Showing papers on "Ricci decomposition published in 2023"

A Note on Some Generalized Curvature Tensor

Abstract: For any semi-Riemannian manifold (M, g) we define some generalized curvature tensor E as a linear combination of Kulkarni-Nomizu products formed by the metric tensor, the Ricci tensor and its square of given manifold. That tensor is closely related to quasi-Einstein spaces, Roter spaces and some Roter type spaces.

On the algebraic classification of the gravitational field in Weyl-Cartan space-times

Abstract: We present a complete algebraic classification for the curvature tensor in Weyl-Cartan geometry, by applying methods of eigenvalues and principal null directions on its irreducible decomposition under the group of global Lorentz transformations, thus providing a full invariant characterisation of all the possible algebraic types of the torsion and nonmetricity field strength tensors in Weyl-Cartan space-times. As an application, we show that in the framework of Metric-Affine Gravity the field strength tensors of a dynamical torsion field cannot be doubly aligned with the principal null directions of the Riemannian Weyl tensor in scalar-flat stationary and axisymmetric space-times.

The Geometry of Curved Spaces

Abstract: The mathematics required to analyse higher dimensional curved spaces and space-times is developed in this chapter. General coordinate transformations, tangent spaces, vectors and tensors are described. Lie derivatives and covariant derivatives are motivated and defined. The concepts of parallel transport and a connection is introduced and the relation between the Levi-Civita connection and geodesics is elucidated. Christoffel symbols the Riemann tensor are defined as well as the Ricci tensor, the Ricci scalar and the Einstein tensor, and their algebraic and differential properties are described (though technical details of the derivationa of the Rimeann tensor are let to an appendix).

Para-Sasaki-like Manifolds with Generalized Symmetric Metric Connection

Abstract: The present paper deals with the generalized symmetric metric connection defined on para-Sasaki-like manifolds. We derive a relation between the Levi-Civita connection and the generalized symmetric metric conneciton on the considered manifold. We investigate the curvature tensor, the Ricci tensor and scalar curvature tensor with respect to the generalized symmetric metric connection. We study para-Sasaki-like solitons on para-Sasaki-like manifolds with the generalized symmetric metric connection. Finally, we construct two examples of para-Sasaki-like manifolds admitting generalized symmetric metric connection and verify our some results.

$\left(LCS\right)_{n}-$Manifolds Admitting Almost $\eta-$Ricci Solitons on Some Special Curvature Tensors

Abstract: In this paper, we consider $\left(LCS\right)_{n}$ manifold admitting almost $\eta-$Ricci solitons by means of curvature tensors. Ricci pseudosymmetry concepts of $\left(LCS\right)_{n}$ manifold admitting $\eta-$Ricci soliton have introduced according to the choice of some special curvature tensors such as pseudo-projective, $W_{1}$, $W_{1}^{\ast}$ and $W_{2}.$ Then, again according to the choice of the curvature tensor, necessary conditions are searched for $\left(LCS\right)_{n}$ manifold admitting $\eta-$Ricci soliton to be Ricci semisymmetric. Then some characterizations are obtained and some classifications have made.

η-Ricci solitons on Lorentzian β-Kenmotsu manifold

Abstract: 휂-Ricci solitons on Lorentzian 훽-Kenmotsu manifold are considered an manifolds satisfying certain curvature Conditions, R(ξ,X).S=0, S(ξ,X).R=0, W2(ξ,X).S=0,S(ξ,X).W2=0 We proved that in Lorentzian β-Kenmotsu manifold (M,φ,ξ,η,푔). Then the existence of an 휂-Ricci solitons implies that M is Einstein manifold and if the Ricci curvature tensor satisfies, S(ξ,X).R=0, then Ricci solitons M is steady. If the condition 휇=0, then 휆=0, which shows that 휆is steady

Generalized conformal curvature tensor of lp-sasakian manifold

Abstract: The object of the present paper is to generalize conformal curvature tensor of LP-Sasakian manifold with the help of a new generalized (0, 2) symmetric tensor Z introduced by Mantica and Suh [7]. Various geometric properties of the generalized conformal curvature tensor of LP-Sasakian manifold have been studied. It is shown that a generalized conformally ϕ-Symmetric LP-Sasakian manifold is an η-Einstein manifold

