Conformal Vector Fields Research Articles

A classification of Bianchi Type I solutions via conformal vector fields and energy conditions in modified teleparallel gravity

The focus of this research is to investigate conformal vector fields (CVFs) of Bianchi type I space–times in modified teleparallel gravity (MTG). In order to determine such vector fields, we make a classification of said space–times. As a result, we discover thirty-one distinct Bianchi type I classes in the MTG. With the aid of conformal symmetry adopting a direct integration approach, we thoroughly scrutinize each case and find that the only class that formulates conformally non-flat space–times admits proper CVFs. In the rest of the cases, space–times either become conformally flat or tend to admit homothetic vector fields (HVFs) or Killing vector fields (KVFs). The overall dimension of CVFs for the Bianchi type I space–times in the MTG has turned out to be three, four, five, six, and fifteen. The physical parameters associated with each solution have also been calculated. Moreover, to discuss physically viably resulting solutions, we have to classify the solutions via energy conditions.

General light-cone gauge approach to conformal fields and applications to scalar and vector fields

Totally symmetric arbitrary spin conformal fields propagating in the flat space of even dimension greater than or equal to four are studied. For such fields, we develop a general ordinary-derivative light-cone gauge formalism and obtain restrictions imposed by the conformal algebra symmetries on interaction vertices. We apply our formalism for the detailed study of conformal scalar and vector fields. For such fields, all parity-even cubic interaction vertices are obtained. The cubic vertices obtained are presented in terms of dressing operators and undressed vertices. We show that the undressed vertices of the conformal scalar and vector fields are equal, up to overall factor, to the cubic vertices of massless scalar and vector fields. Various conjectures about interrelations between the cubic vertices for conformal fields in conformal invariant theories and the cubic vertices for massless fields in Poincaré invariant theories are proposed.

Rigidity of critical metrics for quadratic curvature functionals

In this paper we prove new rigidity results for complete, possibly non-compact, critical metrics of the quadratic curvature functionals Ft2=∫|Ricg|2dVg+t∫Rg2dVg, t∈R, and S2=∫Rg2dVg. We show that (i) flat surfaces are the only critical points of S2, (ii) flat three-dimensional manifolds are the only critical points of Ft2 for every t>−13, (iii) three-dimensional scalar flat manifolds are the only critical points of S2 with finite energy and (iv) n-dimensional, n>4, scalar flat manifolds are the only critical points of S2 with finite energy and scalar curvature bounded below. In case (i), our proof relies on rigidity results for conformal vector fields and an ODE argument; in case (ii) we draw upon some ideas of M.T. Anderson concerning regularity, convergence and rigidity of critical metrics; in cases (iii) and (iv) the proofs are self-contained and depend on new pointwise and integral estimates.

Nearly Kaehler manifolds admitting a closed conformal vector field

Nearly Kaehler manifolds admitting a closed conformal vector field

Conformal vector fields of projectively flat (α,β)-Finsler spaces

In this paper, it is proved that any conformal vector field is homothetic on a locally projectively flat (α,β)-space of non-Randers type in dimension n≥3, and the local solutions of such a vector field are determined. While on a locally projectively flat Randers space, examples show that the conformal vector fields are not necessarily homothetic.

A note on closed vector fields

<abstract><p>Special vector fields, such as conformal vector fields and Killing vector fields, are commonly used in studying the geometry of a Riemannian manifold. Though there are Riemannian manifolds, which do not admit certain conformal vector fields or certain Killing vector fields, respectively. Closed vector fields exist in abundance on each Riemannian manifold. In this paper, we used closed vector fields to study the geometry of the Riemannian manifold. In the first result, we showed that a compact Riemannian manifold $ (M^{n}, g) $ admits a closed vector field $\boldsymbol{\omega }$ with $ \mathrm{div} \boldsymbol{\omega }$ non-constant and an eigenvector of the rough Laplace operator, the integral of the Ricci curvature $ Ric(\boldsymbol{\omega }, \boldsymbol{\omega }) $ has a suitable lower bound that is necessarily isometric to $ S^{n}(c) $ and that the converse holds. In the other result, we found a characterization of an Euclidean space using a closed vector field $\boldsymbol{\omega }$ with non-constant length that annihilates the rough Laplace operator and squared length of its covariant derivative that has a suitable upper bound. Finally, we used the closed vector field provided by the gradient of the non-trivial solution of the Fischer-Marsden equation on a complete and simply connected Riemannian manifold $ (M, g) $ and showed that it is necessary and sufficient for $ (M, g) $ to be isometric to a sphere and that the squared length of the covariant derivative of this closed vector field has a suitable upper bound.</p></abstract>

