Conformal Vector Fields Research Articles

Hyperbolic Ricci solitons on perfect fluid spacetimes

<abstract><p>In the present paper, we investigate perfect fluid spacetimes and perfect fluid generalized Roberston-Walker spacetimes that contain a torse-forming vector field satisfying almost hyperbolic Ricci solitons. We show that the perfect fluid spacetimes that contain a torse-forming vector field satisfy an almost hyperbolic Ricci soliton, and we prove that a perfect fluid generalized Roberston-Walker spacetime satisfying an almost hyperbolic Ricci soliton $ (g, \zeta, \varrho, \mu) $ is an Einstein manifold. Also, we study an almost hyperbolic Ricci soliton $ (g, V, \varrho, \mu) $ on these spacetimes when $ V $ is a conformal vector field, a torse-forming vector field, or a Ricci bi-conformal vector field.</p></abstract>

Some new characterizations of spheres and Euclidean spaces using conformal vector fields

<p>Given a conformal vector field $ X $ defined on an $ n $-dimensional Riemannian manifold $ \left(N^{n}, g\right) $, naturally associated to $ X $ are the conformal factor $ \sigma $, a smooth function defined on $ N^{n} $, and a skew symmetric $ (1, 1) $ tensor field $ \Omega $, called the associated tensor, that is defined using the $ 1 $-form dual to $ X $. In this article, we prove two results. In the first result, we show that if an $ n $-dimensional compact and connected Riemannian manifold $ \left(N^{n}, g\right) $, $ n > 1 $, of positive Ricci curvature admits a nontrivial (non-Killing) conformal vector field $ X $ with conformal factor $ \sigma $ such that its Ricci operator $ Rc $ and scalar curvature $ \tau $ satisfy</p><p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ Rc\left( X\right) = -(n-1)\nabla \sigma \; \; \text{ and }\; \; X(\tau ) = 2\sigma \left( n(n-1)c-\tau \right) $\end{document} </tex-math></disp-formula></p><p>for a constant $ c $, necessarily $ c > 0 $ and $ \left(N^{n}, g\right) $ is isometric to the sphere $ S_{c}^{n} $ of constant curvature $ c $. The converse is also shown to be true. In the second result, it is shown that an $ n $-dimensional complete and connected Riemannian manifold $ \left(N^{n}, g\right) $, $ n > 1 $, admits a nontrivial conformal vector field $ X $ with conformal factor $ \sigma $ and associated tensor $ \Omega $ satisfying</p><p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ Rc\left( X\right) = -div\Omega \; \; \text{ and }\; \; \Omega \left( X\right) = 0, $\end{document} </tex-math></disp-formula></p><p>if and only if $ \left(N^{n}, g\right) $ is isometric to the Euclidean space $ \left(E^{n}, \langle, \rangle \right) $.</p>

The Morse Index of Sacks–Uhlenbeck α‐Harmonic Maps for Riemannian Manifolds

In this paper, first we prove a nonexistence theorem for α‐harmonic mappings between Riemannian manifolds. Second, the instability of nonconstant α‐harmonic maps is studied with regard to the Ricci curvature criterion of their codomain. Then, we estimate the Morse index for measuring the degree of instability of some particular α‐harmonic maps. Furthermore, the notion of α‐stable manifolds and its applications are considered. Finally, we investigate the α‐stability of any compact Riemannian manifolds admitting a nonisometric conformal vector field and any Einstein Riemannian manifold under certain assumptions on the smallest positive eigenvalue of its Laplacian operator on functions.

Quasi-Einstein manifolds admitting a closed conformal vector field

In this article, we investigate quasi-Einstein manifolds admitting a closed conformal vector field. Initially, we present a rigidity result for quasi-Einstein manifolds with constant scalar curvature and carrying a non-parallel closed conformal vector field. Moreover, we prove that quasi-Einstein manifolds admitting a closed conformal vector field can be conformally changed to constant scalar curvature almost everywhere. Finally, we obtain a characterization for quasi-Einstein manifolds endowed with a non-parallel gradient conformal vector field.

