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This paper considers the possibility that abstract thinking and advanced synthesis skills might encourage extraterrestrial civilizations to accept communication with mankind on Earth. For this purpose, a notation not relying upon the use of alphabet and numbers is proposed, in order to denote just some basic geometric structures of current physical theories: vector fields, 1 -form fields, and tensor fields of arbitrary order. An advanced civilization might appreciate the way here proposed to achieve a concise description of electromagnetism and general relativity, and hence it might accept the challenge of responding to our signals. The abstract symbols introduced in this paper to describe the basic structures of physical theories are encoded into black and white bitmap images that can be easily converted into short bit sequences and modulated on a carrier wave for radio transmission. [ABSTRACT FROM AUTHOR]

In this paper, we aim to characterize the sequential warped product κ -almost gradient conformal Ricci–Bourguignon soliton. We derive applications of some vector fields like conformal vector field, torse-forming vector field, torqued vector field on κ -almost conformal Ricci–Bourguignon soliton. The inheritance properties of the Einstein-like sequential warped product κ -almost gradient conformal Ricci–Bourguignon solitons of class types ℙ , , are investigated in this paper. Later we show that if M ̄ = (M 1 × f I M 2 ) × f ̄ I M 3 is a sequential warped product of a complete connected (n − 2) -dimensional Riemannian manifold M 1 and one-dimensional Riemannian manifolds I M 2 and I M 3 admitting κ -almost conformal Ricci–Bourguignon soliton, then (M 1 , g 1) becomes a (n − 2) -dimensional sphere of radius ρ = n − 2 r 1 + (μ − 1 2 (p + 2 n) + ρ R) (4 − n) . We have discovered that for a κ -almost gradient conformal Ricci–Bourguignon soliton sequential warped product, the warping functions are constants under certain conditions. [ABSTRACT FROM AUTHOR]

This research paper explores the properties of quasi-Einstein manifolds with a unit concircular generator vector field within doubly warped product structures. The paper begins by investigating the characteristics of quasi-Einstein manifolds that possess a unit concircular generator vector field. Subsequently, it analyzes the behavior of the Hessian, Riemann, and Ricci vector fields in the context of a doubly warped product. Moreover, the impact of the quasi-Einstein condition on both the doubly warped product and its factor manifolds is studied. Finally, conditions are derived to determine whether a doubly warped product satisfying the quasi-Einstein condition can be classified as a warped or a direct product. By presenting these findings, this study contributes to the comprehension of quasi-Einstein manifolds and their interactions within doubly warped product structures. [ABSTRACT FROM AUTHOR]

This paper presents a methodical dynamical analysis of the Big Bang model, taking into account dark matter and an arbitrary form of dark energy. Nevertheless, why dynamic analysis? The primary advantage of the dynamical analysis approach is that it obviates the necessity to precisely solve the system for every individual dependent variable. Instead, it enables the prediction of the overall evolutions. In this regard, there are some powerful tools of non-dynamical systems (theorems, perturbation, etc.) which can help us a lot. The model under consideration is a nonlinear system with two dimensions (2D). In order to investigate this, we employ a sophisticated form of nonlinear dynamical systems theory in our research. We have delved into the full potential of nonlinear dynamical system theory known until now in our work. This analysis has produced very interesting solutions, which are in quite a contrast with the linear analysis approach. The stability of the system is considered at length, and the corresponding evolutions of the universe are also discussed consequently. A handful number of figures are given to visualize the behavior of the evolutions. The plots that are presented here are vector field plots and a new plotting technique initiated by us and used in our earlier papers. Interestingly, our work indicates the universe might resemble “phantom” evolution. Also, among the fixed points of our model, one is able to escape the singular situation called “Big Rip”. [ABSTRACT FROM AUTHOR]

