Randers Metrics Research Articles

Deflection of charged signals in a dipole magnetic field in Kerr background

This paper investigates charged particle deflection in a Kerr spacetime background with a dipole magnetic field, focusing on the equatorial plane and employing the weak field approximation. We employ the Jacobi–Randers metric to unify the treatment of the gravitational and electromagnetic effects on charged particles. Furthermore, we utilize the Gauss–Bonnet theorem to calculate the deflection angle through curvature integrals. The difference between the prograde and retrograde deflection angles is linked to the non-reversibility of metrics and geodesics in Finsler geometry, revealing that this difference can be considered a Finslerian effect. We analyze the impact of both gravitomagnetic field and dipole magnetic field on particle motion and deflection using the Jacobi–Randers magnetic field. The model considered in this paper exhibits interesting features in the second-order approximation of (M/b), i.e., the ratio between the spacetime mass and impact parameters. When qμ=2MaE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q\\mu =2MaE$$\\end{document}, where q,E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q,\\,E$$\\end{document} are the charge and asymptotic energy of the charged particle and μ,a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu ,\\,a$$\\end{document} are the dipole magnetic moment and spacetime spin, the Jacobi–Randers metric possesses reversible geodesics, leading to equal prograde and retrograde deflection angles. In this case, the gravitomagnetic field and dipole magnetic field cancel each other out, distinguishing it from scenarios involving only the gravitomagnetic field or the dipole magnetic field. We also explore the magnetic field’s impact on gravitational lensing of charged particles.

Developing novel wormhole metrics in finsler-randers geometry using the barthel connection and osculating-riemannian method

In this research, we focus on modelling a wormhole using Finsler-Randers geometry. By applying the Barthel connection from the range of Finslerian connections, we have developed a new wormhole metric, termed the osculating Barthel-Finsler-Randers metric, through the osculating-Riemannian method within the Finsler-Randers framework. The fundamental field equations are formulated in terms of the obtained metric tensor characterised by η(r), which plays a pivotal role in characterizing the anisotropy of the model. To explore the application of Finsler geometry in addressing the existence of a wormhole, we adopt an exponential shape function. Further, in order to streamline the computational aspects of our analysis, and to discuss the effect of anisotropy effect of Finsler Rander geometry, we adopt a specific functional form for the anisotropic parameter η(r)=arn, where α is a constant with implications for the anisotropy, among various possibilities for η(r) a non-zero analytical component of vector field in Randers metric. In addition, we consider three distinct values for the exponent n = −1, 0, and 1 and subsequently, we proceed to develop two distinct models for each of these three values of n with constant and non-constant redshift function. We analyse the physical implications of η(r) on the energy conditions. Our results demonstrate the crucial impact of anisotropy on the stability and structure of wormholes, emphasizing the requirement of exotic matter under specific conditions. This research deepens the understanding of wormhole physics within Finsler geometry, shedding light on the theoretical frameworks essential for the existence and stability of these cosmic structures.

The isoperimetric problem in Randers Poincaré disc

It is known that a simply connected Riemann surface satisfies the isoperimetric equality if and only if it has constant Gaussian curvature. In this paper, we show that the circles centered at origin in the Randers Poincaré disc satisfy the isoperimetric equality with respect to different volume forms however, these Randers metrics do not necessarily have constant (negative) flag curvature. In particular, we show that Osserman’s result [12] of the Riemannian case cannot be extended to the Finsler geometry as such.

Conformally related invariant (α, β)-metrics on homogeneous spaces

In this paper, we give the flag curvature formula of general (α, β)-metrics of Berwald type. We study conformally related (α, β)-metrics, especially general (α, β)-metrics that are conformally related to invariant (α, β)-metrics. Also, a necessary and sufficient condition for a Finsler metric conformally related to an (α, β)-metric is given and conformally related Douglas Randers metrics are studied. Finally, we present some examples of conformally related (α, β)-metrics.

Projective change between G.R.C. exponential metrics and Randers metrics

In this paper, we find the necessary and sufficient conditions under which two classes of (α, β)-metrics, i.e. generalized Randers change (G.R.C.) exponential metric F = α exp(β/α) + ϵβ and Randers metric F˜ = ˜α + β˜ are projectively related. Note that α and ˜α are Riemannian metrics, β and β˜ are 1-forms on a manifold M with dimension n > 2 and ϵ ̸= 0 is a constant. Furthermore, we study this projective relation when F has special curvature properties.

ON h-RANDERS EXPONENTIAL CHANGE OF FINSLER METRIC

Studying an (α, β)-metrics is a central idea in Finsler geometry, whichis a generalization of Randers metric. In this paper, we have derived the Cartanconnection for the Finsler space whose metric is given by h-Randers exponentialchange and also obtained the condition under which the Finslerian hypersurface tobe hyperplane of first, second and third kind.

