Riemann Tensor Research Articles

A study on the tangent bundle with the vertical generalized Berger type deformed Sasaki metric

In this paper, we introduce a vertical generalized Berger type deformed Sasaki metric on the tangent bundle $TM$ over an anti-paraK\"{a}hler manifold as a new natural metric. Firstly, we investigate the Levi-Civita connection of this metric and then we calculate all forms of the Riemannian curvature tensors. Also, we present some results concerning curvature properties. Finally, we study the geometry of $\varphi $-unit tangent bundle equipped with the vertical generalized Berger type deformed Sasaki metric.

Locally Product-like Statistical Manifolds and Their Hypersurfaces

In this paper, almost product-like Riemannian manifolds are investigated. Basic properties on tangential hypersurfaces of almost product-like Riemannian manifolds are obtained. Some examples of tangential hypersurfaces are presented. Some relations involving the Riemannian curvature tensor of a tangential hypersurface are computed.

Constraints on cosmological model in Weyl-Cartan spacetime from astronomical measurements

In order to give some lights on the issue of Hubble tension, alternative to Riemannian spacetime, we investigate a cosmological model in Weyl-Cartan spacetime by astronomical measurements, where a new density parameter Ω ϕ is introduced to denote the non-Riemannian spacetime structure, which scales as ∼(1 + z)6. Based on Markov Chain Monte Carlo estimation and taking astronomical measurements of cosmic microwave background (Planck 2018), baryon acoustic oscillations and Type Ia supernova (Pantheon), model parameters of our considering case are constrained. Our results show that Hubble tension cannot be alleviated in Weyl-Cartan cosmology. However, constraints on the parameter Ω ϕ can not be distinguished from ΛCDM model in 1σ confidence region. Moreover, the space of the Universe is infinite and unbounded, and an upper limit of the model parameter Ω ϕ is also obtained. Additionally, it is worth noting that results obtained in this work are valid for models that the scaling behavior of density parameters is same to our considering case.

Defect Wormholes Are Defective

The various “defect wormholes” developed by Klinkhamer have recently attracted considerable attention—especially in view of the fact that the simplest example, the so-called “vacuum defect wormhole”, was claimed to be an everywhere-vacuum everywhere-Ricci-flat exact solution to the Einstein equations. This claim has been conclusively refuted by Feng, and in the current article, we take a deeper look at exactly what goes wrong. The central issue is this: Although Klinkhamer’s specific representation of the metric gab is smooth (C∞), his inverse metric gab is not even everywhere continuous (C0), being undefined at the wormhole throat. This situation implies that one should very carefully investigate curvature tensors at the throat using the Israel–Lanczos–Sen thin-shell formalism. Doing so reveals the presence of a delta-function energy-condition-violating thin shell of matter at the wormhole throat. The “defect wormholes” are thus revealed to be quite ordinary “cut-and-paste” thin-shell wormholes, but represented in a coordinate system that is unfortunately pathological at exactly the same place that all the interesting physics occurs. To help clarify the situation, we shall focus on the behavior of suitable coordinate invariants—the Ricci scalar, the eigenvalues of the mixed Rab Ricci tensor, and the eigenvalues of the mixed Rabcd Riemann tensor.

Analysis of complexity on the anisotropic charged fluid in modified teleparallel gravity

This article is about the impact of complexity on the anisotropic charged fluid in f(T) gravity, where accountability of the torsion scalar T is offered by the gravitational influence. Our focus is on the anisotropic charged fluid. We took into account the static spherical distribution for the interior region. Foremost, we present the basic formalism of f(T) theory, which comprises the electric charge, and then evaluate the field equations along with the hydrostatic equilibrium equation in this framework. The criterion of complexity is taken up from Herrera’s design (Herrera, 2018), where the complexity factor is attained from the division of the Riemann curvature tensor in an orthogonal direction. We set up the relation between the Misner–Sharp mass function and the hydrostatic equilibrium equation comprising the electric charge in f(T) theory. We match the interior region with the Reissner-Nordström geometry in the exterior direction and investigate the role of the Tolman mass in this background. We carry out the analysis of the structure scalars for charged anisotropic fluid within f(T) gravity and study the impact of the density homogeneity and inhomogeneity along with local anisotropy on the complexity factor YTF. The manuscript concluded with the defined concepts that satisfy the constraint of diminishing complexity in f(T) theory, which comprises the electric charge.