The aim of this paper is to characterize almost co-Kähler manifolds and co-Kähler three-manifolds whose metrices are the gradient ρ -Einstein solitons. At first we prove that a proper (κ ̃ , μ ̃) -almost co-Kähler manifold with κ ̃ < 0 does not admit gradient ρ -Einstein soliton. It is also shown that if a proper -Einstein almost co-Kähler manifold with constant coefficients admits a gradient ρ -Einstein soliton, then either the manifold is a K -almost co-Kähler manifold or the soliton is trivial. Next, we prove that in case of co-Kähler three-manifold the manifold is of constant scalar curvature. Moreover, either the manifold is flat or the gradient of the potential function is collinear with the Reeb vector field ξ. Finally, we construct two examples to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we study classification of magnetic curves corresponding to Killing vector fields of ℍ 3 . First, we solve the geodesic equation analytically. Then we calculate the trajectories generated by all the six Killing vector fields, which are considered as magnetic field vectors, by using perturbation method up to first-order with respect to the strength of the magnetic field. We present a comparison of our solution with the numerical solution for one case. We also prove that 3-dimensional (α) -Kenmotsu manifolds cannot have any magnetic vector field in the direction of their Reeb vector fields. [ABSTRACT FROM AUTHOR]

This paper characterizes (m , ρ) -quasi-Einstein solitons on 3 -dimensional trans-Sasakian manifolds. It has been shown that a closed (m , ρ) -quasi-Einstein soliton on a 3 -dimensional trans-Sasakian manifold is either cosymplectic or Einstein under certain restrictions. Moreover, we investigate (m , ρ) -quasi-Einstein solitons with certain potential vector fields on 3 -dimensional trans-Sasakian manifolds. As an application of this soliton, we also analyze super quasi-Einstein spacetimes whose metrics are associated with (m , ρ) -quasi-Einstein solitons. [ABSTRACT FROM AUTHOR]

In this paper, we thoroughly study ϱ-Ricci–Bourguignon almost soliton and gradient ϱ-Ricci–Bourguignon almost soliton in the paracontact geometry, precisely, on K-paracontact and para-Sasakian manifolds. Here, we prove that if the metric g of the K-paracontact manifold endows a ϱ-Ricci–Bourguignon almost soliton with the nonzero potential vector field V parallel to ξ, then the manifold is an Einstein with Einstein constant − 2n. Next, we show that if a para-Sasakian manifold represents a gradient ϱ-Ricci–Bourguignon almost soliton, then the manifold is an Einstein with constant scalar curvature − 2n(2n + 1). We also discuss ϱ-Ricci–Bourguignon almost soliton on (κ,μ)-paracontact manifold. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study geometrical aspects of an almost Ricci soliton on (wRs)4 perfect fluid spacetime obeying Einstein’s field equation. Among others, we first find the soliton constant and cosmological constant in terms of scalar curvature and potential vector field in (wRs)4 perfect fluid spacetime. Next, we discuss some physical phenomena related to dust fluid, dark fluid, and radiation era in (wRs)4 perfect fluid spacetime admitting an almost Ricci soliton with potential vector field as solenoidal vector field and basic vector field ρ under matter collineation condition. Further, we find an inequality for soliton constant when (wRs)4 perfect fluid spacetime obeys the timelike convergence condition. Finally, we obtain some results in a (wRs)4 perfect fluid spacetime whose metric represents an almost Ricci soliton when basic vector field and potential vector field both are torseforming vector field ρ. Also, for such spacetime we find the soliton constant in terms of cosmological constant, gravitational constant, energy density, and isotropic pressure. We also provide an example of (wRs)4 spacetime whose metric represents an almost Ricci soliton. [ABSTRACT FROM AUTHOR]

This paper delivers the characterization of a pseudo B-symmetric spacetimes and we illustrate that a pseudo B-symmetric spacetime admitting Codazzi type of B-tensor represents a perfect fluid spacetime and if this spacetime admits the time-like convergence criterion, then the pseudo B-symmetric spacetime fulfills cosmic strong energy criterion and contains pure matter. Besides, we find in a pseudo B-symmetric spacetime with Codazzi type of B-tensor the electric part of the Weyl tensor vanishes and has Riemann and Weyl compatible vector fields. Furthermore, it is established that the chosen spacetime with Codazzi type of B-tensor is conformally flat and represents a Robertson–Walker spacetime. Also, we calculate the scale factor Ψ(t) for these spacetimes in a spatially flat Robertson–Walker spacetime. Finally, we study the impact of this spacetime under f(R) gravity scenario and deduce several energy conditions by considering a new model f(ℛ) = e(αℛ) −ln(βℛ) in which α and β are positive constants. [ABSTRACT FROM AUTHOR]

