Killing Vector Field Research Articles

On general solutions of equidistant vector fields on two-dimensional (pseudo-)Riemannian spaces

The paper is devoted to studying equidistant two-dimensional (pseudo-) Riemannian spaces. Embeddings of these spaces in three-dimensional Euclidean and Minkowski spaces as revolution or helical surfaces are given. The general solution of equidistant equations is found beyond these spaces under minimal requirements for the differentiability of the studied objects. These vector fields are associated with Killing vector fields on those spaces.

Generalized Ricci solitons and Einstein metrics on weak $ K $-contact manifolds

<abstract><p>We study so-called "weak" metric structures on a smooth manifold, which generalize the metric contact and $ K $-contact structures and allow a new look at the classical theory. We characterize weak $ K $-contact manifolds among all weak contact metric manifolds using the property well known for $ K $-contact manifolds, as well as find when a Riemannian manifold endowed with a unit Killing vector field is a weak $ K $-contact manifold. We also find sufficient conditions for a weak $ K $-contact manifold with a parallel Ricci tensor or with a generalized Ricci soliton structure to be an Einstein manifold.</p></abstract>

Geometry of foliations of Minkowski spaces

As is well known, foliations of constant curvature and foliations generated by the orbits of Killing vector fields are important classes of foliations from a geometric point of view. The paper studies the geometry of some foliations of the Minkowski space, which arise in a natural way. It is shown that these foliations are foliations of constant curvature. The paper also studies the geometry of some singular foliations generated by the orbits of Killing vector fields.

Локально изометричные римановы аналитические пространства

Classes of locally isometric Riemannian analytic manifolds are studied. A generalization of the concept of completeness is given. We consider the Lie algebra 𝔤 of all Killing vector fields of a Riemannian analytic manifold, its stationary subalgebra 𝔥 the simply connected Lie group 𝐺 corresponding to the Lie algebra 𝔤, and the subgroup 𝐻 corresponding to the Lie subalgebra 𝔥. In the absence of a center in the algebra 𝔤 the concept of a quasi-complete (compressed) manifold is introduced. An oriented Riemannian analytic manifold whose vector field algebra has zero center is said to be quasi-complete if it is non-extendable and does not admit non-trivial orientation-preserving and all Killing vector fields local isometries to itself. The main property of such a manifold is that it is unique in the class of all locally isometric Riemannian analytic manifolds, and any locally given isometry of this manifold 𝑀 into itself can be analytically extended to an isometry 𝑓: 𝑀 ≈ 𝑀. For an arbitrary class of locally isometric Riemannian analytic manifolds, a definition of a pseudocomplete manifold is given, which is complete if a complete manifold exists in the given class. A Riemannian analytic simply connected manifold M is called pseudocomplete if it has the following properties. 𝑀 is non-extendable. There is no locally isometric covering map f; M→N, where N is a simply connected Riemannian analytic manifold and f (M) is an open subset of N not equal to N.

A variational setting for an indefinite Lagrangian with an affine Noether charge

We introduce a variational setting for the action functional of an autonomous and indefinite Lagrangian on a finite dimensional manifold M. Our basic assumption is the existence of an infinitesimal symmetry whose Noether charge is the sum of a one-form and a function on M. Our setting includes different types of Lorentz–Finsler Lagrangians admitting a timelike Killing vector field.

A Gutzwiller Trace Formula for Dirac Operators on a Stationary Spacetime

A Duistermaat–Guillemin–Gutzwiller trace formula for Dirac-type operators on a globally hyperbolic spatially compact stationary spacetime is achieved by generalising the recent construction by Strohmaier and Zelditch (Adv Math 376:107434, 2021) to a vector bundle setting. We have analysed the spectrum of the Lie derivative with respect to a global timelike Killing vector field on the solution space of the Dirac equation and found that it consists of discrete real eigenvalues. The distributional trace of the time evolution operator has singularities at the periods of induced Killing flow on the space of lightlike geodesics. This gives rise to the Weyl law asymptotic at the vanishing period.

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)

On Momentum Operators Given by Killing Vectors Whose Integral Curves Are Geodesics

The paper considers momentum operators on intrinsically curved manifolds. Given that momentum operators are Killing vector fields whose integral curves are geodesics, the corresponding manifold is flat or of the compact type with positive constant sectional curvature and dimensions equal to 1, 3, or 7. Explicit representations of momentum operators and the associated Casimir element are discussed for the 3-sphere S3. It is verified that the structural constants of the underlying Lie algebra are proportional to 2 ℏ/R, where R is the curvature radius of S3 and ℏ is the reduced Planck’s constant. This results in a countable energy and momentum spectrum of freely moving particles in S3. The maximal resolution of the possible momenta is given by the de Broglie wave length, λR=πR, which is identical to the diameter of the manifold. The corresponding covariant position operators are defined in terms of geodesic normal coordinates, and the associated commutator relations of position and momentum are established.