On Possible Minimal Length Deformation of Metric Tensor, Levi-Civita Connection, and the Riemann Curvature Tensor

The minimal length conjecture is merged with a generalized quantum uncertainty formula, where we identify the minimal uncertainty in a particle’s position as the minimal measurable length scale. Thus, we obtain a quantum-induced deformation parameter that directly depends on the chosen minimal length scale. This quantum-induced deformation is conjectured to require the generalization of Riemannian spacetime geometry underlying the classical theory of general relativity to an eight-dimensional spacetime fiber bundle, which dictates the deformation of the line element, metric tensor, Levi-Civita connection, Riemann curvature tensor, etc. We calculate the deformation thus produced in the Levi-Civita connection and find it to explicitly and exclusively depend on the product of the minimum measurable length and the particle’s spacelike four-acceleration vector, L2x¨2. We find that the deformed Levi-Civita connection preserves all properties of its undeformed counterpart, such as torsion freedom and metric compatibility. Accordingly, we have constructed a deformed version of the Riemann curvature tensor whose expression can be factorized in all its terms with different functions of L2x¨2. We also show that the classical four-manifold status of being Riemannian is preserved when the quantum-induced deformation is negligible. We study the dependence of a parallel-transported tangent vector on the spacelike four-acceleration. We illustrate the impact of the minimal-length-induced quantum deformation on the classical geometrical objects of the general theory of relativity using the unit radius two-sphere example. We conclude that the minimal length deformation implies a correction to the spacetime curvature and its contractions, which is manifest in the additional curvature terms of the corrected Riemann tensor. Accordingly, quantum-induced effects endow an additional spacetime curvature and geometrical structure.

Minimal supergeometric quantum field theories

We formulate minimal SuperGeometric Quantum Field Theories (SG-QFTs) that allow for scalar-fermion field transformations in a manifestly reparameterisation covariant manner. First, we discuss the issue of uniqueness in defining the field-space supermetric of the underlying supermanifold, and clarify the fact that different supermetric definitions can lead to distinct theories in the off-shell kinematic region. By adopting natural choices for the field-space supermetric, we then show that scalar fields alone cannot induce a non-trivial field-space Riemannian curvature in the fermionic sector, beyond the one originating from the scalar part of the theory. We present for the first time minimal SG-QFT models that feature non-zero fermionic curvature both in two and four spacetime dimensions. Physical applications of SG-QFTs are discussed.

Impact of charge on the complexity of static cylindrical system

This work examines the complexity of charged static cylindrical object with anisotropic matter configuration in the background of energy–momentum squared gravity. For this purpose, we construct the field equations, the non-conservation equation and investigate the celestial structure. We evaluate the mass functions through both C-energy as well as Tolman mass formulae. Numerous factors, such as anisotropic pressure, inhomogeneous energy density and charge, etc have an impact on the complexity of the structure under consideration. The orthogonal splitting of the Riemann tensor yields two structure scalars. Among these both, the factor incorporating all the physical parameters of the self-gravitating system is named as the complexity factor. The complexity-free constraint is employed to compute charged static solutions for the Gokhroo–Mehra model as well as the polytropic equation of state. We conclude that the inclusion of charge reduces the complexity of anisotropic system.

On the mathematical structure of Einstein field equations and the existence of dark fields

In this work, we examine the possible existence of dark fields by showing that an energy‐momentum tensor associated with a dark field can be introduced into Einstein field equations of general relativity, in which the trace of the energy‐momentum of the dark field is identified with the cosmological constant. The introduction of dark fields into Einstein field equations is made possible by establishing field equations for the Ricci curvature tensor, which have similar mathematical structure to Maxwell field equations of electromagnetism. We also establish a system of field equations for the Riemann curvature tensor and investigate the possibility to represent physical systems consisting of dark fields and observable fields as spaces of constant scalar curvature, which are maximally symmetric spaces that admit the maximal number of Killing vectors. As an illustration, we show that if a dark field is considered as a dark fluid, then the pressure associated with the dark field can take negative values if the cosmological constant is assigned with positive values.

Spherically Symmetric C3 Matching in General Relativity

We study the problem of matching interior and exterior solutions to Einstein’s equations along a particular hypersurface. We present the main aspects of the C3 matching approach that involve third-order derivatives of the corresponding metric tensors in contrast to the standard C2 matching procedures known in general relativity, which impose conditions on the second-order derivatives only. The C3 alternative approach does not depend on coordinates and allows us to determine the matching surface by using the invariant properties of the eigenvalues of the Riemann curvature tensor. As a particular example, we apply the C3 procedure to match the exterior Schwarzschild metric with a general spherically symmetric interior spacetime with a perfect fluid source and obtain that on the matching hypersurface, the density and pressure should vanish, which is in accordance with the intuitive physical expectation.