In this paper, we explore a set of geometric properties of Mixed Super Quasi-Einstein (MSQE) manifolds and provide examples of both Riemannian and Lorentzian MSQE manifolds to demonstrate their existence. Furthermore, we examine MSQE spacetimes in the context of the space-matter tensor, discussing several related properties. Finally, we establish the existence of an MSQE spacetime through a nontrivial example. [ABSTRACT FROM AUTHOR]

In this paper, our aim is to research the change in the magnetic field using the curvature theory of ruled surfaces. For this, we examine magnetic curves of the curve whose existence is guaranteed from the derivative formulas of the rotation frame. The Killing vector fields and Lorentz forces of these magnetic curves are calculated. Additionally, correlations between curvatures are obtained. Flux ruled surfaces are obtained and the conditions for developable are achieved. Finally, examples of the magnetic flux developable ruled surfaces are given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the dual magnetic trajectories traced by the dual Frenet vectors of any dual curves in dual space via the dual Lorentz force, and then we give the characterizations of dual magnetic curves. Moreover, we define the dual flux ruled surfaces corresponding to the Killing vectors on the dual unit sphere and identify the conditions for these surfaces to be minimal or developable. Additionally, we present certain characterizations related to these surfaces. We demonstrate the visual representations of some dual flux ruled surfaces in some examples. [ABSTRACT FROM AUTHOR]

The principal aim of this paper is to find the expression of *-Bach tensors on (κ,μ)′-almost-Kenmotsu manifolds. It is shown that if the metric of a (κ,μ)′-almost-Kenmotsu manifold admits a generalized closed *-Bach soliton, then the metric is *-Bach flat under certain restrictions. We also prove that the metric of a generalized gradient *-Bach soliton on a (κ,μ)′-almost-Kenmotsu manifold is *-Bach flat, for a soliton function which is invariant under the Reeb vector field. Further, we characterize generalized *-Bach solitons with trace-free *-Bach tensors on (κ,μ)′-almost-Kenmotsu manifolds and results are verified by an example. As an application of these solitons, we study generalized almost-Bach solitons whose metrics are associated with the Lorentzian metric of the perfect fluid spacetimes. [ABSTRACT FROM AUTHOR]

We introduce the notion of conformal trajectories in three-dimensional Riemannian manifolds M3. Given a conformal vector field V ∈ 픛(M3), a conformal trajectory of V is a regular curve γ in M3 satisfying ∇γ′γ′ = qV × γ′, for some fixed nonzero constant q ∈ ℝ. In this paper, we study the conformal trajectories in the space forms ℝ3, 핊3 and ℍ3. For (non-Killing) conformal vector fields in 핊3 (respectively, in ℍ3), we prove that conformal trajectories have constant curvature and its torsion is a linear combination of trigonometric (respectively, hyperbolic) functions on the arc-length parameter. In the case of Euclidean space ℝ3, we obtain the same result for the radial vector field and characterizing all conformal trajectories. [ABSTRACT FROM AUTHOR]

In this research paper, we determine the nature of conformal η -Ricci–Bourguignon soliton on a general relativistic spacetime with torse forming potential vector field. Besides this, we evaluate a specific situation of the soliton when the spacetime admitting semi-symmetric energy–momentum tensor with respect to conformal η -Ricci–Bourguignon soliton, whose potential vector field is torse-forming. Next, we explore some characteristics of curvature on a spacetime that admits conformal η -Ricci–Bourguignon soliton. In addition, we turn up some physical perception of dust fluid, dark fluid and radiation era in a general relativistic spacetime in terms of conformal η -Ricci–Bourguignon soliton. Finally, we examine necessary and sufficient conditions for a 1-form η , which is the g -dual of the vector field ξ on general relativistic spacetime to be a solution of the Schrödinger–Ricci equation. [ABSTRACT FROM AUTHOR]

In this paper, we define the notion of Bott connection in the Heisenberg group (ℍ 3 , g) and derive the expression of the Bott connection by using the Levi-Civita connection. Moreover, we derive the expressions of killing vector fields by using the killing equation and obtain some explicit formulas for killing magnetic curves associated to the Bott connection. Furthermore, we give some examples of killing magnetic curves. [ABSTRACT FROM AUTHOR]