On recurrent Riemannian and Ricci curvatures of Finsler metrics

In this paper, we study a property of Riemannian and Ricci curvatures under which it reproduces itself, namely, recurrent Finsler metrics. We prove that if (M,F) is a recurrent Finsler manifold of non-zero isotropic flag curvature, then F is a Landsberg metric. It follows that Every positively complete 2-dimensional Randers metric is recurrent if and only if it is a Riemannian or locally Minkowskian metric. Next, we study two positive (or negative) projectively related Ricci parallel Finsler metrics on a compact manifold. We show that the projective equivalence is trivial and then the Riemannian curvatures are equal. In the same vein, we explore the class of Ricci-recurrent Randers metrics with closed and conformal form, and show that the related Riemannian metric is Ricci-recurrent if and only if the Randers metric is a Berwald metric. Finally, we find the necessary and sufficient condition under which a Kropina metric be Ricci-recurrent, provided that its one-form is closed and conformal.

On L-reducible spherically symmetric Finsler metrics

In this paper, we study one of the oldest open problems in Finsler geometry which was introduced by Matsumoto-Shimada in 1977 about the existence of a concrete L-reducible Finsler metric that is not C-reducible. To spot such a Finsler metric, we study the class of spherically symmetric Finsler metrics. We prove two rigidity theorems for spherically symmetric Finsler metrics. First, we prove that every spherically symmetric Finsler metric is semi-C-reducible. Second, we show that every non-Riemannian spherically symmetric Finsler metric is a generalized L-reducible metric. Finally, under a particular condition, we prove that every non-Riemannian L-reducible spherically symmetric Finsler metric on a manifold of dimension n≥3 must be a Randers metric. This result provides a negative answer to Matsumoto-Shimada's problem in the class of spherically symmetric Finsler metrics.

On mean Cartan torsion of Finsler metrics

UDC 514.7 We prove that Finsler manifolds with unbounded mean Cartan torsion cannot be isometrically imbedded into any Minkowski space. We also study the generalized Randers metrics obtained by the Rizza structure and show that any generalized Randers metric has an unbounded mean Cartan torsion. Then generalized Randers metrics cannot be isometrically imbedded into any Minkowski space. Further, we prove that every generalized Randers metric is quasi-C-reducible. Finally, we show that every generalized Randers metric on 2-dimensional Finsler manifold has a vanishing mean Cartan torsion.

Randers metrics on two-spheres of revolution with simple cut locus

<abstract><p>In the present paper, we study the Randers metric on two-spheres of revolution in order to obtain new families of Finsler of Randers type metrics with simple cut locus. We determine the geodesics behavior, conjugate and cut loci of some families of Finsler metrics of Randers type whose navigation data is not a Killing field and without sectional or flag curvature restrictions. Several examples of Randers metrics whose cut locus is simple are shown.</p></abstract>

On a class of weakly weighted Einstein metrics

The notion of general weighted Ricci curvatures appears naturally in many problems. The N-Ricci curvature and the projective Ricci curvature are just two special ones with totally different geometric meanings. In this paper, we study general weighted Ricci curvatures. We find that Randers metrics of certain isotropic weighted Ricci curvature must have isotropic S-curvature. Then we classify them via their navigation expressions. We also find equations that characterize Randers metrics of almost isotropic weighted Ricci curvature.

Finslerian Viewpoint to the Rectifying, Normal, and Osculating Curves

The theory of Finsler metric was introduced by Paul Finsler, in 1918. The author defines this metric using the Minkowski norm instead of the inner product. Therefore, this geometry is a more general metric and includes the Riemannian metric. In the present work, using the Finsler metric, we investigate the position vector of the rectifying, normal and osculating curves in Finslerian 3-space $\mathbb{F}^{3}$. We obtain the general characterizations of these curves in $\mathbb{F}^{3}$. Furthermore, we show that rectifying curves are extremal curves derived from the Finslerian spherical curve. We also plotted various examples by using the Randers metrics.

Randers metrics based on deformations by gradient winds

Randers metrics based on deformations by gradient winds

On projective Riemann curvature

With a volume form on a manifold, every spray can be deformed to a projective spray. The Riemann curvature of a projective spray is called the projective Riemmann curvature. In this paper, we introduce the notion of projectively R-flat sprays/Finsler metrics. As an application, we investigate and characterize projectively R-flat Randers metrics. Moreover, we obtained a rigidity theorem on a closed Finsler manifold.

