Riemann Tensor Research Articles

Generalized Learning Vector Quantization With Log-Euclidean Metric Learning on Symmetric Positive-Definite Manifold.

In many classification scenarios, the data to be analyzed can be naturally represented as points living on the curved Riemannian manifold of symmetric positive-definite (SPD) matrices. Due to its non-Euclidean geometry, usual Euclidean learning algorithms may deliver poor performance on such data. We propose a principled reformulation of the successful Euclidean generalized learning vector quantization (GLVQ) methodology to deal with such data, accounting for the nonlinear Riemannian geometry of the manifold through log-Euclidean metric (LEM). We first generalize GLVQ to the manifold of SPD matrices by exploiting the LEM-induced geodesic distance (GLVQ-LEM). We then extend GLVQ-LEM with metric learning. In particular, we study both 1) a more straightforward implementation of the metric learning idea by adapting metric in the space of vectorized log-transformed SPD matrices and 2) the full formulation of metric learning without matrix vectorization, thus preserving the second-order tensor structure. To obtain the distance metric in the full LEM learning (LEML) approaches, two algorithms are proposed. One method is to restrict the distance metric to be full rank, treating the distance metric tensor as an SPD matrix, and readily use the LEM framework (GLVQ-LEML-LEM). The other method is to cast no such restriction, treating the distance metric tensor as a fixed rank positive semidefinite matrix living on a quotient manifold with total space equipped with flat geometry (GLVQ-LEML-FM). Experiments on multiple datasets of different natures demonstrate the good performance of the proposed methods.

Analog of the Sommerfeld Law in Quantum Vacuum

The activation temperature T in the de Sitter environment is twice the Gibbons–Hawking temperature, related to the cosmological horizon. We consider the activation temperature as the local temperature of the de Sitter vacuum, and construct the local thermodynamics of the de Sitter state. This thermodynamics includes also the gravitational coupling K and the scalar Riemann curvature mathcal{R} as the thermodynamically conjugate variables. These variables modify the thermodynamics of the Gibbs–Duhem relation in the de Sitter state. The free energy density is proportional to - {{T}^{2}}, which is similar to that in the nonrelativistic Fermi liquids and in relativistic matter with equation of state w = 1. The local entropy is proportional to the local temperature, while the total entropy inside the cosmological horizon is A{text{/}}4G, where A is the area of the horizon. This entropy is usually interpreted as the entropy of the cosmological horizon. We also consider the possible application of the de Sitter thermodynamics to the Schwarzschild–de Sitter black hole and to black and white holes with the de Sitter cores.

Robot grasping and regrasping kinematics using Lie algebra, the geodesic, and Riemann curvature tensor

Robot grasping and regrasping kinematics using Lie algebra, the geodesic, and Riemann curvature tensor

Propagation of anisotropic gravitational and electromagnetic waves at very high energies

We analyze the dispersion relation for an anisotropic gravity-electromagnetic theory at very high energies. In particular for photons of very high energy. We start by introducing the anisotropic gravity-gauge vector field model. It is invariant under spacelike diffeomorphisms, time parametrization, and U(1) gauge transformations. It includes high-order spacelike derivatives as well as polynomial expressions of the Riemann and field strength tensor fields. It is based on the Hořava-Lifshitz anisotropic proposal. We show its consistency, and the stability of the Minkowski ground state. Finally, we determine the exact zone at which the physical degrees of freedom, i.e. the transverse-traceless tensorial degrees of freedom and the transverse vectorial degrees of freedom propagate according to a linear wave equation. This is so, in spite of the fact that there exists in the zone a non-trivial Newtonian background of the same order. The wave equation contains spatial derivatives up to the sixth order, in the lowest order it exactly matches the relativistic wave equation. We then analyze the dispersion relation at very high energies in the context of recent experimental data. The qualitative predictions of the proposed model, concerning the propagation of highly energetic photons, are different from the ones obtained from the modified dispersion relation of the LIV models.

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.