In this paper, we examine curves drawn by a timelike or spacelike curve α on the pseudo-hyperbolic sphere H 0 2 and the unit Lorentzian sphere S 1 2   in three-dimensional Minkowski space ℝ 1 3. We present Lorentz forces and magnetic curves produced by the Frenet frame { T T , N T , B T } on spherical indicatricies in a magnetic field of the curve α. We calculate magnetic vector fields of magnetic curves for T T , N T , B T . Furthermore, we define magnetic flux surfaces constructed by magnetic vector fields along magnetic spherical indicatricies. We obtain developability conditions for these surfaces. Finally, we illustrate magnetic flux surfaces with some interesting examples. [ABSTRACT FROM AUTHOR]

In this paper, we study the semi-symmetry and pseudo-symmetry of almost cosymplectic 3 -manifolds. First, we prove that an H -almost cosymplectic 3 -manifold M is semi-symmetric if and only if it is cosymplectic. Here by an H -almost cosymplectic 3 -manifold, we mean an almost cosymplectic 3 -manifold whose characteristic vector field ξ is a harmonic unit vector field. If an almost cosymplectic 3 -manifold M whose fundamental endomorphism field h is parallel in the direction of the characteristic vector field ξ ( ∇ ξ h = 0), then it is H -almost cosymplectic. In particular, an almost cosymplectic 3 -manifold M satisfying ∇ ξ h = 0 is semi-symmetric if and only if it is cosymplectic. Next, we prove that pseudo-symmetric H -almost cosymplectic 3 -manifolds are certain generalized almost cosymplectic (κ , μ , ν) -spaces. [ABSTRACT FROM AUTHOR]

In this paper, we characterize three-dimensional trans-Sasakian manifolds of type (α , β) admitting *- η -Ricci-solitons and gradient almost *- η -Ricci-solitons. First, we prove that a trans-Sasakian three-manifold of type (α , β) admitting a *- η -Ricci soliton reduces to a β -Kenmostu manifold, provided potential vector field V is Jacobi along a geodesic of ξ. Next, we show that the trans-Sasakian metric as a gradient almost *- η -Ricci soliton is either flat o r *- η -Einstein o r a β -Kenmotsu manifold. [ABSTRACT FROM AUTHOR]

The prime object of this paper is to study the perfect fluid spacetimes obeying f (ℛ) -gravity, when η -Ricci solitons, gradient η -Ricci solitons, gradient Einstein solitons and gradient m -quasi Einstein solitons are its metrics. At first, the existence of the η -Ricci solitons is proved by a non-trivial example. We establish conditions for which the η -Ricci solitons are expanding, steady or shrinking. Besides, in the perfect fluid spacetimes obeying f (ℛ) -gravity, when the potential vector field of η -Ricci soliton is of gradient type, we acquire a Poisson equation. Moreover, we investigate gradient η -Ricci solitons, gradient Einstein solitons and gradient m -quasi Einstein solitons in f (ℛ) -gravity, respectively. As a result, we establish some significant theorems about dark matter era. [ABSTRACT FROM AUTHOR]

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 O or N. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study certain types of metric such as ∗-conformal Ricci–Yamabe soliton (RYS), whose potential vector field is torse-forming on Kenmotsu manifold. Here, we establish the conditions for solitons to be expanding, shrinking or steady and find the scalar curvature when the manifold admits ∗-conformal RYS on Kenmotsu manifold. Next, we developed the nature of the vector field when the manifold satisfies ∗-conformal RYS. Also, we have adorned some applications of torse-forming vector field in terms of ∗-conformal RYS on Kenmotsu manifold. We have also studied infinitesimal CL-transformation and Schouten–van Kampen connection on Kenmotsu manifold, whose metric is ∗-conformal RYS. We present an example of ∗-conformal RYS on three-dimensional Kenmotsu manifold, and verify some of our findings. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the η -Einstein cosymplectic manifolds with the gradient Einstein solitons and the conformal vector fields. It is proven that if an η -Einstein cosymplectic manifold M 2 n + 1 of dimension 2 n + 1 with n > 1 admits a gradient Einstein soliton, then either M 2 n + 1 is Ricci flat or the gradient of Einstein potential function is pointwise collinear with the Reeb vector field of M 2 n + 1 . Also, we prove that if M 2 n + 1 admits a conformal vector field W , then either M 2 n + 1 is Ricci flat or W is homothetic. We also establish that a conformal vector field W on M 2 n + 1 is a strict infinitesimal contact transformation, provided the scalar curvature of M 2 n + 1 is constant. Finally, the non-trivial examples of cosymplectic manifolds admitting the gradient Einstein solitons are given. [ABSTRACT FROM AUTHOR]

