Conformal Vector Fields Research Articles

Conformal vector fields on Finsler square metrics

Abstract: In this paper, we characterise the conformal vector fields on square metric. Further, we obtain the explicit expressions of conformal vector fields on a square metric of weakly isotropic flag curvature with the condition that b=|β|α is constant.

Left-invariant conformal vector fields on non-solvable Lie groups

Left-invariant conformal vector fields on non-solvable Lie groups

A vector field method for some nonlinear Dirac models in Minkowski spacetime

Firstly we use the Lie derivatives of spinors with respect to Killing vector fields and conformal Killing vector fields to obtain the generalized commuting vector field set adapted to the Dirac operator. By the conservation law of the charge current and total angular momentum, as well as the weighted Sobolev estimates, it gives the optimal decay for the linear massless spinor fields and more decay for certain decomposition parts in any dimensions. Moreover, by exploiting the Dirac null structure, we also show the small data global existence and asymptotic decay properties for some well-known nonlinear models with cubic nonlinearities in space dimension two and quadratic nonlinearities in space dimension three.

The navigation problems on a class of conic Finsler manifolds

Based on our deep understanding on the essential relationships between the Zermelo navigation problems and the geometries of indicatrix on Finsler manifolds, we study navigation problems on conic Kropina manifolds. For a conic Kropina metric F(x,y) and a vector field V with F(x,−Vx)≤1 on an n-dimensional manifold M, let F˜=F˜(x,y) be the solution of the navigation problem with navigation data (F,V). We prove that F˜ must be either a Randers metric or a Kropina metric. Then we establish the relationships between some curvature properties of F and the corresponding properties of F˜ when V is a conformal vector field on (M,F), which involve S-curvature, flag curvature and Ricci curvature.

Gap Results for Free Boundary CMC Surfaces in Radially Symmetric Conformally Euclidean Three-Balls

In this work, we consider $$M=(\mathbb {B}_r^3,{\bar{g}})$$ as the Euclidean three-ball with radius r equipped with the metric $${\bar{g}}=e^{2h}\left\langle \,,\right\rangle $$ conformal to the Euclidean metric, where the function $$h=h(x)$$ depends only on the distance of x to the center of $$\mathbb {B}_r^3$$ . We show that if a free boundary CMC surface $$\varSigma $$ in M satisfies a pinching condition on the length of the traceless second fundamental tensor which involves the support function of $$\varSigma $$ , the positional conformal vector field $$\mathbf {x}$$ and its potential function $$\sigma ,$$ then either $$\varSigma $$ is a disk or $$\varSigma $$ is an annulus rotationally symmetric. These results extend to the CMC case and to many other different conformally Euclidean spaces the main result obtained by Li and Xiong (J Geom Anal 28(4):3171–3182, 2018).

Classification of non-conformally flat static plane symmetric perfect fluid solutions via proper conformal vector fields in f(T) gravity

The aim of this paper is to classify non-conformally flat static plane symmetric (SPS) perfect fluid solutions via proper conformal vector fields (CVFs) in [Formula: see text] gravity. For this purpose, first we explore some SPS perfect fluid solutions of the Einstein field equations (EFEs) in [Formula: see text] gravity. Second, we utilize these solutions to find proper CVFs. In this study, we found 16 cases. A detailed study of each case reveals that in three of these cases, the space-times admit proper CVFs whereas in the rest of the cases, either the space-times become conformally flat or they admit proper homothetic vector fields (HVFs) or Killing vector fields (KVFs). The dimension of CVFs for non-conformally flat space-times in [Formula: see text] gravity is four, five or six.

A maximum principle related to volume growth and applications

In this paper, we derive a new form of maximum principle for smooth functions on a complete noncompact Riemannian manifold M for which there exists a bounded vector field X such that $$\langle \nabla f,X\rangle \ge 0$$ on M and $$\rm {div} X\ge af$$ outside a suitable compact subset of M, for some constant $$a>0$$ , under the assumption that M has either polynomial or exponential volume growth. We then use it to obtain some Bernstein-type results for hypersurfaces immersed into a Riemannian manifold endowed with a Killing vector field, as well as to some results on the existence and size of minimal submanifolds immersed into a Riemannian manifold endowed with a conformal vector field.

A conformal characterization of manifolds of constant sectional curvature

It is shown that a connected Riemannian manifold has constant sectional curvature if and only if every one of its points is a non-degenerate maximum of some germ of smooth functions whose Riemannian gradient is a conformal vector field.

Conformal vector fields of static spherically symmetric perfect fluid space-times in modified teleparallel theory of gravity

In this paper, initially we solve the Einstein field equations (EFEs) for a static spherically (SS) symmetric perfect fluid space-times in the [Formula: see text] gravity with the aid of some algebraic techniques. The extracted solutions are then utilized in order to get conformal vector fields (CVFs). It is important to mention that the adopted techniques enable us to obtain various classes of space-times with viable [Formula: see text] gravity models which already exist in the literature. Excluding all such classes, we find that there exist three cases for which the space-times admit proper CVFs, whereas in rest of the cases, CVFs become KVFs. We have also highlighted some physical implications of our obtained results.

