We study the behavior of geodesics on a Randers surface of revolution. The main tool is the extension of Clairaut relation from Riemannian case to the Randers case. Moreover, we show that our Randers surface of revolution can be embedded in a Minkowski space as hypersurface.
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48 results for “Randers surfaces”
Study classifies holonomy groups of projectively flat Randers surfaces.
problem Classifying holonomy groups of Randers surfaces.
method Investigation of holonomy structure of Randers surfaces, classification of holonomy groups.
result Holonomy group of simply connected non-Riemannian projectively flat Finsler two-manifolds is maximal and isomorphic to the circle diffeomorphism group.
The contribution of this paper is two-fold. The first one is to derive a simple formula of the mean curvature form for a hypersurface in the Randers space with a Killing field, by considering the Busemann-Hausdorff measure and Holmes-Thompson measure simultaneously. The second one is to obtain the explicit local expres…
Study on cut locus structure of a specific Randers surface.
problem Analyzing the cut locus of a Randers rotational 2-sphere.
method Examined Gaussian curvature monotonicity and its effect on cut locus.
result Cut locus properties depend on Gaussian curvature monotonicity.
In this paper, we study Clifford-Wolf translations of homogeneous Randers metrics on spheres. It turns out that we can present a complete description of all the Clifford-Wolf translations of all the homogeneous Randers metrics on spheres. The most important point of this paper is that a new phenomena surfaces. Namely, …
New asymmetric metric on Teichmüller space for surfaces.
problem Defining a new asymmetric weak metric on Teichmüller space.
method Introducing Teichmüller-Randers metric as an asymmetric deformation of the Teichmüller metric.
result Teichmüller geodesics become unique Teichmüller-Randers geodesics under certain conditions.
The article proves Randers Poincaré disc satisfies isoperimetric equality.
problem Extending Riemannian isoperimetric equality to Finslerian case.
method Analyzes Randers Poincaré disc with different volume forms.
result Osserman's result cannot be extended to Finslerian case.
In 2D, Finsler metrics with 3+ projective fields are projectively equivalent to Randers.
problem Characterizing Finsler metrics with multiple projective vector fields.
method Analyzing the Lie algebra of projective vector fields and showing equivalence to Randers metrics.
result A complete list of 2D Finsler metrics with at least 3 projective fields up to equivalence.
Study geodesics on a special cylinder with arbitrary wind.
problem Global behavior of geodesics on a Randers metric cylinder.
method Solve Zermelo's navigation problem to define the Randers metric; analyze geodesics, conjugate, and cut loci.
result Characterize geodesics and their properties on the base manifold.
The paper proves rigidity results for non-positively curved homogeneous Finsler metrics.
problem Rigidity of non-positively curved homogeneous Finsler metrics.
method Analyzes and proves rigidity results for specific Finsler metrics with non-positive flag curvature.
result Homogeneous Finsler spaces with non-positive flag curvature and isotropic S-curvature are either Riemannian or locally Minkowskian.
The paper examines Randers metrics with isotropic scalar curvature properties.
problem Characterizing Randers metrics with specific scalar curvature properties.
method Analyzes properties of Randers metrics with isotropic scalar curvature.
result Proves that Randers metrics with weakly isotropic scalar curvature have isotropic -curvature and are either Minkowskian or Riemannian.
Two-dimensional metrics related by conformal transformations are also Randers.
problem Understanding the relationship between conformally related Douglas metrics.
method Analyzing two-dimensional metrics and their conformal transformations.
result Two-dimensional conformally related Douglas metrics are specifically Randers.
Study of Randers metrics on spheres with simple cut loci.
problem Understanding Randers metrics on spheres and their cut loci.
method Analyzing geodesics, conjugate, and cut loci of Finsler metrics of Randers type.
result Found new families of Randers metrics with simple cut loci.
Paper classifies Randers metrics based on Ricci curvature properties.
problem Investigating isotropic projective Ricci curvature in Randers metrics.
method Classification of Randers metrics based on isotropic projective Ricci curvature properties.
result Randers metric of isotropic projective Ricci curvature is reversible if and only if it is of square projective Ricci curvature.
Study connects stationary spacetimes with pre-Randers metrics using Fermat's principle.
problem Understanding causal structures in stationary spacetimes.
method Relativistic Fermat's principle linking stationary spacetimes and pre-Randers metrics.
result Description of causal ladder in terms of pre-Randers metric elements.
The paper studies projectively and dually flat Finsler spaces with Randers changes.
problem Characterizing projectively and dually flat Finsler spaces with Randers changes.
method Analyzing Randers changes of special (α, β)-metrics, finding fundamental and inverse metric tensors, and establishing necessary conditions.
result Conditions for Randers changes of (α, β)-metrics to be projectively and locally dually flat.