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.

Killing submersions and magnetic curves

We investigate magnetic curves in Killing submersions and we show that the bundle curvature is constant along magnetic curves with respect to the Killing vector field if and only if all vertical tubes derived from these magnetic curves are of constant mean curvature. Then we examine magnetic Jacobi fields along horizontal Killing magnetic curves in the total space of a Killing submersion. We generalize some important results obtained by O'Neill for horizontal geodesics to horizontal magnetic curves. Finally, we study magnetic Jacobi fields along horizontal Killing magnetic trajectories in 3-dimensional Sasakian space forms.

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 η-Ricci Solitons on Riemannian Submersions under Canonical Variation

This research article endeavors to discuss the attributes of Riemannian submersions under the canonical variation in terms of the conformal η-Ricci soliton and gradient conformal η-Ricci soliton with a potential vector field ζ. Additionally, we estimate the various conditions for which the target manifold of Riemannian submersion under the canonical variation is a conformal η-Ricci soliton with a Killing vector field and a φ(Ric)-vector field. Moreover, we deduce the generalized Liouville equation for Riemannian submersion under the canonical variation satisfying by a last multiplier Ψ of the vertical potential vector field ζ and show that the base manifold of Riemanian submersion under canonical variation is an η Einstein for gradient conformal η-Ricci soliton with a scalar concircular field γ on base manifold. Finally, we illustrate an example of Riemannian submersions between Riemannian manifolds, which verify our results.

First law of mechanics for spinning compact binaries: Dipolar order

Building upon the Noether charge formalism of Iyer and Wald, we derive a variational formula for spacetimes admitting a Killing vector field, for a generic energy-momentum distribution with compact support. Applying this general result to the particular case of a binary system of spinning compact objects moving along an exactly circular orbit, modelled using the multipolar gravitational skeleton formalism, we derive a first law of compact binary mechanics at dipolar order. We prove the equivalence of this new result with the canonical Hamiltonian first law previously derived for binary systems of spinning compact objects, for spins colinear with the orbital angular momentum. This paper paves the way to an extension of the first law of binary mechanics to the next quadrupolar order, thereby accounting for the spin-induced and tidally-induced deformability of the compact bodies.

A note on classification of Kantowski–Sachs and Bianchi type III solutions in f(T) gravity via conformal vector fields

Conformal symmetries act as a source to investigate and classify exact solutions of the Einstein field equations (EFEs) via conformal vector fields (CVFs). It is well known that such classification leads to an important class of symmetries known as Killing symmetry which is the source of generating conservation laws of physics. In this paper, first we explore various classes of Kantowski–Sachs (KS) and Bianchi type III solutions in [Formula: see text] gravity by adopting some algebraic techniques. Utilizing the above-mentioned technique, we come to know that there exist 30 cases where the KS and Bianchi type III space-times admit solutions in [Formula: see text] gravity. Inspecting all the classes precisely, we familiarize that 25 solutions formulate non-conformally flat metrics whereas rest of the five solutions tend to formulate conformally flat metrics. We utilize the resulting solutions in finding the CVFs via direct integration approach. After a detailed study, we found that in two cases, the space-times admit proper CVFs, whereas in rest of the cases, the space-times either become conformally flat or it admit homothetic vector fields (HVFs) or Killing vector fields (KVFs). The overall dimension of CVFs for the space-times under consideration has turned out to be four, five, six or fifteen.

Geometrical Aspects of Motion of Charged Particles in Magnetic and Killing Magnetic Fields and Their Corresponding Trajectories in Anti-De Sitter 3-Space

We inquire the motion of charged particles varying under the effect of Lorentz force produced by magnetic and Killing magnetic fields in anti-de Sitter 3-space . Primarily, we characterize magnetic trajectories in in terms of their Frenet apparatus. Thereafter, utilizing a geometrical model of split-quaternions for (where corresponds to a subspace of the Lie group H of split-quaternions) we find 6 independent unit Killing vector fields on which constitutes a basis for the corresponding 6D Lie algebra i(). We conclude with characterizations of Killing magnetic trajectories in with respect to quasi-slope, curvature and torsion or pseudo-torsion (depending on the situation).