Conformal vector fields on almost Kenmotsu manifolds

Conformal vector fields on almost Kenmotsu manifolds

Generalized Minkowski Type Integral Formulas for Compact Hypersurfaces in Pseudo-Riemannian Manifolds

We obtain some generalized Minkowski type integral formulas for compact Riemannian (resp., spacelike) hypersurfaces in Riemannian (resp., Lorentzian) manifolds in the presence of an arbitrary vector field that we assume to be timelike in the case where the ambient space is Lorentzian. Some of these formulas generalize existing formulas in the case of conformal and Killing vector fields. We apply these integral formulas to obtain interesting results concerning the characterization of such hypersurfaces in some particular cases such as when the ambient space is Einstein admitting an arbitrary (in particular, conformal or Killing) vector field, and when the hypersurface has a constant mean curvature.

ON THE GEOMETRY OF SPACELIKE MEAN CURVATURE FLOW SOLITONS IMMERSED IN A GRW SPACETIME

Abstract We investigate geometric aspects of complete spacelike mean curvature flow solitons of codimension one in a generalized Robertson–Walker (GRW) spacetime $-I\times _{f}M^n$ , with base $I\subset \mathbb R$ , Riemannian fiber $M^n$ and warping function $f\in C^\infty (I)$ . For this, we apply suitable maximum principles to guarantee that such a mean curvature flow soliton is a slice of the ambient space and to obtain nonexistence results concerning these solitons. In particular, we deal with entire graphs constructed over the Riemannian fiber $M^n$ , which are spacelike mean curvature flow solitons, and we also explore the geometry of a conformal vector field to establish topological and further rigidity results for compact (without boundary) mean curvature flow solitons in a GRW spacetime. Moreover, we study the stability of spacelike mean curvature flow solitons with respect to an appropriate stability operator. Standard examples of spacelike mean curvature flow solitons in GRW spacetimes are exhibited, and applications related to these examples are given.

Shadows of Kerr–Vaidya-like black holes

In this work, we study the shadow boundary curves of rotating time-dependent black hole solutions which have well-defined Kerr and Vaidya limits. These solutions are constructed by applying the Newman–Janis algorithm to a spherically symmetric seed metric conformal to the Vaidya solution with a mass function that is linear in Eddington–Finkelstein coordinates. Equipped with a conformal Killing vector field, this class of solution exhibits separability of null geodesics, thus allowing one to develop an analytic formula for the boundary curve of its shadow. We find a simple power law describing the dependence of the mean radius and asymmetry factor of the shadow on the accretion rate. Applicability of our model to recent Event Horizon Telescope observations of M87 and Sgr A is also discussed.

Almost Ricci–Bourguignon Solitons on Doubly Warped Product Manifolds

This study aims at examining the effects of an almost Ricci–Bourguignon soliton structure on the base and fiber factor manifolds of a doubly warped product manifold. First, a number of preconditions and sufficiency criteria for an almost Ricci–Bourguignon soliton doubly warped product are addressed. Additionally, an almost Ricci–Bourguignon soliton on doubly warped product manifolds admitting a conformal vector field is taken into consideration. Finally, how the almost Ricci–Bourguignon soliton behaves in doubly warped product space–times is examined.

Conformal motions of static cylindrically symmetric solutions in f(T, B) gravity

In this paper, we investigate the conformal vector fields (CVFs) of static cylindrically symmetric (SCS) spacetimes in [Formula: see text] gravity, a novel modified teleparallel gravity that connects both [Formula: see text] and [Formula: see text] theories via a boundary term [Formula: see text]. Initially, we employ an algebraic approach to extract some classes that represent SCS solutions in [Formula: see text] gravity. Applying the above-mentioned approach, we find solutions in nine cases. We also investigate the CVFs using the direct integration technique. Following a thorough investigation, we discover that the spacetimes admit proper CVFs in only one case. In rest of the eight cases, the spacetimes either become conformally flat or they admit homothetic vector fields (HVFs) or Killing vector fields (KVFs). The overall dimension of CVFs for the spacetimes under consideration has come out to be 5, 6 or 15. To complete the study, we also compute the energy density and pressure of each model. Corresponding to the resulting models, we discuss the energy conditions in order to check the physical sound of the solutions.