Normal paracontact metric space form on $W_0$-curvature tensor

In this article, normal paracontact metric space forms are investigated on $W_0$-curvature tensor. Characterizations of normal paracontact space forms are obtained on $W_0$-curvature tensor. Special curvature conditions established with the help of Riemann, Ricci, and concircular curvature tensors are discussed on $W_0$-curvature tensor. Through these curvature conditions, some important characterizations of normal paracontact metric space forms are obtained. Finally, the need for further research is discussed.

Riemannian maps whose base manifolds admit a Ricci soliton

In this paper, we study Riemannian maps whose base manifolds admit a Ricci soliton and give a non-trivial example of such a Riemannian map. First, we find Riemannian curvature tensor for the base manifolds of Riemannian map $F$. Further, we obtain the Ricci tensor and calculate the scalar curvature of the base manifold. Moreover, we obtain necessary conditions for the leaves of $rangeF_\ast$ to be Ricci soliton, almost Ricci soliton, and Einstein. We also obtain necessary conditions for the leaves of $(rangeF_\ast)^\bot$ to be Ricci soliton and Einstein. Also, we calculate the scalar curvatures of $rangeF_\ast$ and $(rangeF_\ast)^\bot$ by using Ricci soliton. Finally, we study the harmonicity and biharmonicity of such a Riemannian map. We obtain a necessary and sufficient condition for such a Riemannian map between Riemannian manifolds to be harmonic. We also obtain necessary and sufficient conditions for a Riemannian map from a Riemannian manifold to a space form that admits Ricci soliton to be harmonic and biharmonic.

A New Way to Construct the Riemann Curvature Tensor Using Geometric Algebra and Division Algebraic Structure

A New Way to Construct the Riemann Curvature Tensor Using Geometric Algebra and Division Algebraic Structure

Born reciprocity and relativistic generalized uncertainty principle in Finsler structure: Fundamental tensor in discretized curved spacetime

The main findings of the generalized uncertainty principle (GUP), the phenomenological approach, for instance, the emergence of a minimal measurable length uncertainty, are obtained in various versions from theories of quantum gravity, such as string theory, loop quantum gravity, doubly special relativity and black hole physics. GUP counts for impacts of relativistic energies and finite gravitational fields on the fundamental theories of quantum mechanics (QM), the noncommutation and measurement uncertainty. Utilizing GUP in reconciling principles of general relativity (GR) and QM, thereby enables to draw convincing conclusions about quantum gravity. To resolve the shortcuts reported with the nonrelativistic three-dimensional GUP, namely, violation of Lorentz covariance, dependence on frame of reference, and violation of the linear additional law of momenta, we introduce relativistic four-dimensional generalized uncertainty principle (RGUP) to curved spacetime. To unify GR and QM, we apply the Born reciprocity principle (BRP), distance-momentum duality symmetry and RGUP to estimate the fundamental tensor in discretized curved spacetime. To this end, we generalize Riemann geometry. The Finsler geometry, which is characterized by manifold and Finsler structure, allows to directly apply RGUP to the Finsler structure of a free particle so that [Formula: see text] can be expressed as [Formula: see text], from which the metric tensor in discretized Riemann spacetime could be deduced. We conclude that [Formula: see text] is homogeneous with degree [Formula: see text] in [Formula: see text], while [Formula: see text] is [Formula: see text]-homogeneous resulting in [Formula: see text]. Despite, the astonishing similarity with the conformal transformation, know as Weyl tensor, this study suggests that principles of QMs could be unambiguously imposed on the resulting fundamental tensor. Also, we conclude that the features of Finsler geometry assumed in this study are likely the ones of the duel Hamilton geometry.