In this paper, we investigate conformal η-Ricci–Yamabe soliton on semiconformally flat Kählerian Norden spacetime in conjunction with the Kählerian Norden torse-forming vector field. Furthermore, we examine various physical aspects of such soliton on special fluid spacetime, such as dust fluid, dark fluid, radiation era, viscous fluid, and heat flux. In addition, we light up the harmonic features of conformal η-Ricci-Yamabe soliton on semiconformally flat Kählerian Norden spacetime. Finally, we also deduce the necessary and sufficient conditions for a 1-form η, which is the g-dual of the vector field ξ on such a spacetime, to be a solution of the Schrödinger-Ricci equation. [ABSTRACT FROM AUTHOR]

A generalized Robertson–Walker spacetime is not, in general, a perfect fluid spacetime and the converse is not, in general, true. In this paper, we show that if a perfect fluid spacetime admits an η – ρ -Einstein soliton, then the integral curves generated by the velocity vector field u are geodesics and the acceleration vector vanishes. Also, we show that if a perfect fluid spacetime with Killing velocity vector field admits a gradient η – ρ -Einstein soliton, then the spacetime represents either dark matter era or, the acceleration vector vanishes. Next, we show that if a generalized Robertson–Walker spacetime admits an η – ρ -Einstein soliton, then it becomes a perfect fluid spacetime. Finally, we prove that in a dark matter fluid with vanishing vorticity satisfying gradient η – ρ -Einstein soliton in f (ℛ) -gravity with constant Ricci scalar, either the energy density and isotropic pressure are constant or, the potential function ψ remains invariant under the velocity vector u. Also, for two models f (ℛ) = e α ℛ + α log ℛ and f (ℛ) = ℛ n (α = constant and ℛ is the Ricci scalar of the spacetime), various energy conditions in terms of the Ricci scalar are examined and state that the Universe is in an accelerating phase and satisfies the weak, null and dominant energy conditions, but violate the strong energy condition. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate Lie symmetries including Killing, homothetic and conformal symmetries of Lemaitre–Tolman–Bondi (LTB) spacetime metric. To find all LTB metrics admitting these three types of symmetries, we have analyzed the set of symmetry equations by a Maple algorithm that provides some restrictions on the functions involved in LTB metric under which this metric admits the three mentioned symmetries. The solution of symmetry equations under these restrictions leads to the explicit form of symmetries. The stress–energy tensor is calculated for all the obtained metrics in order to discuss their physical significance. It is noticed that most of these metrics satisfy certain energy conditions and correspond to anisotropic fluids. [ABSTRACT FROM AUTHOR]

In this paper, we give some characterizations by considering ∗-Ricci soliton as a Kenmotsu metric. We prove that if a Kenmotsu manifold represents an almost ∗-Ricci soliton and the potential vector field V is a Jacobi along the Reeb vector field, then it is a steady ∗-Ricci soliton. Next, we show that a Kenmotsu matric endowed an almost ∗-Ricci soliton which is Einstein metric if it is η -Einstein or the potential vector field V is collinear to the Reeb vector field or V is an infinitesimal contact transformation. [ABSTRACT FROM AUTHOR]

In this paper, using the Finslerian settings, we study the existence of parallel one forms (or, equivalently parallel vector fields) on a Riemannian manifold. We show that a parallel one form on a Riemannian manifold M is a holonomy invariant function on the tangent bundle T M with respect to the geodesic spray. We prove that if the metrizability freedom of the geodesic spray of (M , F) is 1 , then the (M , F) does not admit a parallel one form. We investigate a sufficient condition on a Riemannian manifold to admit a parallel one form. As by-product, we relate the existence of a proper affine Killing vector field by the metrizability freedom. We establish sufficient conditions for the existence of a parallel one form on a Finsler manifold. By counter-examples, we show that if the metrizability freedom is greater than 1, then the manifold (Riemannian or Finslerian) does not necessarily admit a parallel one form. Various special cases and examples are studied and discussed. [ABSTRACT FROM AUTHOR]