Strain, disfigurement and flexure tensors in continuum kinematics

In the differential geometric formulation of nonlinear elasticity, the strain tensor can intrinsically be regarded as a measure of change of the metric of the Riemannian manifold that embodies the continuum undergoing a deformation. Here, this classical concept is revisited and complemented with two newly defined higher order deformation measures, namely the disfigurement and flexure tensors, which respectively measure the change of the Levi-Civita connexion and the Riemann curvature of the continuum. The strain, disfigurement and flexure tensors can, thereby, roughly indicate the alteration of the geometry of a deforming body in the zeroth, first and second orders, respectively. In this light, one may predict that the disfigurement and flexure tensors should be related to the first and second covariant derivatives of the strain tensor. While this is true, the exact strain-disfigurement-flexure relations, as derived in this paper, are found to be involved and non-trivial, offering a deeper understanding of the deformation. In our discussions, distance-preserving, geodesic-preserving and curvature-preserving deformations are studied as three special cases to help better interpret the above-mentioned deformation measures. Some related results have also been found as by-products that might be of current interest or prove useful in future investigations.

Holographic aspects of non-minimal R F F AdS black brane

In this paper, we study the holographic dual to an asymptotically anti-de Sitter black brane in an Einstein–Yang–Mills model with a non-minimal coupling between the Riemann and Yang–Mills fields. First, we construct a planar black hole solution of this model up to the first order of the non-minimal coupling of the Yang–Mills field with the Riemann–Christoffel tensor, denoted as q 2. Then, we calculate the color non-abelian direct current conductivity and the ratio of shear viscosity to entropy density for this solution. Our result for the shear viscosity η to entropy density s ratio saturates the Kovtun, Son, and Starinets bound, which is proportional to . However, our result for the conductivity is new up to the first order of q 2.

On the Geometry of the Riemannian Curvature Tensor of Nearly Trans-Sasakian Manifolds

This paper presents the results of fundamental research into the geometry of the Riemannian curvature tensor of nearly trans-Sasakian manifolds. The components of the Riemannian curvature tensor on the space of the associated G-structure are counted, and the components of the Ricci tensor are calculated. Some identities are obtained that are satisfied by the Riemannian curvature tensors and the Ricci tensor. A number of properties are proved that characterize nearly trans-Sasakian manifolds with a closed contact form. The structure of nearly trans-Sasakian manifolds with a closed contact form is obtained. Several classes are singled out in terms of second-order differential-geometric invariants, and their local structure is obtained. The k-nullity distribution of a nearly trans-Sasakian manifold is studied.

Quantum Black Holes in Conformal Dilaton–Higgs Gravity on Warped Spacetimes

A promising method for understanding the geometric properties of a spacetime in the vicinity of the horizon of a Kerr-like black hole can be developed by applying the antipodal boundary condition on the two opposite regions in the extended Penrose diagram. By considering a conformally invariant Lagrangian on a Randall–Sundrum warped five-dimensional spacetime, an exact vacuum solution is found, which can be interpreted as an instanton solution on the Riemannian counterpart spacetime, R+2×R1×S1, where R+2 is conformally flat. The antipodal identification, which comes with a CPT inversion, is par excellence, suitable when quantum mechanical effects, such as the evaporation of a black hole by Hawking radiation, are studied. Moreover, the black hole paradoxes could be solved. By applying the non-orientable Klein surface, embedded in R4, there is no need for instantaneous transport of information. Further, the gravitons become “hard” in the bulk, which means that the gravitational backreaction on the brane can be treated without the need for a firewall. By splitting the metric in a product ω2g˜μν, where ω represents a dilaton field and g˜μν the conformally flat “un-physical” spacetime, one can better construct an effective Lagrangian in a quantum mechanical setting when one approaches the small-scale area. When a scalar field is included in the Lagrangian, a numerical solution is presented, where the interaction between ω and Φ is manifest. An estimate of the extra dimension could be obtained by measuring the elapsed traversal time of the Hawking particles on the Klein surface in the extra dimension. Close to the Planck scale, both ω and Φ can be treated as ordinary quantum fields. From the dilaton field equation, we obtain a mass term for the potential term in the Lagrangian, dependent on the size of the extra dimension.

On recurrent Riemannian and Ricci curvatures of Finsler metrics

In this paper, we study a property of Riemannian and Ricci curvatures under which it reproduces itself, namely, recurrent Finsler metrics. We prove that if (M,F) is a recurrent Finsler manifold of non-zero isotropic flag curvature, then F is a Landsberg metric. It follows that Every positively complete 2-dimensional Randers metric is recurrent if and only if it is a Riemannian or locally Minkowskian metric. Next, we study two positive (or negative) projectively related Ricci parallel Finsler metrics on a compact manifold. We show that the projective equivalence is trivial and then the Riemannian curvatures are equal. In the same vein, we explore the class of Ricci-recurrent Randers metrics with closed and conformal form, and show that the related Riemannian metric is Ricci-recurrent if and only if the Randers metric is a Berwald metric. Finally, we find the necessary and sufficient condition under which a Kropina metric be Ricci-recurrent, provided that its one-form is closed and conformal.