Classification of exact Bianchi type V cosmological solutions via conformal vector fields admitting energy conditions in [formula omitted] gravity

The aim of this paper is to discuss exact Bianchi-type V cosmological solutions via conformal vector fields (CVFs) admitting energy conditions within the framework of f(T) gravity. For this study, we first derive the Einstein field equations (EFEs) by considering the Bianchi type V space–times in f(T) gravity. We formulate the exact Bianchi type V metrics in f(T) gravity by imposing certain constraints on space–time components. In particular, we emphasize extracting the power law, exponential, and singular solutions. We plot the dynamical parameters of the obtained solutions versus cosmic time t to observe their behavior graphically. We analyze the metrics via CVFs in order to sort out conservation laws. We also discuss several cosmological constraints for each of the obtained solutions. Furthermore, to check the physical soundness of the resultant metrics, we find several constraints under which the solutions admit energy conditions.

Conformal motions of anisotropic exact Bianchi type II models admitting energy conditions in f(T) gravity

The purpose of this paper is to explore the anisotropic exact Bianchi type II solutions in [Formula: see text] gravity, where [Formula: see text] denotes the torsion scalar. We utilize the solutions to discuss conformal vector fields (CVFs) and energy conditions. In the first slot of this study, we find the CVFs. The CVFs being a generalization of the Killing vector fields (KVFs) are affiliated with the conservation laws of physics. Corresponding to the obtained solutions, we observe law of conservation of (linear or generalized) momentum. In the second slot, we derive the constraints under which the solution classes can admit certain energy conditions.

Vanishing Love numbers of black holes in general relativity: From spacetime conformal symmetry of a two-dimensional reduced geometry

We study the underlying structure of the vanishing of the Love numbers of both Schwarzschild and Kerr black holes in terms of spacetime conformal symmetry in a unified manner for the static spin-$s$ fields. The perturbations can be reduced with the harmonic decomposition to a set of infinite static scalar fields in a two-dimensional anti--de Sitter spacetime $({\text{AdS}}_{2})$. In the reduced system, each scalar field is paired with another, implying that all multipole modes of the perturbation can be regarded as symmetric partners, which can be understood from the property of the supersymmetry algebra. The generator of the supersymmetric structure is constructed from a closed conformal Killing vector field of ${\text{AdS}}_{2}$. The associated conserved quantity allows one to show no static response, i.e., vanishing of the Love and dissipation numbers. We also discuss the vanishing Love numbers of the Kerr black hole with the nonzero dissipation numbers for the nonaxisymmetric perturbations in terms of a radial constant found in a parallel manner as the axisymmetric field case even though the interpretation for the structure is controversial. The symmetric structure corresponds to the ``ladder'' symmetry in [Hui et al., J. Cosmol. Astropart. Phys. 01 (2022) 032] although the geometrical origin is different. Our ladder operator includes the generators of hidden symmetries in previous works.

On complete Kählerian manifolds endowed with closed conformal vector fields

Let {overline{M}}^{2n}, n>1, be a complete, noncompact Kählerian manifold, endowed with a nontrivial closed conformal vector field xi having at least one singular point. Under a reasonable set of conditions, we show that xi has just one singular point p and that {overline{M}}{setminus }{p} is isometric to a one dimensional cone over a simply connected Sasakian manifold N diffeomorphic to {mathbb {S}}^{2n-1}.As a straightforward consequence, we conclude that if the addition of a single point to the Kählerian cone of a (2n-1)-dimensional Sasakian manifold N has the structure of a complete, noncompact, 2n-dimensional Kählerian manifold whose metric extends that of the cone, and such that the canonical vector field of the cone extends to it having a singularity at the extra point, then N is isometric to mathbb S^{2n-1}, endowed with an appropriate Sasakian structure.