On the Jacobi Stability of Two SIR Epidemic Patterns with Demography

In the present work, two SIR patterns with demography will be considered: the classical pattern and a modified pattern with a linear coefficient of the infection transmission. By reformulating of each first-order differential systems as a system with two second-order differential equations, we will examine the nonlinear dynamics of the system from the Jacobi stability perspective through the Kosambi–Cartan–Chern (KCC) geometric theory. The intrinsic geometric properties of the systems will be studied by determining the associated geometric objects, i.e., the zero-connection curvature tensor, the nonlinear connection, the Berwald connection, and the five KCC invariants: the external force εi—the first invariant; the deviation curvature tensor Pji—the second invariant; the torsion tensor Pjki—the third invariant; the Riemann–Christoffel curvature tensor Pjkli—the fourth invariant; the Douglas tensor Djkli—the fifth invariant. In order to obtain necessary and sufficient conditions for the Jacobi stability near each equilibrium point, the deviation curvature tensor will be determined at each equilibrium point. Furthermore, we will compare the Jacobi stability with the classical linear stability, inclusive by diagrams related to the values of parameters of the system.

Hawking temperature as the total Gauss–Bonnet invariant of the region outside a black hole

We provide two novel ways to compute the surface gravity (kappa ) and the Hawking temperature (T_{H}) of a stationary black hole: in the first method T_{H} is given as the three-volume integral of the Gauss–Bonnet invariant (or the Kretschmann scalar for Ricci-flat metrics) in the total region outside the event horizon; in the second method it is given as the surface integral of the Riemann tensor contracted with the covariant derivative of a Killing vector on the event horizon. To arrive at these new formulas for the black hole temperature (and the related surface gravity), we first construct a new differential geometric identity using the Bianchi identity and an antisymmetric rank-2 tensor, valid for spacetimes with at least one Killing vector field. The Gauss–Bonnet tensor and the Gauss–Bonnet scalar play a particular role in this geometric identity. We calculate the surface gravity and the Hawking temperature of the Kerr and the extremal Reissner–Nordström holes as examples.

Boundary and bulk notions of transport in the AdS3/CFT2 correspondence

We construct operators in holographic two-dimensional conformal field theory, which act locally in the code subspace as arbitrary bulk spacelike vector fields. Key to the construction is an interplay between parallel transport in the bulk spacetime and in kinematic space. We outline challenges, which arise when the same construction is extended to timelike vector fields. We also sketch several applications, including boundary formulations of the bulk Riemann tensor, dreibein, and spin connection, as well as an application to holographic complexity.

On the Geometry of Tangent Bundle and Unit Tangent Bundle with Deformed-Sasaki Metric

Let $(M^{m}, g)$ be a Riemannian manifold and $TM$ its tangent bundle equipped with a deformed Sasaki metric. In this paper, firstly we investigate all forms of Riemannian curvature tensors of $TM$ (Riemannian curvature tensor, Ricci curvature, sectional curvature and scalar curvature). Secondly, we study the geometry of unit tangent bundle equipped with a deformed Sasaki metric, where we presented the formulas of the Levi-Civita connection and also all formulas of the Riemannian curvature tensors of this metric.

On the Symmetries of a Kaehler Manifold

The natural symmetries of Riemannian manifolds are described by the symmetries of its Riemann curvature tensor. In that sense, the most symmetric manifolds are the constant sectional curvature ones. Its natural generalizations are locally symmetric manifolds, semisymmetric manifolds, and pseudosymmetric manifolds. The analogous generalizations of constant holomorphic sectional curvature Kaehler manifolds are locally symmetric Kaehler manifolds, semisymmetric Kaehler manifolds, and holomorphically pseudosymmetric Kaehler manifolds. Do they differ in some way from their Riemannian analogues? Yes, we prove they can all be characterized only in terms of holomorphic planes. Furthermore, the concept of holomorphically pseudosymmetric Kaehler manifold is different from the classical notion of pseudosymmetric Riemannian manifold proposed by Deszcz. We study some relations between both definitions of pseudosymmetry and the so called double sectional curvatures in the sense of Deszcz. We also present a geometric interpretation of the complex Tachibana tensor and a new characterization of constant holomorphic sectional curvature Kaehler manifolds.