Vector and Tensor

Abstract: The purpose of this chapter is to present part of basic properties of vector and tensor operations, which is frequently used in the rest of the book. At first, the laws of vector addition, scalar multiplication, scalar product, vector product and triple products, etc. are presented. Next, the definitions and attributes of dual vector are introduced in the same manner. Then, the tensor definition and its calculation are discussed. The second order tensor is selected as a special case to express the tensor manipulations in detail. Many types of tensors are involved, such as rotation tensor, curvature tensor, and the metric tensor for surface and three-dimensional mapping. Finally, the motion tensor is introduced with the aid of geometric description of infinitesimal motion for a line vector.

Class Nearly kahler Manifold of W – Projective Curvature Tensor

Abstract: The current study deals with new three classes of the manifold of w – projective curvature tensor. The aim of this paper to calculate differential - geometrical and topological properties closest for new classes , and , through it ,an equivalence relationship was obtained between these classes and one of or more the tensor compounds and the components of curvature tensor and with adjoint G-structure space. Finally, we discover a relationship between , , with each other.

Back to almost Ricci solitons

Abstract: In the paper, we study complete almost Ricci solitons using the concepts and methods of geometric dynamics and geometric analysis. In particular, we characterize Einstein manifolds in the class of complete almost Ricci solitons. Then, we examine compact almost Ricci solitons using the orthogonal expansion of the Ricci tensor, this allows us to substantiate the concept of almost Ricci solitons.

Kropina Metrics with Isotropic Scalar Curvature

Abstract: In this paper, we study Kropina metrics with isotropic scalar curvature. First, we obtain the expressions of Ricci curvature tensor and scalar curvature. Then, we characterize the Kropina metrics with isotropic scalar curvature on by tensor analysis.

Coarse Ricci curvature of weighted Riemannian manifolds

Abstract: We show that the generalized Ricci tensor of a weighted complete Riemannian manifold can be retrieved asymptotically from a scaled metric derivative of Wasserstein 1-distances between normalized weighted local volume measures. As an application, we demonstrate that the limiting coarse curvature of random geometric graphs sampled from Poisson point process with non-uniform intensity converges to the generalized Ricci tensor.

A note on some generalized curvature tensor

Abstract: For any semi-Riemannian manifold (M,g) we define some generalized curvature tensor as a linear combination of Kulkarni-Nomizu products formed by the metric tensor, the Ricci tensor and its square of given manifold. That tensor is closely related to quasi-Einstein spaces, Roter spaces and some Roter type spaces.

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

Abstract: 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 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) its Weyl tensor is always zero, so the inertia tensor is completely characterized by its (rank-2) Ricci contraction. So unlike in general relativity, the Weyl tensor does not cause any qualitatively new phenomenology for rigid-body dynamics in $n \geq 4$ dimensions.

Geometrical Properties of a Point-like Global Monopole Spacetime

Abstract: The objective of the paper is to study the geometric properties of the point-like global monopole (briefly, PGM) spacetime, which is a static and spherically symmetric solution of Einstein field equation. It is shown that PGM spacetime admits various types of pseudosymmetric structures, such as, pseudosymmetry due to Weyl conformal curvature tensor, pseudosymmetry due to concircular curvature tensor, pseudosymmetry due to conharmonic curvature tensor, Ricci generalized conformal pseudosymmetric due to projective curvature tensor, Ricci generalized projective pseudosymmetric etc. Moreover, it is proved that PGM spacetime is 2-quasi Einstein, generalized quasi-Einstein, Einstein manifold of degree 2 and its Weyl conformal curvature 2-forms are recurrent. It is also shown that the stress energy momentum tensor of the PGM spacetime realizes several types of pseudosymmetry, and its Ricci tensor is compatible for Riemann curvature, Weyl conformal curvature, projective curvature, conharmonic curvature and concircular curvature. Further, it is shown that PGM spacetime admits motion, curvature collineation and Ricci collineation. Also, the notion of curvature inheritance (resp., curvature collineation) for the (1,3)-type curvature tensor is not equivalent to the notion of curvature inheritance (resp., curvature collineation) for the (0,4)-type curvature tensor as it is shown that such distinctive properties are possessed by PGM spacetime. Hence the notions of curvature inheritance defined by Duggal [1] and Shaikh and Datta [2] are not equivalent.