We consider a triality between the Zermelo navigation problem, the geodesic flow on a Finslerian geometry of Randers type, and spacetimes in one dimension higher admitting a timelike conformal Killing vector field. From the latter viewpoint, the data of the Zermelo problem are encoded in a (conformally) Painleve-Gullst…
The paper classifies isoparametric hypersurfaces in Randers space forms.
problem Classifying isoparametric hypersurfaces in Randers space forms.
method Anisotropic submanifolds and isoparametric hypersurfaces in Randers space forms (N,F) with (h,W) were studied.
result The isoparametric hypersurfaces in a Randers space form (N,F) are the same as in Riemannian space, despite different isoparametric functions.
By a Randers' structure on a manifold we mean a Finsler structure , where is a Riemannian structure and is a 1-form on . This structure was first introduced by Randers ~\cite{[8]} from the standpoint of general relativity. In this paper, we replace by a Finsler structure, calling the resulti…
Defines a new Randers metric based on an existing one.
problem No specific problem stated; focuses on defining a new metric.
method Defines a new left-invariant Randers metric based on an existing one .
result Shows that is of Berwald (Douglas) type if and only if is of Berwald (Douglas) type.
The paper studies Randers and equigeodesics on compact homogeneous manifolds.
problem Characterizing equigeodesics on compact homogeneous manifolds.
method Analyzing different types of equigeodesics (Riemannian, Finsler, Randers, ) on compact homogeneous manifolds.
result Randers and equigeodesics are equivalent on compact homogeneous manifolds, and a criterion is found.
The paper classifies Randers metrics with scalar flag curvature.
problem Classifying Finsler metrics of scalar flag curvature.
method Investigating Randers metrics under the condition that β is a Killing 1-form.
result Obtained necessary conditions for Randers metrics to be of scalar flag curvature.
Study on Randers metrics on tangent Lie groups and their geometric properties.
problem Characterizing Randers metrics of Berwald type on tangent Lie groups.
method Analyzing the relations between flag curvature and sectional curvature.
result Identifying all 3-dimensional Lie groups with tangent bundles admitting Berwald type Randers metrics.
Sub-Randers metricsmath.DG
Introduce sub-Randers metrics by adding a one-form to a sub-Riemannian metric
problem Define a new class of sub-Finsler metrics
method Derive equations for sub-Randers normal geodesics
result Prove a Hopf-Rinow type theorem for sub-Randers manifolds
We determine all Finsler metrics of Randers type for which the Riemannian part is a scalar multiple of the Euclidean metric, on an open subset of the Euclidean plane, whose geodesics are circles. We show that the Riemannian part must be of constant Gaussian curvature, and that for every such Riemannian metric there is …
The paper classifies special Randers metrics on Lie groups.
problem Classifying Randers metrics of Douglas type on Lie groups.
method Classification through invariant hyper-Hermitian metrics.
result Formulas for flag curvature and same sign curvature in some directions.
In this paper, we study Randers metrics and find a condition on Ricci tensor of these metrics to be Berwaldian. This generalize Shen's Theorem which says: every R-°at complete Randers metric is locally Minkowskian. Then we find a necessary and sufficient condition on Ricci tensor under which a Randers metric of scalar …
The paper explores conditions for Randers metrics to have compatible linear connections.
problem Conditions for Randers metrics to have compatible linear connections.
method Solving constrained optimization problems for tensors.
result Necessary and sufficient conditions for a Randers metric to be a generalized Berwald metric.
A left invariant Z-Randers metric on the five-dimensional Heisenberg group is a left invariant Randers metric with deformation vector from the center of the Heisenberg algebra. In this note we prove that for every left invariant Z-Randers metric on the five-dimensional Heisenberg group there exist flags of strictly neg…
Study on conditions for Randers metrics to be of constant Ricci curvature.
problem Conditions for Randers metrics to be of constant Ricci curvature.
method Analysis of sufficient and necessary conditions for Randers metrics with and without strong convexity.
result Classification of Randers metrics with and .
Study examines geometry of special metric on Heisenberg group.
problem Geometry of left-invariant Randers metrics on the Heisenberg group.
method Investigation of left-invariant Randers metrics on the Heisenberg group.
result Analysis of the geometry of these metrics.
