Riemann Tensor Research Articles

HYPERBOLIC KENMOTSU MANIFOLD ADMITTING A NEW TYPE OF SEMI-SYMMETRIC NON-METRIC CONNECTION

In this work, we study a new type of semi-symmetric non-metric connection on hyperbolic Kenmotsu manifold. Some Riemannian curvature’s characteristics on hyperbolic Kenmotsu manifold are investigated. The properties of semi-symmetric, locally ϕ-symmetric and Ricci semi-symmetric hyperbolic Kenmotsu manifold endowed with a new type of semi-symmetric nonmetric connection are evaluated. A semi-symmetric and Ricci semi-symmetric hyperbolic Kenmotsu manifold with a semi-symmetric non-metric connection is also demonstrated, the Ricci soliton of data (g1,ξ,λ) is shrinking. Finally, we demonstrate our results with a 3-dimensional example.

Characterization of Almost \(\eta\) -Ricci Solitons With Respect to Schouten-van Kampen Connection on Sasakian Manifolds

In this paper, we investigate Sasakian manifolds that admit almost \(\eta\) -Ricci solitons with respect to the Schouten-van Kampen connection using certain curvature tensors. Concepts of Ricci pseudosymmetry for Sasakian manifolds admitting \(\eta\)-Ricci solitons are introduced based on the selection of specific curvature tensors such as Riemann, concircular, projective, pseudo-projective, M-projective, and W2 tensors. Subsequently, necessary conditions are established for a Sasakian manifold admitting \(\eta\)-Ricci soliton with respect to the Schouten-van Kampen connection to be Ricci semisymmetric, based on the choice of curvature tensors. Characterizations are then derived, and classifications are made under certain conditions.

Classification of generalised higher-order Einstein-Maxwell Lagrangians

We classify all higher-order generalised Einstein-Maxwell Lagrangians that include terms linear in the curvature tensor and quadratic in the derivatives of the electromagnetic field strength tensor. Using redundancies due to the Bianchi identities, dimensionally dependent identities and boundary terms, we show that a general Lagrangian of this form can always be reduced to a linear combination of only 21 terms, with coefficients that are arbitrary functions of the two scalar invariants derived from the field strength. We give an explicit choice of basis where these 21 terms include 3 terms linear in the Riemann tensor and 18 terms quadratic in the derivatives of the field strength.

Observational constraints for cubic gravity theory based on third order contractions of the Riemann tensor

The paper studies different observational features in the case of a specific cubic gravity theory, based on third order contractions of the Riemann tensor. Considering viable cosmic chronometers data, baryon acoustic oscillations, and supernovae, we analyze the viability of such a theoretical model, obtaining specific constraints for different parameters in the current scenario. It is shown that the present extension of the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM cosmological model is compatible with recent data sets. The results indicate that the dark energy equation of state is exhibiting a phantom regime in the near past in the case of the best fitted values, a behavior which is in agreement with various phenomenological studies.

Two loop renormalization of scalar theories using a geometric approach

We derive a general formula for two-loop counterterms in Effective Field Theories (EFTs) using a geometric approach. This formula allows the two-loop results of our previous paper to be applied to a wide range of theories. The two-loop results hold for loop graphs in EFTs where the interaction vertices contain operators of arbitrarily high dimension, but at most two derivatives. We also extend our previous one-loop result to include operators with an arbitrary number of derivatives, as long as there is at most one derivative acting on each field. The final result for the two-loop counterterms is written in terms of geometric quantities such as the Riemann curvature tensor of the scalar manifold and its covariant derivatives. As applications of our results, we give the two-loop counterterms and renormalization group equations for the O(n) EFT to dimension six, the scalar sector of the Standard Model Effective Field Theory (SMEFT) to dimension six, and chiral perturbation theory to order p6.

A Survey on Riemannian Curvature Tensor for Certain Classes of Almost Contact Metric Manifolds

A Survey on Riemannian Curvature Tensor for Certain Classes of Almost Contact Metric Manifolds

Parallel and totally umbilical hypersurfaces of the four‐dimensional Thurston geometry Sol04$\text{Sol}^4_0$

AbstractWe study hypersurfaces of the four‐dimensional Thurston geometry , which is a Riemannian homogeneous space and a solvable Lie group. In particular, we give a full classification of hypersurfaces whose second fundamental form is a Codazzi tensor—including totally geodesic hypersurfaces and hypersurfaces with parallel second fundamental form—and of totally umbilical hypersurfaces of . We also give a closed expression for the Riemann curvature tensor of , using two integrable complex structures.

Compact stellar model with vanishing complexity under Vaidya–Tikekar background geometry

We make use of the condition of vanishing complexity, based on the current definition proposed by Herrera (Phys Rev D 97:044010, 2018), to find exact interior solutions to the Einstein equations for describing compact stellar objects. In the framework of general relativity, the complexity factor is an outcome of the orthogonal splitting of the Riemann tensor from which structure scalars are obtained. By using the Vaidya–Tikekar (V–T) metric ansatz (J Astrophys Astron 3:325, 1982) for the spacetime of a static spherically symmetric matter distribution, we model superdense, relativistic stars. The interior spacetime is matched to the exterior Schwarzschild solution across the boundary of the star where the radial pressure vanishes. The physical viability of the model has been tested following the current data corresponding to the pulsar 4U1820-30\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ 4U 1820-30 $$\\end{document}. The stability of the model fulfilled the given criteria, namely the Tolman–Oppenheimer–Volkoff equation, the adiabatic index and the causality conditions.