Conformal vector fields on vector bundle manifolds with spherically symmetric metrics

Conformal vector fields on vector bundle manifolds with spherically symmetric metrics

Conformal vector fields on Finsler manifolds

In this paper, we give an equivalent characterization of conformal vector fields on a Finsler manifold [Formula: see text], whose metric [Formula: see text] is defined by a Riemannian metric [Formula: see text] and a 1-form [Formula: see text]. This characterization contains all related results in [Z. Shen and Q. Xia, On conformal vector fields on Randers manifolds, Sci. China Math. 55(9) (2012) 1869–1882; Z. Shen and M. Yuan, Conformal vector fields on some Finsler manifolds, Sci. China Math. 59(1) (2016) 107–114; X. Cheng, Y. Li and T. Li, The conformal vector fields on Kropina manifolds, Diff. Geom. Appl. 56 (2018) 344–354] as special cases. Further, we determine conformal fields on some Finsler manifolds [Formula: see text] when [Formula: see text] is of constant sectional curvature and [Formula: see text] is a conformal 1-form with respect to [Formula: see text].

Critical point equation and closed conformal vector fields

AbstractIn this article, we study the critical points of the total scalar curvature functional restricted to the space of metrics with constant scalar curvature of unitary volume, for simplicity, CPE metrics. Here, we prove that a CPE metric admitting a non‐trivial closed conformal vector field must be isometric to a round sphere metric, which provides a partial answer to the CPE conjecture.

Conformal vector fields of some vacuum classes of static spherically symmetric space-times in f(T,B) gravity

The aim of this paper is to investigate the conformal vector fields (CVFs) for some vacuum classes of static spherically symmetric space-times in [Formula: see text] gravity. First, we have explored the space-times by solving the Einstein field equations in [Formula: see text] gravity. These solutions have been obtained by imposing various conditions on the space-time components and selecting separable form of the bivariate function [Formula: see text]. Second, we find the CVFs of the obtained space-times via direct integration approach. The overall study reveals that there exist 17 cases. From these 17 cases, the space-times in five cases admit proper CVFs whereas in rest of the 12 cases, CVFs become Killing vector fields (KVFs). We have also calculated the torsion scalar and boundary term for each of the obtained solutions.

Sharp Estimates for the First Stability Eigenvalue of Surfaces in the Presence of a Closed Conformal Vector Field

The purpose of this article is to find out sharp estimates for the first eigenvalue of the stability operator for compact surfaces with constant mean curvature immersed into a Riemannian 3-manifold having a nowhere vanishing closed conformal vector field, and characterize the cases where the estimates are reached.

Classification of proper non-static cylindrically symmetric perfect fluid space-times via conformal vector fields in f(R) gravity

In this paper, we classify proper non-static cylindrically symmetric (CS) perfect fluid space-times via conformal vector fields (CVFs) in the [Formula: see text] gravity. In order to classify the space-times, we use the algebraic and direct integration approaches. In the process of classification, there exist 23 cases for which the considered space-times become proper non-static. By studying each case in detail, we find that the conformal vector fields are of dimensions two, three and fifteen in the [Formula: see text] gravity.

English

We study conformal and Killing vector fields on the unit tangent bundle, over a Riemannian manifold, equipped with an arbitrary pseudo-Riemannian g-natural metric. We characterize the conformal and Killing conditions for classical lifts of vector fields and we give a full classification of conformal fiber-preserving vector fields on the unit tangent bundle endowed with an arbitrary pseudo-Riemannian Kaluza-Klein type metric.

Classification of proper teleparallel conformal symmetry of spherically symmetric static spacetimes using diagonal tetrads

We study proper teleparallel conformal vector fields in spherically symmetric static spacetimes. The main objective of this paper is to present the classification for the above-mentioned spacetimes. The problem has been examined by two methods: direct integration technique and diagonal tetrads. We show that the spherically symmetric static spacetimes do not admit proper teleparallel conformal vector field, so are actually the teleparallel killing vector fields.

G‐natural symmetries on tangent bundles

AbstractThe study of symmetries is a well known research topic in differential geometry with relevant physical interpretations. Given a Riemannian manifold , we consider pseudo‐Riemannian g‐natural metrics G on its tangent bundle and characterize conformal, homothetic and Killing vector fields of obtained from natural lifts of vector fields of M.

M-Quasi-Einstein Metrics Satisfying Certain Conditions on the Potential Vector Field

In this paper we study Riemannian manifolds $$(M^n, g)$$ admitting an m-quasi-Einstein metric with V as its potential vector field. We derive an integral formula for compact m-quasi-Einstein manifolds and prove that the vector field V vanishes under certain integral inequality. Next, we prove that if the metrically equivalent 1-form $$V^{\flat }$$ associated with the potential vector field is a harmonic 1-form, then V is an infinitesimal harmonic transformation. Moreover, if M is compact then it is Einstein. Some more results were obtained when (i) V generates an infinitesimal harmonic transformation, (ii) V is a conformal vector field.

Existence of conformal vector fields of Bianchi type I space-times in f(R) gravity

In this paper, Bianchi type I space-times in the [Formula: see text] theory of gravity are classified via conformal vector fields using algebraic and direct integration techniques. In this classification, we show that the conformal vector fields are of dimension four, five, six or fifteen. Additionally, we found that non-conformally flat Bianchi type I space-times admit conformal vector fields of dimension four, five or six. In the case of conformally flat or flat space-times, the dimension of the conformal vector fields is fifteen.