Generalized m-Quasi-Einstein manifolds admitting a closed conformal vector field

Generalized m-Quasi-Einstein manifolds admitting a closed conformal vector field

Lightlike Hypersurfaces in Semi-Riemmanian Manifolds Admitting Affine Conformal Vector Fields

Lightlike hypersurfaces with integrable screen distributions are very important as far as lightlike geometry is concerned. They include, among others, screen conformal and screen totally umbilic ones. In this paper, we show that any lightlike hypersurface of a semi-Riemannian manifold admitting a certain closed affine conformal vector field has an integrable screen distribution. Several examples are furnished in support of the main results.

Relativistic spacetimes admitting almost Schouten solitons

In this paper, we investigate almost Schouten solitons and almost gradient Schouten solitons in spacetimes of general relativity. At first, it is proven that if a generalized Robertson–Walker spacetime permits an almost Schouten soliton, then it becomes a perfect fluid spacetime as well as the spacetime represents a dark matter era. Besides this, we investigate almost gradient Schouten solitons in generalized Robertson–Walker spacetimes. Moreover, a spacetime obeying almost Schouten solitons whose potential vector field is a non-homothetic conformal vector field is of Petrov type [Formula: see text] or [Formula: see text].

Holomorphically Planar Conformal Vector Field On Almost $\alpha $-Cosymplectic $(\kappa ,\mu )-$ Spaces

The aim of the present paper is to study holomorphically planar conformal vector (HPCV) fields on almost α−α−cosymplectic (κ,μ)−(κ,μ)−spaces. This is done assuming various conditions such as i) UU is pointwise collinear with ξξ ( in this case, the integral manifold of the distribution DD is totally geodesic, or totally umbilical), ii) MM has a constant ξ−ξ−sectional curvature (under this condition the integral manifold of the distribution DD is totally geodesic (or totally umbilical) or the manifold is isometric to sphere S2n+1(√c)S2n+1(c) of radius 1√c1c), iii) MM an almost α−α−cosymplectic (κ,μ)−(κ,μ)−spaces ( in this case the manifold has constant curvature, or the integral manifold of the distribution DD is totally geodesic(or totally umbilical) or UU is an eigenvector of h).h).

Conformal vector field and gradient Einstein solitons on η-Einstein cosymplectic manifolds

In this paper, we characterize the [Formula: see text]-Einstein cosymplectic manifolds with the gradient Einstein solitons and the conformal vector fields. It is proven that if an [Formula: see text]-Einstein cosymplectic manifold [Formula: see text] of dimension [Formula: see text] with [Formula: see text] admits a gradient Einstein soliton, then either [Formula: see text] is Ricci flat or the gradient of Einstein potential function is pointwise collinear with the Reeb vector field of [Formula: see text]. Also, we prove that if [Formula: see text] admits a conformal vector field [Formula: see text], then either [Formula: see text] is Ricci flat or [Formula: see text] is homothetic. We also establish that a conformal vector field [Formula: see text] on [Formula: see text] is a strict infinitesimal contact transformation, provided the scalar curvature of [Formula: see text] is constant. Finally, the non-trivial examples of cosymplectic manifolds admitting the gradient Einstein solitons are given.

Constructing a family of conformally flat scalar field models

Using purely geometrical methods we present a mechanism to solve the scalar field equations of motion (non-minimally coupled with gravity) in a spherically symmetric background. We found that the full set of spacetimes, which are of Petrov type O (conformally flat) and admit a gradient conformal vector field, can be determined completely. It is shown that the full group of scalar field equations reduced to a single equation that depends only on the distance leaving the metric function (equivalently the functional form of the scalar field or the potential) freely chosen. Depending on the structure of the metric or the potential V (as a function of φ) a solution can be found either analytically or via numerical integration. We provide physically sound examples and prove that (Anti)-de Sitter fits this scheme. We also reconstruct a recently found solution (Strumia and Tetradis 2022 J. High Energy Phys. JHEP09(2022)203) representing an expanding scalar bubble with metric that has a singularity and corresponds to what is termed as Anti-de Sitter crunch.