(μ_{m_1 m_2…m_n }\) is recurence tensors fields.

On Generalized Recurrent Finsler Spaces of Higher Order with Berwald’s Curvature Tensor

In this present paper, we introduced a Finsler space Fn which projective curvature tensor K^i_ijkh satisfies the generalized five recurrence property with respect to Berwald’s connection parameters G^i_kh, we have derived some theorems and some identities along with few more related results in a recurrent and generalized 5-recurrent Finsler space with Berwald’s curvature tensor field.

Holonomic and Non-Holonomic Geometric Models Associated to the Gibbs–Helmholtz Equation

By replacing the internal energy with the free energy, as coordinates in a “space of observables”, we slightly modify (the known three) non-holonomic geometrizations from Udriste’s et al. work. The coefficients of the curvature tensor field, of the Ricci tensor field, and of the scalar curvature function still remain rational functions. In addition, we define and study a new holonomic Riemannian geometric model associated, in a canonical way, to the Gibbs–Helmholtz equation from Classical Thermodynamics. Using a specific coordinate system, we define a parameterized hypersurface in R4 as the “graph” of the entropy function. The main geometric invariants of this hypersurface are determined and some of their properties are derived. Using this geometrization, we characterize the equivalence between the Gibbs–Helmholtz entropy and the Boltzmann–Gibbs–Shannon, Tsallis, and Kaniadakis entropies, respectively, by means of three stochastic integral equations. We prove that some specific (infinite) families of normal probability distributions are solutions for these equations. This particular case offers a glimpse of the more general “equivalence problem” between classical entropy and statistical entropy.

Norden Golden Manifolds with Constant Sectional Curvature and Their Submanifolds

This paper discusses the Norden golden manifold having a constant sectional curvature. First, it is shown that if a Norden golden manifold has a constant real sectional curvature, the manifold is flat. For this reason, the notions of holomorphic-like sectional curvature and holomorphic-like bisectional curvature on the Norden golden manifold are investigated, but it is seen that these notions do not work on the Norden golden manifold. This shows the need for a new concept of sectional curvature. In this direction, a new notion of sectional curvature (Norden golden sectional curvature) is proposed, an example is given, and if this new sectional curvature is constant, the curvature tensor field of the Norden golden manifold is expressed in terms of the metric tensor field. Since the geometry of the submanifolds of manifolds with constant sectional curvature has nice properties, the last section of this paper examines the semi-invariant submanifolds of the Norden golden space form.

Dirac gauge theory for topological spinors in 3+1 dimensional networks

Gauge theories on graphs and networks are attracting increasing attention not only as approaches to quantum gravity but also as models for performing quantum computation. Here we propose a Dirac gauge theory for topological spinors in dimensional networks associated to an arbitrary metric. Topological spinors are the direct sum of 0-cochains and 1-cochains defined on a network and describe a matter field defined on both nodes and links of a network. Recently in Bianconi (2021 J. Phys. Complex. 2 035022) it has been shown that topological spinors obey the topological Dirac equation driven by the discrete Dirac operator. In this work we extend these results by formulating the Dirac equation on weighted and directed dimensional networks which allow for the treatment of a local theory. The commutators and anti-commutators of the Dirac operators are non vanishing an they define the curvature tensor and magnetic field of our theory respectively. This interpretation is confirmed by the non-relativistic limit of the proposed Dirac equation. In the non-relativistic limit of the proposed Dirac equation the sector of the spinor defined on links follows the Schrödinger equation with the correct giromagnetic moment, while the sector of the spinor defined on nodes follows the Klein–Gordon equation and is not negligible. The action associated to the proposed field theory comprises of a Dirac action and a metric action. We describe the gauge invariance of the action under both Abelian and non-Abelian transformations and we propose the equation of motion of the field theory of both Dirac and metric fields. This theory can be interpreted as a limiting case of a more general gauge theory valid on any arbitrary network in the limit of almost flat spaces.