The objective of this paper is to deal with Kenmotsu manifolds admitting η -Ricci-Yamabe solitons. First, it is proved that if a Kenmotsu manifold M which admits an η -Ricci-Yamabe soliton, then the manifold M is Einstein and is of constant scalar curvature. Then, some important characterizations, which classify Kenmotsu manifolds admitting such solitons, are obtained and an example given which supports our results. [ABSTRACT FROM AUTHOR]

There are two smooth functions σ and ρ associated to a nontrivial concircular vector field v on a connected Riemannian manifold (M , g) , called potential function and connecting function. In this paper, we show that presence of a timelike nontrivial concircular vector field influences the geometry of generalized Robertson–Walker space-times. We use a timelike concircular vector field v on an n -dimensional connected conformally flat Lorentzian manifold, n > 2 , to find a characterization of generalized Robertson–Walker space-time with fibers Einstein manifolds. It is interesting to note that for n = 4 the concircular vector field annihilates energy-momentum tensor and also that in this case the potential function σ is harmonic. In the second part of this paper, we show that presence of a nontrivial concircular vector field v with connecting function ρ on a complete and connected n -dimensional conformally flat Riemannian manifold (M , g) , n > 2 , with Ricci curvature Ric (v , v) non-negative, satisfying n (n − 1) ρ + τ = 0 , is necessary and sufficient for (M , g) to be isometric to either a sphere S n (c) or to the Euclidean space E n , where τ is the scalar curvature. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study conformal almost Ricci solitons within the framework of Kenmotsu manifolds. First, we demonstrate that if the potential vector field is Jacobi along the Reeb vector field, then the soliton reduces to a conformal Ricci soliton. If the manifold is η -Einstein Kenmotsu manifold, we show that either the manifold is of constant scalar curvature or the potential vector field is an infinitesimal contact transformation. In addition, if we consider the soliton vector field as a contact vector field, then either the gradient of λ is pointwise collinear with the Reeb vector field or the manifold becomes η -Einstein. Lastly, we develop an example of a conformal almost Ricci soliton on the Kenmotsu manifold. [ABSTRACT FROM AUTHOR]

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 ξ. 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 η -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. [ABSTRACT FROM AUTHOR]

The paper aims to investigate curvature inheritance (CI) symmetry in M-projectively flat spacetimes. It is shown that the CI symmetry in M-projectively flat spacetime is a conformal motion. We have proved that M-projective curvature tensor follows the symmetry inheritance property along a vector field ξ , when spacetime admits the conditions of both CI symmetry and conformal motion or motion along the vector field ξ. Also, we have derived some results for M-projectively flat spacetime with perfect fluid following the Einstein field equations (EFEs) with a cosmological term and admitting the CI symmetry along the vector field ξ. We have shown that an M-projectively flat perfect fluid spacetime obeying the EFEs with a cosmological term and admitting the CI symmetry along a vector field ξ is either a vacuum or satisfies the vacuum-like equation of state. We have also shown that such spacetimes with the energy–momentum tensor of an electromagnetic field distribution do not admit any curvature symmetry of general relativity. Finally, an example of M-projectively flat spacetime has been exhibited. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate proper conformal vector fields (CVFs) of Bianchi type-I perfect fluid space-times in the f (R , T) theory of gravity, where R and T are the Ricci scalar and the trace of energy–momentum tensor, respectively. To seek the proper CVFs, we perform a classification of the space-times by imposing different restrictions on the space-time components. The whole classification consists of 12 cases. By studying each case thoroughly, we conclude that in the f (R , T) theory of gravity, the overall dimension of CVFs for the Bianchi type-I space-times turns out to be 3, 4, 5 and 15. [ABSTRACT FROM AUTHOR]

The spherical indicatrix of a ruled surface is the image curve of its rulings on the sphere S 2 . In this paper, we present Lorentz forces and magnetic curves produced by the Frenet frame { T e , N e , B e } of the spherical indicatrix of a ruled surface in a magnetic field. We calculate magnetic vector fields of magnetic curves for T e , N e , B e . Furthermore, we define magnetic flux surfaces constructed by magnetic vector fields along magnetic spherical indicatrix. We obtain developability conditions for these surfaces. Finally, we give some examples to show magnetic flux surfaces. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study certain types of metrics such as conformal η -Ricci soliton and Yamabe soliton in general relativistic spacetime. Here, we have shown the nature of the soliton when the spacetime with semisymmetric energy–momentum tensor admits conformal η -Ricci soliton, whose potential vector field is torse-forming. We have studied certain curvature conditions on the spacetime that admits conformal η -Ricci soliton. Also, we have enriched the importance of the Laplace equation on the spacetime admitting conformal η -Ricci soliton. Next, we have given some applications of physical connection of dust fluid, dark fluid and radiation era on general relativistic spacetime admitting conformal η -Ricci soliton and Yamabe soliton. [ABSTRACT FROM AUTHOR]