Аналитическое продолжение локально однородного и неоднородного риманова многообразия

Let’s consider the Lie algebra 𝔤 of all Killing vector fields of a Riemannian analytic manifold, its stationary subalgebra 𝔥, the simply connected Lie group 𝐺 corresponding to the Lie algebra 𝔤 and the subgroup 𝐻 corresponding to the Lie subalgebra 𝔥. The set of left adjacent classes 𝐺𝐻⁄ forms a homogeneous manifold if and only If 𝐻 is closed in 𝐺. We study the properties of the Lie algebra 𝔤 and its subalgebra 𝔥 under which 𝐻 is closed in 𝐺. The following category of Riemannian analytic manifolds is also studied. The objects of this category are oriented Riemannian analytic manifolds having open subsets isometric to each other and, consequently, the same algebra 𝔤 of Killing vector fields. It is assumed that the algebra 𝔤 has no center. Morphisms of this category are locally isometric maps 𝑓:𝑀⟶𝑁 preserving orientation and Killing vector fields. Moreover, the maps 𝑓 are defined on the entire manifold 𝑀 with the exception of the set 𝑆 of codimension at least two, consisting of fixed points of orientation-preserving isometries between open subsets of the manifold 𝑀. This category has a univer-sally attractive object. This is a so-called quasi-complete manifold, which by definition is unextendable manifold that does not admit nontrivial orientation-preserving and vector Killing fields isometries between its open subsets. For an arbitrary Riemannian analytic metric, a pseudo-field Riemannian analytic manifold is defined. This is a simply connected manifold 𝑀 for which there is no locally isometric map 𝑓:𝑀⟶𝑁 define on the whole 𝑀 and preserving orientation and killing vector fields. Where 𝑁 is a simply connected Riemannian analytic manifold other than 𝑀.

On some non-static plane symmetric perfect fluid solutions in [formula omitted] gravity

We address the problem of finding non-static plane symmetric perfect fluid solutions in the f(R,T) theory of gravity, where R and T respectively are the Ricci scalar and trace of the energy momentum tensor. In order to obtain solutions, a classification has been proposed for such spacetimes. Solutions are then calculated in each of these classes, which include special Bianchi-Type I and static plane symmetric geometries. The static geometries are no surprise, but importantly, non-static spacetime metrics have also been found. It is well known that the class of isometries are the source of deriving the conservation laws. Therefore, we work out the Killing symmetries to determine the Lie symmetry groups admitted by the obtained spacetimes. As a result of our analysis, we find that the obtained spacetimes admit 10, 7, 6, 5, 4 and 3 Killing vector fields.

Analytical field equation and wave function solutions of the Bianchi type I universe in vacuum f(R) gravity

It has been proposed that f(R) gravity is a viable proxy for General Relativity and a means for an alternative to dark energy. Within this context, we present the analytical solutions to the Bianchi type I universe. For this spacetime, the Killing and homothetic vector fields categorize the space into fourteen classes. Noether’s theorem is used to determine the first integrals associated with each isometry for every class and we use the geometry of the Noether point symmetries of the space to establish potential functions. We also quantize the Lagrangian of the spacetime to derive the Wheeler–DeWitt equation and successively determine invariant solutions for the wave function of the universe from the Lie point symmetries admitted by the equation. Finally, for all cases, exact classical solutions to the field equations are found.

Conformal Vector Fields and Null Hypersurfaces

We give conditions for a conformal vector field to be tangent to a null hypersurface. We particularize to two important cases: a Killing vector field and a closed and conformal vector field. In the first case, we obtain a result ensuring that a null hypersurface is a Killing horizon. In the second one, the vector field gives rise to a foliation of the manifold by totally umbilical hypersurfaces with constant mean curvature which can be spacelike, timelike or null. We prove several results which ensure that a null hypersurface with constant null mean curvature is a leaf of this foliation.

Lie Algebra of Killing Vector Fields and its Stationary Subalgebra

Let 𝔤 be the Lie algebra of all Killing vector fields on a locally homogeneous, analytic Riemannian manifold M, 𝔥 be a stationary subalgebra of 𝔤, G be the simply connected group generated by the algebra 𝔤, H be the subgroup of G generated by the subalgebra 𝔥, 𝔷 be the center of the algebra g, r be its radical, and [𝔤; 𝔤] be its commutator subgroup. If dim (𝔥 ∩ (𝔷 + [𝔤, 𝔤])) = dim (𝔥 ∩ [𝔤, 𝔤]), then H is closed in G. If for any semisimple subalgebra 𝔭 ⊂ 𝔤 satisfying the condition 𝔭 + 𝔯 = 𝔤, the relation (𝔭 + 𝔷) ∩ 𝔥 = 𝔭 ∩ 𝔥 holds, then H is closed in G. We also examine the analytic continuation of the given local, analytic Riemannian manifold.