Tensor Decompositions and Their Properties

In the present paper, we study two different approaches of tensor decomposition. The first part aims to study some properties of tensors that result from the fact that some components are vanishing in certain coordinates. It is proven that these conditions allow tensor decomposition, especially (1, σ), σ=1,2,3 tensors. We apply the results for special tensors such as the Riemann, Ricci, Einstein, and Weyl tensors and the deformation tensors of affine connections. Thereby, we find new criteria for the Einstein spaces, spaces of constant curvature, and projective and conformal flat spaces. Further, the proof of the theorem of Mikeš and Moldobayev is repaired. It has been used in many works and it is a generalization of the criteria formulated by Schouten and Struik. The second part deals with the properties of a special differential operator with respect to the general decomposition of tensor fields on manifolds with affine connection. It is shown that the properties of special differential operators are transferred to the components of a given decomposition.

Geometry of $\ast$-$k$-Ricci-Yamabe soliton and gradient $\ast$-$k$-Ricci-Yamabe soliton on Kenmotsu manifolds

The goal of the current paper is to characterize the $\ast$-$k$-Ricci-Yamabe soliton within the framework on Kenmotsu manifolds. Here, we have shown the nature of the soliton and found the scalar curvature when the manifold admits the $\ast$-$k$-Ricci-Yamabe soliton on the Kenmotsu manifold. Next, we have evolved the characterization of the vector field when the manifold satisfies the $\ast$-$k$-Ricci-Yamabe solitons. Also we have embellished some applications of vector field as torse-forming in terms of $\ast$-$k$-Ricci-Yamabe soliton on Kenmotsu manifold. Then, we studied the gradient $\ast$-$k$-Ricci-Yamabe soliton to yield the nature of the Riemannian curvature tensor. We have developed an example of a $\ast$-$k$-Ricci-Yamabe soliton on a 5-dimensional Kenmotsu manifold to prove our findings.

A Gaussian kernel for Kendall’s space of [formula omitted]-D shapes

In this paper, we develop an approach to exploit kernel methods with data lying on the m-D Kendall shape space. When data arise in a finite-dimensional curved Riemannian manifold, as in this case, the usual Euclidean computer vision and machine learning algorithms must be treated carefully. A good approach is to use positive definite kernels on manifolds to embed the manifold with its corresponding metric in a high-dimensional reproducing kernel Hilbert space, where it is possible to utilize algorithms developed for linear spaces. Different Gaussian kernels can be found in the literature on the 2-D Kendall shape space to perform this embedding. The main novelty of this work is to provide a Gaussian kernel for the m-D Kendall shape space. This new Kernel coincides in the case m=2 with the Gaussian kernels most widely used in the Kendall planar shape space and allows to define an embedding of the m-D Kendall shape space into a reproducible kernel Hilbert space. As far as we know, the complexity of the m-D Kendall shape space has meant that this embedding has not been addressed in the literature until now. This methodology will be tested on a machine learning problem with a simulated and a real data set.

Some Curvature Properties of Finsler Warped Product Metrics

The class of warped product metrics can often be interpreted as key space models for the general theory of relativity and theory of space-time. In this paper, we first obtain the PDE characterization of Finsler warped product metrics with a vanishing Riemannian curvature. Moreover, we obtain equivalent conditions for locally Minkowski Finsler warped product spaces. Finally, we explicitly construct two types of non-Riemannian examples.

Ricci-cubic holographic dark energy

In this work, we propose the Ricci-cubic holographic dark energy model. The model is inspired by the cubic curvature invariant formed by the contraction of three Riemann tensors. A combination of Ricci scalar and the cubic invariant is used to describe the infrared cutoff of the holographic dark energy. Such a construction is extremely useful since the evolution does not depend on the past or future features of the universe, but completely on its present features. Moreover, the use of invariants makes the theory more fundamental in nature. We have constructed the model and studied its cosmological features. The analytical solutions of various cosmological parameters such as the density parameter, equation of state parameter, and deceleration parameter are extracted and their behaviour is studied. It is seen that the holographic dark energy model can exhibit all the cosmological epoch, sequentially starting from radiation in the early universe, followed by matter, and finally the dark energy dominated epoch at late times. The equation of state parameter shows that the model can exhibit quintessence nature, phantom-divide crossing, and even phantom nature depending on the choice of parameter spaces.