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 f (T) 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 f (T) 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. [ABSTRACT FROM AUTHOR]

In this paper, we present a new geometric interpretation of the notion of the Heisenberg antiferromagnetic spin for quasi flows of normal magnetic particles with quasi frame in 3D space. Moreover, we investigate integrability conditions for quasi new frame fields. Therefore, we determine necessary and sufficient condition for given normal magnetic particles with flow in space. Additionally, we present some new constructions for evolution of particle by Heisenberg antiferromagnetic model and we give total magnetic phase for new quasi vector fields in 3D space. [ABSTRACT FROM AUTHOR]

In this paper, we study geometric phases for an electric field in Minkowski 3-space. In particular, we consider Bishop 2-type frames of a spacelike and a timelike curve in Minkowski 3-space and investigate three geometric phases for three cases of electric fields with respect to the Bishop 2-type frames along a spacelike and a timelike curve. Also, we compute spatial and time evolutions of the complex frame connected with electric fields for Bishop 2-type in terms of modified Hasimoto transformations. As a result, we give a modified Fermi–Walker parallel derivative and a Rytov parallel transportation of electric fields. Finally, we discuss Lorentz force of a magnetic field as a Killing vector field along a spacelike and timelike curve in Minkowski 3-space. [ABSTRACT FROM AUTHOR]

The aim of this paper is to find some important classes of Einstein manifolds using conformal η -Ricci solitons and conformal η -Ricci almost solitons. We prove that a Kenmotsu metric as conformal η -Ricci soliton is Einstein if it is η -Einstein or the potential vector field V is infinitesimal contact transformation or collinear with the Reeb vector field ξ. Next, we prove that a Kenmotsu metric as gradient conformal η -Ricci almost soliton is Einstein if the Reeb vector field leaves the scalar curvature invariants. Finally, we construct some examples to illustrate the existence of conformal η -Ricci soliton, gradient almost conformal η -Ricci soliton on Kenmotsu manifold. [ABSTRACT FROM AUTHOR]

In this paper, we have studied different energy conditions for some metrics arising from the classification of Kantowski–Sachs spacetimes via homothetic vector fields (HVFs). The homothetic equations for Kantowski–Sachs spacetimes are analyzed using a Maple algorithm which transforms these equations to the reduced involutive form (Rif), giving all possible cases in the form of a tree, known as Rif tree, where the branches of the resulting tree give the potential spacetime metrics which may possess proper HVFs. The set of homothetic equations is integrated for all branches of the Rif tree and the explicit form of HVFs is obtained in ease case. For all the obtained metrics, we have found the bounds for different energy conditions. The stability of these metrics is also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we classify dust static plane symmetric (SPS) space-times via proper curvature collineations (CCs) in the f (R) gravity using the rank of 6 × 6 Riemann matrix and direct integration technique. Classifying the above mentioned space-times, we find that there arise six cases corresponding to specific values of the function f (R). Inspecting all the cases, we reach at the fact that there exist two cases where the space-times admit proper CCs which form an infinite dimensional vector space (IDVS). In rest of the four cases, the space-time becomes either flat or CCs become the killing vector fields (KVFs). [ABSTRACT FROM AUTHOR]

Let M be an almost cosymplectic manifold such that the Reeb vector field is Killing. In this paper, it is proved that if the metric of M satisfying the η -Einstein condition is an η -Ricci soliton, then either M is Ricci flat or the potential vector field is an infinitesimal contact transformation. Also, a concrete example of cosymplectic 3 -manifold admitting non-trivial Ricci and η -Ricci solitons is constructed. [ABSTRACT FROM AUTHOR]

