Conformal Vector Fields Research Articles

Conformal vector fields on almost co-Kähler manifolds

Abstract In this paper, we characterize almost co-Kähler manifolds with a conformal vector field. It is proven that if an almost co-Kähler manifold has a conformal vector field that is collinear with the Reeb vector field, then the manifold is a K-almost co-Kähler manifold. It is also shown that if a (κ, μ)-almost co-Kähler manifold admits a Killing vector field V, then either the manifold is K-almost co-Kähler or the vector field V is an infinitesimal strict contact transformation, provided that the (1,1) tensor h remains invariant under the Killing vector field.

Characterization of Almost Yamabe Solitons and Gradient Almost Yamabe Solitons with Conformal Vector Fields

In this paper, some sufficient conditions of almost Yamabe solitons are established, such that the solitons are Yamabe metrics, by which we mean metrics of constant scalar curvature. This is achieved by imposing fewer topological constraints. The properties of the conformal vector fields are exploited for the purpose of establishing various necessary criteria on the soliton vector fields of gradient almost Yamabe solitons so as to obtain Yamabe metrics.

Conformal vector fields of Bianchi type-I spacetimes

This paper aims to investigate Conformal Vector Fields (CVFs) of Bianchi type-I spacetimes. A set of 10-coupled Partial Differential Equations (PDEs) is obtained from the conformal Killing equations. These equations are solved by using direct integration techniques to explore the components of CVFs. Utilizing these components, we get a system of three integrability conditions. Finally, we achieve CVFs along with conformal factors for unique possibilities of unknown metric functions from the solution of these conditions. From our results, it is examined that Bianchi type-I spacetimes admit five or fifteen CVFs for specific choices of metric functions.

Mapping relativistic to ultra/non-relativistic conformal symmetries in 2D and finite sqrt{Toverline{T}} deformations

The conformal symmetry algebra in 2D (Diff(S1)⊕Diff(S1)) is shown to be related to its ultra/non-relativistic version (BMS3≈GCA2) through a nonlinear map of the generators, without any sort of limiting process. For a generic classical CFT2, the BMS3 generators then emerge as composites built out from the chiral (holomorphic) components of the stress-energy tensor, T and overline{T} , closing in the Poisson brackets at equal time slices. Nevertheless, supertranslation generators do not span Noetherian symmetries. BMS3 becomes a bona fide symmetry once the CFT2 is marginally deformed by the addition of a sqrt{Toverline{T}} term to the Hamiltonian. The generic deformed theory is manifestly invariant under diffeomorphisms and local scalings, but it is no longer a CFT2 because its energy and momentum densities fulfill the BMS3 algebra. The deformation can also be described through the original CFT2 on a curved metric whose Beltrami differentials are determined by the variation of the deformed Hamiltonian with respect to T and overline{T} . BMS3 symmetries then arise from deformed conformal Killing equations, corresponding to diffeomorphisms that preserve the deformed metric and stress-energy tensor up to local scalings. As an example, we briefly address the deformation of N free bosons, which coincides with ultra-relativistic limits only for N = 1. Furthermore, Cardy formula and the S-modular transformation of the torus become mapped to their corresponding BMS3 (or flat) versions.

SOME VECTORS FIELDS ON THE TANGENT BUNDLE WITH A SEMI-SYMMETRIC METRIC CONNECTION

Let $M$ is a (pseudo-)Riemannian manifold and $TM$ be its tangent bundlewith the semi-symmetric metric connection $\overline{\nabla }$. In thispaper, we examine some special vector fields, such as incompressible vectorfields, harmonic vector fields, concurrent vector fields, conformal vectorfields and projective vector fields on $TM$ with respect to thesemi-symmetric metric connection $\overline{\nabla }$ and obtain someproperties related to them.

Exploring conformal vector fields of Bianchi type-I perfect fluid solutions in f(T) gravity

In this paper, first, we deduce various classes of Bianchi type-I perfect fluid solutions in the [Formula: see text] gravity. In order to obtain the above-mentioned classes, we use some algebraic techniques that help to formulate 18 cases. As an application, we bifurcate the obtained classes according to their conformal vector fields (CVFs). Inspecting each class using the process of direct integration, we come to know that in one case, the space-time admits proper CVFs whereas in rest of the cases, obtained metrics either become conformally flat or admit underlying symmetries of the CVFs i.e. homothetic vector fields (HVFs) or Killing vector fields (KVFs). The overall dimension of CVFs for the Bianchi type-I perfect fluid space-times in [Formula: see text] gravity turns out to be three, four, five, six and fifteen.