On general (α,β)-metric with isotropic Berwald curvature

In this paper, we study a special class of Finsler metrics: general (α,β)-metric. We show that every general (α,β)-metric on Mn with isotropic Berwald curvature and n≥3 must be a Berwald metric or a Randers metric.

Boundary rigidity for Randers metrics


 If a non-reversible Finsler norm is the sum of a reversible Finsler norm and a closed 1-form, then one can uniquely recover the 1-form up to potential fields from the boundary distance data. We also show a boundary rigidity result for Randers metrics where the reversible Finsler norm is induced by a Riemannian metric which is boundary rigid. Our theorems generalize Riemannian boundary rigidity results to some non-reversible Finsler manifolds. We provide an application to seismology where the seismic wave propagates in a moving medium.

On non-positively curved homogeneous Finsler metrics

In this paper, we prove two rigidity results for non-positively curved homogeneous Finsler metrics. Our first main result yields an extension of Hu-Deng's well-known result proven for the Randers metrics. Indeed, we prove that every connected homogeneous Finsler space with non-positive flag curvature and isotropic S-curvature is Riemannian or locally Minkowskian. We extend the Szabó's rigidity theorem for Berwald surfaces and show that homogeneous isotropic Berwald metrics with non-positive flag curvature are Riemannian or locally Minkowskian. Our second main result is to show that every homogeneous (α,β)-metric with non-positive flag curvature and almost isotropic S-curvature is Riemannian or locally Minkowskian.

On geodesically reversible Finsler manifolds

A Finsler manifold is said to be geodesically reversible if the reversed curve of any geodesic remains a geometrical geodesic. Well-known examples of geodesically reversible Finsler metrics are Randers metrics with closed [Formula: see text]-forms. Another family of well-known examples are projectively flat Finsler metrics on the [Formula: see text]-sphere that have constant positive curvature. In this paper, we prove some geometrical and dynamical characterizations of geodesically reversible Finsler metrics, and we prove several rigidity results about a family of the so-called Randers-type Finsler metrics. One of our results is as follows: let [Formula: see text] be a Riemannian–Finsler metric on a closed surface [Formula: see text], and [Formula: see text] be a small antisymmetric potential on [Formula: see text] that is a natural generalization of [Formula: see text]-form (see Sec. 1). If the Randers-type Finsler metric [Formula: see text] is geodesically reversible, and the geodesic flow of [Formula: see text] is topologically transitive, then we prove that [Formula: see text] must be a closed [Formula: see text]-form. We also prove that this rigidity result is not true for the family of projectively flat Finsler metrics on the [Formula: see text]-sphere of constant positive curvature.

Geometry of left-invariant Randers metric on the Heisenberg groups

PurposeIn this paper, we consider the Heisenberg groups which play a crucial role in both geometry and theoretical physics.Design/methodology/approachIn the first part, we retrieve the geometry of left-invariant Randers metrics on the Heisenberg group H2n+1, of dimension 2n + 1. Considering a left-invariant Randers metric, we give the Levi-Civita connection, curvature tensor, Ricci tensor and scalar curvature and show the Heisenberg groups H2n+1 have constant negative scalar curvature.FindingsIn the second part, we present our main results. We show that the Heisenberg group H2n+1 cannot admit Randers metric of Berwald and Ricci-quadratic Douglas types. Finally, the flag curvature of Z-Randers metrics in some special directions is obtained which shows that there exist flags of strictly negative and strictly positive curvatures.Originality/valueIn this work, we present complete Reimannian geometry of left invarint-metrics on Heisenberg groups. Also, some geometric properties of left-invarainat Randers metrics will be studied.

Optimal navigation strategies for microswimmers on curved manifolds

Finding the fastest path to a desired destination is a vitally important task for microorganisms moving in a fluid flow. We study this problem by building an analytical formalism for overdamped microswimmers on curved manifolds and arbitrary flows. We show that the solution corresponds to the geodesics of a Randers metric, which is an asymmetric Finsler metric that reflects the irreversible character of the problem. Using the examples of spherical and toroidal surfaces, we demonstrate that the swimmer performance that follows this “Randers policy” always beats a more direct policy. Moreover, our results show that the relative gain grows significantly when specific structures related to either the geometry or the flow are exploited by the swimmer. A study of the shape of isochrones reveals features such as self-intersections, cusps, and abrupt nonlinear effects. Our work provides a link between microswimmer physics and geodesics in generalizations of general relativity.Received 15 October 2020Revised 2 February 2021Accepted 29 April 2021DOI:https://doi.org/10.1103/PhysRevResearch.3.023125Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasSwimmingPhysical SystemsActive matterMicroswimmersTechniquesGeometryBiological Physics