Conformal vector fields on f-cosymplectic manifolds

In this paper, at first we characterize f-cosymplectic manifolds admitting conformal vector fields. Next, we establish that if a 3-dimensional f -cosymplectic manifold admits a homothetic vector field V, then either the manifold is of constant sectional curvature ?f?r, V is an infinitesimal contact transformation. Furthermore, we also investigate Ricci-Yamabe solitons with conformal vector fields on f-cosymplectic manifolds. At last, two examples are constructed to validate our outcomes

Quasi-Einstein manifolds admitting conformal vector fields

Quasi-Einstein manifolds admitting conformal vector fields

Hyperbolic Ricci soliton on warped product manifolds

In this paper, we investigate hyperbolic Ricci soliton as the special solution of hyperbolic geometric flow on warped product manifolds. Then, especially, we study these manifolds admitting either a conformal vector field or a concurrent vector field. Also, the question that:? whether or not a hyperbolic soliton reduces to an Einstein manifold?? is considered and answered. Finally, we obtain some necessary conditions for generalized Robertson-Walker space-time to be a hyperbolic Ricci soliton.

Conformal vector fields on almost cosymplectic 3-manifolds

Conformal vector fields on almost cosymplectic 3-manifolds

Conformal vector fields on N(k)-paracontact and para-Kenmotsu manifolds

Conformal vector fields on N(k)-paracontact and para-Kenmotsu manifolds

Conformal Flatness and Conformal Vector Fields on Umbilically Synchronized Space-times

Conformal Flatness and Conformal Vector Fields on Umbilically Synchronized Space-times

Characterization of sequential warped product gradient Ricci-Bourguignon soliton

In this paper, westudy characterization of sequential warped product gradient Ricci-Bourguignon soliton. We derive applications of some vector fields like torse-forming vector field, torqued vector field, conformal vector field on Ricci-Bourguignon soliton. We show that for torse-forming vector field, a Ricci- Bourguignon soliton becomes an almost quasi-Einstein manifold. Next, we obtain the inheritance properties of the Einstein-like sequential warped product gradient Ricci-Bourguignon almost soliton of class type P,A,B. We prove that, when the manifold is complete, the potential function depends only on M1 and M3 must be an Einstein manifold. We also present for a gradient Ricci-Bourguignon soliton sequential warped product, the warping functions are constants under some certain conditions.

Conformal vector fields on lcK manifolds

Conformal vector fields on lcK manifolds

Conformal vector fields and geometric solitons on the tangent bundle with the ciconia metric

Let (X2k, F, g) be an almost anti-paraHermitian manifold with an almost paracomplex structure F and a Riemannian metric g and let TX be its tangent bundle with the ciconia metric g?. The purpose of this paper is divided into two folds. The first one is to examine the curvature properties of the tangent bundle TX with the ciconia metric g?. The second one is to study conformal vector fields and almost Ricci and Yamabe solitons on the tangent bundle TX according to the ciconia metric g?.