Conformal Killing graphs in foliated Riemannian spaces with density: rigidity and stability

In this paper we investigate the geometry of conformal Killing graphs in a Riemannian manifold $\overline{M}_f^{ n+1}$ endowed with a weight function $f$ and having a closed conformal Killing vector field $V$ with conformal factor $\psi_V$, that is, graphs constructed through the flow generated by $V$ and which are defined over an integral leaf of the foliation $V^{\perp}$ orthogonal to $V$. For such graphs, we establish some rigidity results under appropriate constraints on the $f$-mean curvature. Afterwards, we obtain some stability results for $f$-minimal conformal Killing graphs of $ \overline{M}_f^{ n+1}$ according to the behavior of $ \psi_V$. Finally, related to conformal Killing graphs immersed in $\overline{M}_f^{n+1}$ with constant $f$-mean curvature, we study the strong stability.

Lie groups with conformal vector fields induced by derivations

A pseudo-Riemannian Lie group (G,〈⋅,⋅〉) is a connected and simply connected Lie group with a left-invariant pseudo-Riemannian metric of type (p,q). This paper is to study pseudo-Riemannian Lie groups with non-Killing conformal vector fields induced by derivations which is an extension from non-Killing left-invariant conformal vector fields. First we prove that a Riemannian (i.e. type (n,0)), Lorentzian (i.e. type (n−1,1)) or trans-Lorentzian (i.e. type (n−2,2)) Lie group with such a vector field is solvable. Then we construct non-solvable unimodular pseudo-Riemannian Lie groups with such vector fields for any min⁡(p,q)≥3. Finally, we give the classification for the Riemannian and Lorentzian cases.

Skew-symmetric endomorphisms in : a unified canonical form with applications to conformal geometry

We show the existence of families of orthonormal, future directed bases which allow to cast every skew-symmetric endomorphism of () in a single canonical form depending on a minimal number of parameters. This canonical form is shared by every pair of elements in differing by an orthochronous Lorentz transformation, i.e. it defines the orbits of the orthochronous Lorentz group O +(1, n) under the adjoint action on its algebra. Using this form, we obtain the quotient topology of . From known relations between and the conformal Killing vector fields (CKVFs) of the sphere , a canonical form for CKVFs follows immediately. This form is used to find adapted coordinates to an arbitrary CKVF that covers all cases at the same time. We do the calculation for even n and obtain the case of odd n as a consequence. Finally, we employ the adapted coordinates to obtain a wide class of TT-tensors for n = 3, which provide Cauchy data at conformally flat null infinity . Specifically, this class of data is characterized for generating Λ > 0-vacuum spacetimes with two-symmetries, one of which axial, admitting a conformally flat . The class of data is infinite dimensional, depending on two arbitrary functions of one variable as well as a number of constants. Moreover, it contains the data for the Kerr–de Sitter spacetime, which we explicitly identify within.

A note on classification of locally rotationally symmetric Bianchi type I space–times via conformal vector fields in f(T,B) gravity

The aim of this paper is to classify locally rotationally symmetric (LRS) Bianchi type I space-times via conformal vector fields (CVFs) in the [Formula: see text] gravity. In a first step, we find the space-times that are solutions of some [Formula: see text] gravity model. Furthermore, we classify the resulting space-times via CVFs. During the classification, there arise eight cases. Reviewing each case thoroughly, we find that in two cases, the space-times confess proper CVFs. The space-times in rest of the six cases either become conformally flat or tends to admit homothetic vector fields (HVFs) or the Killing vector fields (KVFs).

Ricci solitons on singly warped product manifolds and applications

The purpose of this article is to study implications of a Ricci soliton warped product manifold to its base and fiber manifolds. First, it is proved that if a warped product manifold is Ricci soliton then its factors are Ricci soliton. Then we study Ricci soliton on warped product manifolds admitting either a conformal vector field or a concurrent vector field. Finally, we study Ricci soliton on some warped product space-times.

SINGULAR FOLIATION GENERATED BY AN ORBIT OF FAMILY OF VECTOR FIELDS

Geometry of orbit is a subject of many investigations because it has important role in many branches of mathematics such as dynamical systems, control theory. In this paper it is studied geometry of orbits of conformal vector fields. It is shown that orbits of conformal vector fields are integral submanifolds of completely integrable distributions. Also for Euclidean space it is proven that if all orbits have the same dimension they are closed subsets.

Conformal Vector Fields and the De-Rham Laplacian on a Riemannian Manifold

We study the effect of a nontrivial conformal vector field on the geometry of compact Riemannian spaces. We find two new characterizations of the m-dimensional sphere Sm(c) of constant curvature c. The first characterization uses the well known de-Rham Laplace operator, while the second uses a nontrivial solution of the famous Fischer–Marsden differential equation.

Conformal vector fields for some vacuum classes of pp-waves space-times in ghost free infinite derivative gravity

The aim of this paper is to find conformal vector fields (CVFs) for some vacuum classes of the pp-waves space-times in the ghost free infinite derivative gravity (IDG). In order to find the CVFs of the above-mentioned space-times in the IDG, first, we deduce various classes of solutions by employing a classification procedure that in turn leads towards 10 cases. By reviewing each case thoroughly by direct integration technique, we find that there exists only one case for which the space-time admits proper CVFs whereas in rest of the cases, the space-time either becomes flat or it admits homothetic vector fields (HVFs) or Killing vector fields (KVFs). The overall dimension of CVFs for the pp-waves space-times in the IDG has turned out to be one, two, seven or fifteen.