Role of Strong Matter‐Field Coupling on Axial Complex Systems

AbstractEven though extended gravity theories have come up with different healthy measures of complexity for spherical and non‐spherical systems, the study of complexity factors for axially symmetric bodies under the influence of strong matter‐field coupling is still up for debate. The formation of complexity factors for static axially symmetric bodies is induced by the cracking of the Riemann curvature tensor in an orthogonal manner as initiated by Herrera et al. [Phys. Rev. D 99, 044049(2019)]. We provide the continuation of this work in the gravitational theory, where that indicates the powerful matter‐field coupling due to the bilateral influence of the curvature and matter tensors. We proceed our analysis by considering axial geometry confined with anisotropic fluid constituents. In this context, we compute the field equations for the system under discussion. To examine the source's energy content, two dynamical equations that correspond to independent coordinates are constructed. The curvature tensor is split into tensorial quantities to formulate the scalar functions (structure scalars), which involve the physical attributes of the source in the domain of high curvature regimes. Among all, three of the scalars 's are nominated as complexity factors, whose inclusion enhances the inhomogeneity of the propagating system. The vanishing complexity factors criteria are set out for the specific form of the cosmic model that provides the favorable setting to minimize the system's complexity and boost its stability. Furthermore, some analytical insights under various backgrounds of the matter spheroids are presented.

A two-field dark energy model with cubic contractions of the Riemann tensor

The paper proposes a novel cosmological model based on a two-field scenario, where the two fields are independently coupled with a specific invariant, based on cubic contractions of the Riemann tensor. After obtaining the modified Friedmann equations and the specific Klein–Gordon relations, the investigation studies the corresponding physical features by adopting the linear stability theory in a quintom scenario case where the coupling functions and potential energies have an exponential behavior. The corresponding investigation revealed a high complexity of the phase space structure, with various stationary points which can explain different aspects in the evolution of our Universe, the matter dominated epoch, and the late time accelerated stage. The present model can represent a viable solution to the dark energy problem due to the high complexity of the phase space and the existence of the scaling solutions.

Stability of non-degenerate Ricci-type Palatini theories

We study the stability of theories where the gravitational action has arbitrary algebraic dependence on the three first traces of the Riemann tensor: the Ricci tensor, the co-Ricci tensor, and the homothetic curvature tensor. We collectively call them Ricci-type tensors. We allow arbitrary coupling to matter. We consider the case when the connection is unconstrained, and the cases when either torsion or non-metricity is assumed to vanish. We find which combinations of Ricci-type tensors lead to new degrees of freedom around Minkowski and FLRW space, and when there are ghosts. None of the theories with new degrees of freedom are healthy, except the previously known case when torsion is zero and the action depends only on the Ricci tensor. We find that projective invariance is not a sufficient condition for a theory to be ghost-free.

English

In this study, we state some basic results about the geometry of the special linear group SL(2,R), seen as a subset of  in terms of the left invariant fields, such as bracketing, Levi Civita connection ∇ and Riemann curvature tensor R, we give some basic theorems for Mannheim partner curves  in the special linear group. We also find the relations between the curvatures and torsions of these associated curves and we give necessary and sufficient conditions for a given curve to be a Mannheim partner curve of another given curve through a relation between its curvature and torsion. Key words: Special linear group, Mannheim curves, Mannheim, partner curves.

On Some Role of Riemannian Geometry

This paper examines the mathematical significance of Riemannian geometry, including surfaces, Riemannian curvature, Gauss curvature, manifolds, geodesics, and the relationships between them. It also explores the applications of Riemannian geometry in various concepts.

Relativistic theory of elastic bodies in the presence of gravitational waves

The equations of motion governing small elastic oscillations of materials, induced by gravitational waves, are derived from the general framework of Carter and Quintana. In transverse-traceless gauge, no bulk forces are present, and the gravitational wave is found to act as an effective surface traction. For thin rods, an equivalent description is given, in which there is no surface traction, but a bulk acceleration, which is related to the Riemann curvature of the gravitational wave. The resulting equations are compared to those of the Synge–Bennoun elasticity theory.