Decomposition of Riemannian Recurrent Curvature Tensor Manifolds of First Order

Takano [6] premeditated decomposition of curvature tensor in a recurrent Riemannian space. After that, Negi and Bisht [3] defined and deliberated decomposition of recurrent curvature tensor fields in a Kaehlerian manifolds of first order. We have calculated the decomposition of Riemannian recurrent curvature tensor manifolds of first order and some theorems established using the decomposition tensor field.

Some characterizations on indefinite space forms admitting almost contact structures

In this paper, the projective curvature tensor field of a generalized indefinite Sasakian space form is investigated. Some results dealing with projectively semi-symmetric and [Formula: see text]-projectively semi-symmetric generalized indefinite Sasakian space forms are obtained. Furthermore, biharmonic Legendre Frenet curves are discussed on these manifolds.

The Properties of Projective, Concircular and Conharmonic Curvature Tensor Fields on a Complex Sasakian Manifold

In this article, the properties of projective, concircular and conharmonic curvature tensor fields on a complex Sasakian manifold are investigated.

Notes on the second-order tangent bundles with the deformed Sasaki metric

The paper deals with the second-order tangent bundle T2MT2M with the deformed Sasaki metric ¯gg¯ over an n−dimensional Riemannian manifold gover an n−dimensional Riemannian manifold (M,g)(M,g). We calculate all Riemannian curvature tensor fields of the deformed Sasaki metric ¯gg¯ and search Einstein property of and search Einstein property of T2MT2M. Also the weakly symmetry properties of the deformed Sasaki metric are presented.

Meta‐Golden Riemannian manifolds

The logarithmic spiral in nature has been given as an example of the golden ratio until now. But recently, it has been shown that this is not true, and the logarithmic spiral has actually been shown to provide the so‐called Meta‐Golden‐Chi ratio. Inspiring from Meta‐Golden‐Chi ratio, we introduce almost Meta‐Golden manifolds, give a characterization, provide examples, and investigate certain properties of Meta‐Golden structure. The integrability of almost Meta‐Golden structure is checked, and the relation of Meta‐Golden structure with curvature tensor field is examined. In addition, if the Meta‐Golden manifold has constant curvature, it is shown that such manifold is a flat space under certain conditions. This implies that there is a need for a new concept of sectional curvature in Meta‐Golden Riemannian manifolds.

Classification of Holomorphic Functions as Pólya Vector Fields via Differential Geometry

We review Pólya vector fields associated to holomorphic functions as an important pedagogical tool for making the complex integral understandable to the students, briefly mentioning its use in other dimensions. Techniques of differential geometry are then used to refine the study of holomorphic functions from a metric (Riemannian), affine differential or differential viewpoint. We prove that the only nontrivial holomorphic functions, whose Pólya vector field is torse-forming in the cannonical geometry of the plane, are the special Möbius transformations of the form f(z)=b(z+d)−1. We define and characterize several types of affine connections, related to the parallelism of Pólya vector fields. We suggest a program for the classification of holomorphic functions, via these connections, based on the various indices of nullity of their curvature and torsion tensor fields.

Minimum Time Problem Controlled by Affine Connection

Geometrically, the affine connection is the main ingredient that underlies the covariant derivative, the parallel transport, the auto-parallel curves, the torsion tensor field, and the curvature tensor field on a finite-dimensional differentiable manifold. In this paper, we come up with a new idea of controllability and observability of states by using auto-parallel curves, and the minimum time problem controlled by the affine connection. The main contributions refer to the following: (i) auto-parallel curves controlled by a connection, (ii) reachability and controllability on the tangent bundle of a manifold, (iii) examples of equiaffine connections, (iv) minimum time problem controlled by a connection, (v) connectivity by stochastic perturbations of auto-parallel curves, and (vi) computing the optimal time and the optimal striking time. The connections with bounded pull-backs result in bang–bang optimal controls. Some significative examples on bi-dimensional manifolds clarify the intention of our paper and suggest possible applications. At the end, an example of minimum striking time with simulation results is presented.