Quaternions are widely used in physics. Quaternions, an extension of complex numbers, are closely related to many fundamental concepts (e.g. Pauli matrices) in physics. The aim of this study is the geometric structure underlying the quaternions used in physics. In this paper, we have investigated a new structure of unit speed associated curves such as spatial quaternionic and quaternionic osculating direction curves. For this, we have assumed that the vector fields χ (ϱ) = υ 1 (ϱ) (ϱ) + υ 2 (ϱ) (ϱ) , where υ 1 2 (ϱ) + υ 2 2 (ϱ) = 1 for the spatial quaternionic curve ψ and χ (ϱ) = λ 1 (ϱ) ⊤ (ϱ) + λ 2 (ϱ) η (ϱ) + λ 3 β 2 (ϱ) , where λ 1 2 (ϱ) + λ 2 2 (ϱ) + λ 3 2 (ϱ) = 1 for the quaternionic curve ϕ. Then, we have given the relationship between (spatial) quaternionic (OD)-curves and Mannheim curve pair. Moreover, we have examined in which cases the (spatial) quaternionic (OD)-curve can be helix or slant helix. So, considering that helices also take place in electron physics, it is thought that this study will create a bridge between physics and geometry. Finally, we have given the examples and draw the figures of curves in the examples. [ABSTRACT FROM AUTHOR]

The main intention of this paper is to analyze integrability for the derivative formulas of the rotation minimizing frame in the Lorentz–Minkowski 3-space. As far as we know, no one has yet given a method to study their integrability in the Lorentz–Minkowski 3-space. So, we introduce the coordinate system in order to provide a tool for studying the integrability. As an application, the position vectors of some special curves having an important place in mathematical and physical research are obtained in the natural representation form. Finally, we support our work with examples. [ABSTRACT FROM AUTHOR]

In this paper, we have studied Noether symmetries of the general Bianchi type I spacetimes. The Lagrangian associated with the most general Bianchi type I metric is used to find the set of Noether symmetry equations. These equations are analyzed using an algorithm, developed in Maple, to get all possible Bianchi type I metrics admitting different Noether symmetries. The set of Noether symmetry equations is then solved for each metric to obtain the Noether symmetry algebras of dimensions 4, 5, 6 and 9. [ABSTRACT FROM AUTHOR]

In this paper, we study Clairaut Riemannian maps whose total manifolds admit a Ricci soliton and give a nontrivial example of such Clairaut Riemannian maps. First, we calculate Ricci tensors and scalar curvature of total manifolds of Clairaut Riemannian maps. Then we obtain necessary conditions for the fibers of such Clairaut Riemannian maps to be Einstein and almost Ricci solitons. We also obtain a necessary condition for vector field α ̇ to be conformal, where α is a geodesic curve on total manifold of Clairaut Riemannian map. Further, we show that if total manifolds of Clairaut Riemannian maps admit a Ricci soliton with the potential mean curvature vector field of ker F ∗ then the total manifolds of Clairaut Riemannian maps also admit a gradient Ricci soliton and obtain a necessary and sufficient condition for such maps to be harmonic by solving Poisson equation. [ABSTRACT FROM AUTHOR]

The main scope of this paper is to examine null Cartan curves especially the ones with constant torsion. In accordance with this scope, the position vector of a null Cartan curve is stated by a linear combination of the vector fields of its pseudo-orthogonal frame with differentiable functions. However, the most important difference that distinguishes this study from the other studies is that the Bertrand curve couples (timelike, spacelike or null) of null Cartan curves are also examined. Consequently, it is seen that all kinds of Bertrand couples of a given null Cartan curve with constant curvature functions have also constant curvature functions. This result is the most valuable result of the study, but allows us to introduce a transformation on null Cartan curves. Then, it is proved that aforesaid transformation is a Backlund transformation which is well recognized in modern physics. Moreover, motion of an inextensible null Cartan curve is investigated. By considering time evolution of null Cartan curve, the angular momentum vector is examined. And three different situations are given depending on the character of the angular momentum vector Ω. In the case of τ t = 0 , we discuss the solution of the system which is obtained by compatibility conditions. Finally, we provide the relation between torsion of the curve and the velocity vector components of the moving curve  C. [ABSTRACT FROM AUTHOR]

In this paper, we show that the velocity vector field of classical Bernoulli–Euler elastic curve is harmonic in R n space. We propose a new characterization for classical Bernoulli–Euler elastic curves and plot graphs of examples that satisfy this characterization. [ABSTRACT FROM AUTHOR]