Minimal Length Deformation of Spacetime Curvature

Abstract: When generalized noncommutative Heisenberg algebra accommodating impacts of finite gravitational fields as specified by loop quantum gravity, doubly–special relativity, and string theory, for instance, is thoughtfully applied to the eight-dimensional manifold (M8), the generalization of the Riemannian manifold becomes imminent. By constructing the deformed affine connections on a four-dimensional Riemannian manifold, we have determined the minimal length deformation of Riemann curvature tensor and its contractions, the Ricci curvature tensor, and Ricci scalar. Consequently, we have been able to construct the deformed Einstein tensor. As in Einstein’s classical theory of general relativity, we have proved that the covariant derivative of the deformed Einstein tensor vanishes, as well. We conclude that the minimal length correction suggests a correction to the spacetime curvature which is manifest in the additional curvature terms of the corrected Riemann tensor and its contractions. Accordingly, the spacetime curvature endows additional curvature and geometrical structure.

Normal paracontact metri̇c space form on w_0- curvature tensor

Abstract: 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, concircular curvature tensors are discussed on W_0-curvature tensor. With the help of these curvature conditions, important characterizations of normal paracontact metric space forms are obtained.

Back to Almost Ricci Solitons

Abstract: In the paper, we study complete almost Ricci solitons using the concepts and methods of geometric dynamics and geometric analysis. In particular, we characterize Einstein manifolds in the class of complete almost Ricci solitons. Then, we examine compact almost Ricci solitons using the orthogonal expansion of the Ricci tensor, this allows us to substantiate the concept of almost Ricci solitons.

Conformal η-Ricci soliton in Lorentzian para Kenmotsu manifolds

Abstract: The objective of the present paper is to study conformal η-Ricci soliton on Lorentzian Para-Kenmotsu manifolds with some curvature conditions. We study Concircular curvature tensor, Quasi conformal curvature tensor, Codazi type of Ricci tensor and cyclic parallel Ricci tensor in Lorentzian Para-Kenmotsu manifolds. At last we give examples of such manifolds.

A Note on the Geometry of RW Space-Times

Abstract: A conformally flat GRW space-time is a perfect fluid RW space-time. In this note, we investigated the influence of many differential curvature conditions, such as the existence of recurrent and semi-symmetric curvature tensors. In each case, the form of the Ricci curvature tensor, the energy–momentum tensor, the energy density, the pressure of the fluid, and the equation of state are determined and interpreted. For example, it is demonstrated that a Ricci semi-symmetric RW space-time reduces to Einstein space-time or a Ricci recurrent RW space-time, and the perfect fluid space-time is referred to as Yang pure space-time or dark matter era.

Spacetime Admitting Semiconformal Curvature Tensor in f(ℛ) Modify Gravity

Abstract: The primary goal of this paper is to examine spacetimes admitting semiconformal curvature tensor in [Formula: see text] modify gravity. The semiconformal flatness of general spacetime and spacetime in [Formula: see text] gravity with perfect fluid, has been analyzed. For this consideration, we generate the forms of isotropic pressure [Formula: see text] and energy density [Formula: see text]. After that, a few energy conditions are taken into account. Finally, we study the divergence-free semiconformal curvature tensor in [Formula: see text] gravity in presence of perfect fluid. We emphasize that for recurrent or bi-recurrent energy–momentum tensor, Ricci tensor of this spacetime is semi-symmetric and consequently, the resulting spacetimes either accomplish inflation or possess fixed isotropic pressure and energy density.

On semi-Riemannian manifolds satisfying some generalized Einstein metric conditions

Abstract: The difference tensor R.C-C.R of a semi-Riemannian manifold (M,g), dim M > 3, formed by its Riemannian-Christoffel curvature tensor R and the Weyl conformal curvature tensor C, under some assumptions, can be expressed as a linear combination of (0,6)-Tachibana tensors Q(A,T), where A is a symmetric (0,2)-tensor and T a generalized curvature tensor. These conditions form a family of generalized Einstein metric conditions. In this survey paper we present recent results on manifolds and submanifolds, and in particular hypersurfaces, satisfying such conditions.