In the present paper, the flag curvature of invariant Randers metrics on homogeneous spaces and Lie groups is studied. We first give an explicit formula for the flag curvature of invariant Randers metrics arising from invariant Riemannian metrics on homogeneous spaces and, in special case, Lie groups. We then study Ran…
The paper finds Einstein-Randers metrics on specific homogeneous spaces.
problem Finding metrics on specific homogeneous spaces.
method Proved existence of Einstein metrics and then showed existence of Non-Riemannian Einstein-Randers metrics.
result Specific homogeneous spaces admit Non-Riemannian Einstein-Randers metrics.
Randers pp-wavesgr-qc
Study of Randers spacetimes and their Finsler gravity solutions.
problem Analyzing Finsler gravity field equations for Randers spacetimes.
method Examined Berwald-type Randers spacetimes and Finsler gravity field equations, showing equivalence to Einstein gravity.
result Found exact solutions for vacuum Finsler gravity are composed of pp-waves and 1-forms.
Characterizes C-projective vector fields on Randers spaces.
problem Characterizing C-projective vector fields on Randers spaces.
method Using a non-Riemannian quantity ${fΞ}$, it is shown that ${fΞ}$ is invariant for C-projective vector fields and the dimension of the algebra of C-projective vector fields is at most .
result An -dimensional Randers space has a C-projective algebra of maximum dimension if and only if it is locally Minkowskian or (up to re-scaling) locally isometric to the generalized Funk metric.
We correct a mistake in Shen Yibing, Yu Yaoyong, On Projectively Related Randers Metrics, International Journal of Mathematics 19}(2008), no. 5, 503--520, and prove the natural generalization of the projective Lichnerowicz-Obata conjecture for Randers metrics.
Geodesics in Randers spaces of constant curvature are classified.
Study on isoperimetric problem in Randers planes achieving maximum area.
problem Isoperimetric problem in Randers planes.
method Analyzing circles centered at the origin for maximum area.
result Circles centered at the origin achieve local maximum area.
Our paper is devoted to the study of the holonomy groups of Finsler surfaces using the methods of infinite dimensional Lie theory. The notion of infinitesimal holonomy algebra will be introduced, by the smallest Lie algebra of vector fields on an indicatrix, containing the curvature vector fields and their horizontal c…
In this paper we argue that when gauge invariance is taken into consideration, there is no consistent geometric framework of Finsler class that can accommodate Randers type spaces. In this context, an alternative non-Finslerian framework for Randers spacetimes compatible with gauge invariance is introduced.
In this article we review the recent results about the flag curvature of invariant Randers metrics on homogeneous manifolds and by using a counter example we show that the formula which obtained for the flag curvature of these metrics is incorrect. Then we give an explicit formula for the flag curvature of invariant Ra…
The paper explores almost Ricci solitons on Finsler spaces, proving conditions for their existence.
problem Characterizing almost Ricci solitons on Finsler measure spaces.
method Introducing and investigating gradient almost Ricci solitons, proving conditions for existence.
result Conditions for the existence of gradient almost Ricci solitons on Finsler measure spaces.
We obtain some results in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure (Causality) of standard stationary spacetimes on and Randers metrics on . In particular, for stationary spacetimes, we give a simple characterization of when they are causally conti…
Characterizes two-dimensional generalized Berwald metrics with vanishing S-curvature.
problem Characterizing metrics with specific curvature properties.
method Analyzing two-dimensional generalized Berwald -metrics with vanishing S-curvature.
result Provides a generalization of Szabó rigidity theorem for -metrics.
In the year 1984 Shibata investigated the theory of a change which is called a -change of a Finsler metric. On the other hand in 1985 a systematic study of geometry of hypersurfaces in Finsler spaces was given by Matsumoto. In the present paper is to devoted to the study of a condition for a Randers conformal chang…
Wind Riemannian structures generalize Randers metrics and are classified for constant flag curvature.
problem Classifying wind Riemannian structures of constant flag curvature.
method Using the natural data for Zermelo navigation problem, constructing WRS from a Riemannian metric and a vector field (wind).
result Local and global classification of wind Riemannian structures of constant flag curvature.
The paper studies weighted Ricci curvatures and characterizes Randers metrics.
problem Characterizing Randers metrics with weighted Ricci curvatures.
method General weighted Ricci curvatures and characterization of Randers metrics.
result Characterization of Randers metrics with almost isotropic weighted Ricci curvatures.
In 2D, Finsler metrics are Douglas and generalized Berwald if they are Berwald or Randers.
problem Characterizing Finsler metrics in 2D that are both Douglas and generalized Berwald.
method Proof that in dimension two, a Finsler metric is both Douglas and generalized Berwald if and only if it is Berwald or a Randers metric with specific properties.
result Finsler metrics in 2D are Douglas and generalized Berwald if they are Berwald or Randers metrics with specific properties.