Nonlinear automorphism of the conformal algebra in 2D and continuous sqrt{Toverline{T}} deformations

The conformal algebra in 2D (Diff(S1)⨁Diff(S1)) is shown to be preserved under a nonlinear map that mixes both chiral (holomorphic) generators T and overline{T} . It depends on a single real parameter and it can be regarded as a “nonlinear SO(1, 1) automorphism.” The map preserves the form of the momentum density and naturally induces a flow on the energy density by a marginal sqrt{Toverline{T}} deformation. In turn, the general solution of the corresponding flow equation of the deformed action can be analytically solved in closed form, recovering the nonlinear automorphism. The deformed CFT2 can also be described through the original theory on a field-dependent curved metric whose lapse and shift functions are given by the variation of the deformed Hamiltonian with respect to the energy and momentum densities, respectively. The conformal symmetries of the deformed theories can then also be seen to arise from diffeomorphisms that fulfill suitably deformed conformal Killing equations. Besides, Cardy formula is shown to map to itself under the nonlinear automorphism. As a simple example, the deformation of N free bosons is briefly addressed, making contact with recent related results and the dimensional reduction of the ModMax theory. Furthermore, the nonlinear map between the conformal algebra in 2D and its ultra/non-relativistic versions (BMS3≈CCA2≈GCA2), including the corresponding finite sqrt{Toverline{T}} deformation, are recovered from a limiting case of the nonlinear automorphism. The extension to a three-parameter nonlinear ISO(1, 1) automorphism of the conformal algebra, and a discrete nonlinear automorphism of BMS3 are also briefly discussed.

Conformal Killing Vector Fields of Riemannian Manifolds

The main aim of this article to study about vector fields of manifold and how these vector fields will be Killing and Conformal Killing vector fields. Conformal transformation of Weyl rescaling which is conformally related to metrices from g to g, Levi-Civita connection Δ , Lie derivative, torsion with tensor concept of manifold N in a multi-linear map have been treated in this paper. Finally, we have been proved Example 3.02and established the theorem 6.02 on Conformal Killing vector fields. Dhaka Univ. J. Sci. 70(2): 18-22, 2022 (July)

Symmetries of locally rotationally symmetric Bianchi type V spacetime

In this paper, we classify the metric of locally rotationally symmetric Bianchi type V spacetime through its symmetries, including Killing, homothetic and conformal vector fields. In order to find all possible Bianchi type V metrics possessing these symmetries, first we analyze the symmetry equations by an algorithm developed in Maple which restricts the metric functions to satisfy certain conditions. These conditions are then used to solve the set of symmetry equations, giving the final form of symmetry vector fields. To add some physical significance of the obtained metrics, the components of energy-moments tensor are found. It is observed that most of these metrics represent perfect fluids satisfying certain energy conditions.

Classification of static spherically symmetric perfect fluid space-times via conformal vector fields in f(T) gravity

In this paper, we classify static spherically symmetric (SS) perfect fluid space-times via conformal vector fields (CVFs) in f(T) gravity. For this analysis, we first explore static SS solutions by solving the Einstein field equations in f(T) gravity. Secondly, we implement a direct integration technique to classify the resulting solutions. During the classification, there arose 20 cases. Studying each case thoroughly, we came to know that in three cases the space-times under consideration admit proper CVFs in f(T) gravity. In one case, the space-time admits proper homothetic vector fields, whereas in the remaining 16 cases either the space-times become conformally flat or they admit Killing vector fields.

Conformal Ricci soliton and almost conformal Ricci soliton in paracontact geometry

In this paper, we study conformal Ricci soliton and almost conformal Ricci soliton within the framework of paracontact manifolds. Here, we have shown the characteristics of the soliton vector field and the nature of the manifold if para-Sasakian metric satisfies conformal Ricci soliton. We also demonstrate the feature of the soliton vector field V and scalar curvature when the para-Sasakian manifold admitting conformal Ricci soliton and vector field is pointwise collinear with the characteristic vector field [Formula: see text]. Next, we prove that if a K-paracontact manifold confesses a gradient conformal Ricci soliton, then it is Einstein. Next, we show that a para-Sasakian metric reveals with an almost conformal Ricci soliton that is either Einstein or [Formula: see text]-Einstein metric if the soliton vector field V is an infinitesimal contact transformation. Lastly, we decorate an example of conformal Ricci soliton on para-Sasakian manifold.