Isotropization and complexity shift of gravitationally decoupled charged anisotropic sources

This article focuses on investigating the role of decoupling in isotropizing anisotropic, self-gravitational charged sources with spherical symmetry through a well-known gravitational technique, known as minimal geometric deformation (MGD). This technique separates the given system into two gravitational systems: the Einstein-Maxwell system and the gravitational system governed by additional source. We employ novel approaches, including the zero complexity factor and isotropization techniques, to construct various charged compact star models using the Tolman IV as the seed source within the framework of the MGD scheme. The term complexity factor emerges as one of the structure-defining scalar quantities resulting from the orthogonal splitting of the Riemann–Christoffel curvature tensor, as proposed by Herrera (Phys Rev D 97(4):044010, 2018). This scalar function, denoted as YTF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Y_{TF}$$\\end{document}, is associated with the fundamental structural characteristics of self-gravitational compact configurations. Our approach is innovative in that it derives the deformation functions by imposing the requirement of YTF=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Y_{TF}=0$$\\end{document} and employs isotropization techniques for electrically charged anisotropic configurations.

Complexity of dynamical spherical system in [formula omitted] gravity

This paper presents complexity measure of a dynamical spherical configuration with anisotropic distribution in energy–momentum squared gravity. The self-gravitating bodies become complex due to non-uniform energy density, asymmetrical pressure, heat loss and contribution of modified terms. By orthogonally decomposing the Riemann tensor, we analyze the structure scalars and obtain the complexity factor that accounts for all the fundamental characteristics of the system. Furthermore, by using the homologous mode as the simplest pattern of evolution, we study the dynamics of the celestial configuration. We also discuss dissipative/non-dissipative fluids in the context of homologous and complexity-free cases. Finally, we investigate a criterion for which the complexity-free condition remains stable throughout evolutionary process. It is concluded that the contribution of product as well as squared components of the considered framework leads to a more complex system.

Effect of local fractional derivatives on Riemann curvature tensor

In this paper, we give a main example indicating the ineffectiveness of the local fractional derivatives on the Riemann curvature tensor that is a common tool in calculating curvature of a Riemannian manifold. For this, first we introduce a general local fractional derivative operator that involves the mostly used ones in the literature as conformable, alternative, truncated M− and V−fractional derivatives. Then, according to this general operator, a particular Riemannian metric on the real affine space Rn that is different from the Euclidean one is defined. In conclusion, our main example states that the Riemann curvature tensor of Rn endowed with this particular metric is identically 0, that is, one is locally isometric to Euclidean space.

Noncommutative branch‐cut quantum gravity with a self‐coupling inflation scalar field: Dynamical equations

AbstractThis article focuses on a recently developed formulation based on the noncommutative branch‐cut cosmology, the Wheeler‐DeWitt (WdW) equation, the Hořava–Lifshitz quantum gravity, chaotic and the coupling of the corresponding Lagrangian approach with the inflaton scalar field. Assuming a mini‐superspace of variables obeying the noncommutative Poisson algebra, we examine the impact of the inflaton scalar field on the evolutionary dynamics of the branch‐cut Universe scale factor, characterized by the dimensionless helix‐like function . This scale factor characterizes a Riemannian foliated spacetime that topologically overcomes the primordial singularities. We take the Hořava–Lifshitz action modeled by branch‐cut quantum gravity as our starting point, which depends on the scalar curvature of the branched Universe and its derivatives and which preserves the diffeomorphism property of General Relativity, maintaining compatibility with the Arnowitt–Deser–Misner formalism. We then investigate the sensitivity of the scale factor of the branch‐cut Universe's dynamics.

Curvature analysis of concircular trajectories in doubly warped product manifolds

<p>The aim of this research paper was to explore the various characteristics of the doubly warped product manifold, focusing particularly on aspects such as the Hessian, Riemannian curvature, Ricci curvature, and concircular curvature tensor components. By examining the necessary conditions that would classify the manifold as Riemann-flat, Ricci-flat, and concircularly-flat, the study aimed to expand our understanding of these concepts. To achieve this, the research incorporated the application of these findings to a generalized Robertson-Walker doubly warped product manifold scenario. This approach allowed us to identify and analyze the specific circumstances under which the manifold displayed concircular flatness.</p>

О некоторых тензорах 6-мерных уплощающихся эрмитовых подмногообразий алгебры Кэли