Curvature Potential Unveiled Topological Defect Attractors

We consider the theoretical and positional assembling of topological defects (TDs) in effectively two-dimensional nematic liquid crystal films. We use a phenomenological Helfrich–Landau–de Gennes-type mesoscopic model in which geometric shapes and nematic orientational order are expressed in terms of a curvature tensor field and a nematic tensor order parameter field. Extrinsic, intrinsic, and total curvature potentials are introduced using the parallel transport concept. These potentials reveal curvature seeded TD attractors. To test ground configurations, we used axially symmetric nematic films exhibiting spherical topology.

Gradient solitons on statistical manifolds

We provide necessary and sufficient conditions for some particular couples (g,∇) of pseudo-Riemannian metrics and affine connections to be statistical structures if we have gradient almost Einstein, almost Ricci, almost Yamabe solitons, or a more general type of solitons on the manifold. In particular cases, we establish a formula for the volume of the manifold and give a lower and an upper bound for the norm of the Ricci curvature tensor field.

Some Problems Concerning with Sasaki Metric on the Second-Order Tangent Bundles

In this paper, we consider a second-order tangent bundle equipped with
 Sasaki metric over a Riemannian manifold. All forms of curvature tensor
 fields are computed. We obtained the relation between the scalar curvature
 of the base manifold and the scalar curvature of the second-order tangent
 bundle and presented some geometric results concerning with kinds of
 curvature tensor fields. Also, we search the weakly symmetry property of the
 second-order tangent bundle. Finally, we end our paper with statistical
 structures on the second-order tangent bundle.

On Holomorphic poly-Norden Manifolds

In this paper, we investigated a new manifold with a poly-Norden structure, which is inspired by the positive root of the equation $x^{2}-mx-1=0$. We call this new manifold as holomorphic poly-Norden manifolds. We examine some properties of the Riemann curvature tensor and give an example of this manifold. Then, we define a different connection on this manifold which is named the semisymmetric metric poly F-connection and study some properties of the curvature and torsion tensor field according to this connection.

Metric f-Contact Manifolds Satisfying the (κ, μ)-Nullity Condition

We prove that if the f-sectional curvature at any point of a ( 2 n + s ) -dimensional metric f-contact manifold satisfying the ( κ , μ ) nullity condition with n > 1 is independent of the f-section at the point, then it is constant on the manifold. Moreover, we also prove that a non-normal metric f-contact manifold satisfying the ( κ , μ ) nullity condition is of constant f-sectional curvature if and only if μ = κ + 1 and we give an explicit expression for the curvature tensor field in such a case. Finally, we present some examples.

On Ricci curvature of submanifolds in statistical manifolds of constant (quasi-constant) curvature

In 1999, B. Y. Chen established a sharp inequality between the Ricci curvature and the squared mean curvature for an arbitrary Riemannian submanifold of a real space form. This inequality was extended in 2015 by M. E. Aydin et al. to the case of statistical submanifolds in a statistical manifold of constant curvature, obtaining a lower bound for the Ricci curvature of the dual connections. Also, the similar inequality for submanifolds in statistical manifolds of quasi-constant curvature studied by H. Aytimur and C. Ozgur in their recent article. In the present paper, we give a different proof of the same inequality but working with the statistical curvature tensor field, instead of the curvature tensor fields with respect to the dual connections. A geometric inequality can be treated as an optimization problem. The new proof is based on a simple technique, known as Oprea's optimization method on submanifolds, namely analyzing a suitable constrained extremum problem. We also provide some examples. This paper finishes with some conclusions and remarks.

Chen Inequalities for Statistical Submanifolds of Kähler-Like Statistical Manifolds

We consider Kähler-like statistical manifolds, whose curvature tensor field satisfies a natural condition. For their statistical submanifolds, we prove a Chen first inequality and a Chen inequality for the invariant δ ( 2 , 2 ) .