Conformally Killing Fields on 2-Symmetric Five-Dimensional Lorentzian Manifolds

The papers of many mathematicians are devoted to the study of conformally Killing vector fields. Being a natural generalization of the concept of Killing vector fields, these fields generate a Lie algebra corresponding to the Lie group of conformal transformations of the manifold. Moreover, they generate the class of locally conformally homogeneous (pseudo) Riemannian manifolds studied by V.V. Slavsky and E.D. Rodionov. Ricci solitons, which R. Hamilton first considered, are another important area of research. Ricci solitons are a generalization of Einstein's metrics on (pseudo) Riemannian manifolds. The Ricci soliton equation has been studied on various classes of manifolds by many mathematicians. In particular, a general solution of the Ricci soliton equation was found on 2-symmetric Lorentzian manifolds of low dimension, and the solvability of this equation in the class of 3-symmetric Lorentzian manifolds was proved. The Killing vector fields make it possible to find the general solution of the Ricci soliton equation in the case of the constancy of the Einstein constant in the Ricci soliton equation. However, the role of the Killing fields is played by conformally Killing vector fields for different values of the Einstein constant. In this paper, we investigate conformal Killing vector fields on 5-dimensional 2-symmetric Lorentzian manifolds. The general solution of the conformal analog of the Killing equation on five-dimensional locally indecomposable 2-symmetric Lorentzian manifolds is described in local coordinates, discovered by A.S. Galaev and D.V. Alekseevsky.

Conformal Vector Fields and Ricci Soliton Structures on Natural Riemann Extensions

The framework of the paper is the phase universe, described by the total space of the cotangent bundle of a manifold M, which is of interest for both mathematics and theoretical physics. When M carries a symmetric linear connection, then $$T^*M$$ is endowed with a semi-Riemannian metric, namely the classical Riemann extension, introduced by Patterson and Walker and then by Willmore. We consider here a generalization provided by Sekizawa and Kowalski of this metric, called the natural Riemann extension, which is also a metric of signature (n, n). We give the complete classification of conformal and Killing vector fields with respect to an arbitrary natural Riemann extension. Ricci soliton is a topic that has been increasingly studied lately. Necessary and sufficient conditions for the phase space to become a Ricci soliton (or Einstein) are given at the end.

Natural Ricci Solitons on Tangent and Unit Tangent Bundles

Considering pseudo-Riemannian $g$-natural metrics on tangent bundles, we prove that the condition of being Ricci soliton is hereditary in the sense that a Ricci soliton structure on the tangent bundle gives rise to a Ricci soliton structure on the base manifold. Restricting ourselves to some class of pseudo-Riemannian $g$-natural metrics, we show that the tangent bundle is a Ricci soliton if and only if the base manifold is flat and the potential vector field is a complete lift of a conformal vector field. We give then a classification of conformal vector fields on a flat Riemannian manifold. When unit tangent bundles over a constant curvature Riemannian manifold are endowed with pseudo-Riemannian Kaluza-Klein type metric, we give a classification of Ricci soliton structures whose potential vector fields are fiber-preserving, inferring the existence of some of them which are non Einstein.

Variational Geometry for Surfaces in Conformally Flat Space

In this paper, close surfaces are considered in 3-dimensional harmonic conformally flat space in point of the variation. It is shown that if the conformal vector field be tangent to surface and the sign of the mean curvature does not change then surface is minimal. Also, it is determined that the critical point of mean curvature functional is homeomorphic to the sphere. Furthermore, the Euler-Lagrange equations associated to the mean curvature and Willmore functionals are determined.

P1-Curvature tensor in the space time of general relativity

The P1 - curvature tensor defined from W3 - curvature tensor has been studied in the space time of general relativity. The Bianchi like differential identity is satisfied by P1 - tensor if and only if the Ricci tensor is of Codazzi type. It is shown that Einstein like field equations can be expressed with the help of the contracted part of P1 - tensor, which is conserved if the energy momentum tensor is Codazzi type. Considering P1 -flat space time satisfying Einstein’s field equations with cosmological term, the existence of Killing vector field ξ is shown if and only if the Lie derivative of the energy-momentum tensor vanishes with respect to ξ, as well as admitting a conformal Killing vector field is established if and only if the energy-momentum tensor has the symmetry inheritance property. Finally for a P1 - flat perfect fluid space time satisfying Einstein’s equations with cosmological term, some results are obtained.

Some Results on Cosymplectic Manifolds Admitting Certain Vector Fields

The purpose of present paper is to study cosymplectic manifolds admitting certain special vector fields such as holomorphically planar conformal (in short HPC) vector field. First, we prove that an HPC vector field on a cosymplectic manifold is also a Jacobi-type vector field. Then, we obtain the necessary conditions for such a vector field to be Killing. Finally, we give an important characterization for a torse-forming vector field on such a manifold given as to be recurrent.