In the present note, we consider six-dimensional Hermitian planar sub­manifolds of Cayley algebra. The almost Hermitian structure on such a six-dimensional submanifold is induced by means of so-called Brown — Gray three-fold vector cross products in Cayley algebra. The six-dimen­sio­nal Hermitian planar submanifolds of the octave algebra contain all six-dimensional Kählerian submanifolds of Cayley algebra. However, there exist non-Kählerian six-dimensional Hermitian planar submanifolds in the octave algebra. The components of the tensor of the Riemannian curvature for a six-dimensional almost Hermitian planar submanifold of Cayley algebra are computed. Remark that the tensor of Riemannian curvature plays a fun­damental role in geometry of almost Hermitian manifolds. Knowing all components of the tensor of the Riemannian curvature for a six-dimen­sio­nal almost Hermitian planar submanifold of the octave algebra, it is possible to study so-called Gray’s identities for this submanifold. The components of the Ricci tensor and of the tensor of conformal curvature (known also as Weyl tensor) for a six-dimensional almost Her­mitian planar submanifold of Cayley algebra are also computed.

The Disformal Transformation of the Einstein-Hilbert Action

Abstract: A disformal transformation is a generalization of the well-known conformal transformation commonly elaborated in mainstream graduate texts on general relativity. This transformation is one of the most important operations in serious attempts in the literature to address pressing problems about the universe such as dark energy and dark matter. In this work, we derive the disformal transformation of the Einstein-Hilbert action following effectively the same logic as that for the conformal transformation. The resulting action, however, contains “anomalous” terms that could be construed as leading to equations of motion that could go beyond second order in spacetime derivatives, signaling instability of the transformed action. We demonstrate that these terms can be manipulated by way of decomposing the Riemann curvature tensor and shifting derivative indices through integration by parts, to end up with a manifestly stable action. Keywords: Disformal transformation, Conformal transformation, Einstein-Hilbert action, Ostrogradsky instability, Horndeski theory. PACS: 04.20.−q, 04.50.Kd, 04.20.Fy.

On Tangent Bundles of Submanifolds of a Riemannian Manifold Endowed with a Quarter-Symmetric Non-metric Connection

The object of this article is to study a quarter-symmetric non-metric connection in the tangent bundle and induced metric and connection on submanifold of co-dimension 2 and hypersurface concerning the quarter-symmetric non-metric connection in the tangent bundle. The Weingarten equations concerning the quarter-symmetric non-metric connection on a submanifold of co-dimension 2 and the hypersurface in the tangent bundle are obtained. Finally, authors deduce the Riemannian curvature tensor and Gauss and Codazzi equations on a submanifold of co-dimension 2 and hypersurface of the Riemannian manifold concerning the quarter-symmetric non-metric connection in the tangent bundle.

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.

Design of 3D anisotropic Voronoi porous structure driven by stress field

Porous structures exhibit excellent mechanical properties and are widely utilized in engineering design. However, there is still substantial potential for enhancing the mechanical performance of these structure. By customizing the anisotropic design of each pore in the porous structure, the strength and stiffness of the structure can be significantly enhanced. In this work, we present a novel method for designing 3D anisotropic Voronoi porous structures (AVPSs). To construct these structures, we propose a novel algorithm for generating 3D anisotropic Voronoi diagrams (AVDs) based on Riemannian distance. The geometries of Voronoi cells are controlled to conform with both the stress distribution and the principal stress directions. The distribution of Voronoi cells are graded using a stress-based site sampling strategy, the orientation of Voronoi cells are modified by introducing stress-mapped Riemann tensor fields. The AVPSs are then modeled by computing implicit surfaces based on AVDs. The AVPSs conform closely to the geometric profile of the input model by clipping AVDs using the tetrahedral mesh. Furthermore, the results obtained from numerical simulations and experimental tests indicate that the AVPSs possess both high strength and high stiffness.

Path integral factorization and the gravitational effective action

We discuss the factorization and continuity properties of fields in the Euclidean gravitational path integral with higher dimension operators constructed from powers of the Riemann tensor. We construct the boundary terms corresponding to the microcanonical ensemble and show that the saddle point approximation to the path integral with a quasilocal energy constraint generally yields a saddle point with discontinuous temperature. This extends a previous result for the Euclidean Schwarzschild–de Sitter geometry in Einstein gravity and shows that it is robust against at least some types of quantum corrections from heavy fields. As an application, we compute the entropy of SdS in D = 4 using the BTZ method. Our result matches the entropy calculated using Wald’s formula.

CONFORMALLY FLAT KÄHLERIAN SPACES

We study the geometric properties of Kähler spaces admitting conformal mappings, other than homothetic mappings, to flat pseudo-Riemannian spaces. It is proved that there are no non-flat conformally flat Kählerian spaces with dimensions other than four. It is shown that four-dimensional conformally flat spaces can be embedded in six-dimensional flat pseudo-Riemannian spaces. An idempotent covariantly stable metric tensor is constructed in conformally flat Kählerian spaces and thus it is proved that these spaces are reduced pseudo-Riemannian spaces. The study is carried out locally, using tensor methods without restrictions on the signature and signification of the metric tensor of the Kählerian space. In this paper, we make extensive use of a special operation (conjugation) introduced for Kählerian spaces and its properties for Riemann and Ricci tensors.