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.
Homogeneous Finsler spheres with constant curvature have specific geodesic properties.
problem Existence and properties of homogeneous Finsler spheres with constant flag curvature.
method Proofs and analysis of geodesic properties on homogeneous Finsler spheres.
result Homogeneous Finsler spheres with constant flag curvature are either Riemannian or Randers.
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, …
The paper studies Randers and (α,β) equigeodesics on compact homogeneous manifolds.
problem Characterizing equigeodesics on compact homogeneous manifolds.
method Analyzing different types of equigeodesics (Riemannian, Finsler, Randers, (α,β)) on compact homogeneous manifolds. result Randers and (α,β) equigeodesics are equivalent on compact homogeneous manifolds, and a criterion is found. The paper extends geodesic orbit sphere classification to Finsler geometry.
problem Classifying geodesic orbit spheres in Finsler geometry.
method Generalized from Riemannian to Finsler geometry, proving constant curvature conditions.
result Geodesic orbit Finsler spheres with constant flag curvature are Randers.
The study finds infinitely many periodic orbits just above a critical value on a 2-sphere.
problem Finding periodic orbits just above a critical value on a 2-sphere.
method Introduced a new critical value c∞(L) and showed its strict inequality to the Mañé critical value c(L), proving the existence of infinitely many periodic orbits on energy levels e∈(c(L),c∞(L)). result Infinitely many periodic orbits exist on energy levels just above the Mañé critical value.
Study isoparametric hypersurfaces in a Randers sphere with constant flag curvature.
problem Characterize isoparametric hypersurfaces in a Randers sphere of constant flag curvature.
method Analyze isoparametric hypersurfaces for the standard round sphere and show their behavior under navigation.
result Provide a classification of special isoparametric hypersurfaces and their ambient metrics.
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.
By using Stationary-to-Randers correspondence (SRC), a characterization of light and time-convexity of the boundary of a region of a standard stationary (n+1)-spacetime is obtained, in terms of the convexity of the boundary of a domain in a Finsler n or (n+1)-space of Randers type. The latter convexity is analyzed in d…
A Clifford-Wolf translation of a connected Finsler space is an isometry which moves each point the sam distance. A Finsler space (M,F) is called Clifford-Wolf homogeneous if for any two point x1,x2∈M there is a Clifford-Wolf translation ρ such that ρ(x1)=x2. In this paper, we study Clifford-Wolf transl…
The paper studies geodesics and isoparametric functions on Finsler spheres.
problem Analyzing geodesics and isoparametric functions on Finsler spheres.
method Global expressions of geodesics and isoparametric functions derived using navigation and Cartan-Münzner polynomials.
result Construction of isoparametric families and focal submanifolds.
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 S-curvature and are either Minkowskian or Riemannian. An isometry of a Finsler space is called Clifford-Wolf translation (CW-translation) if it moves all points the same distance. A Finsler space (M,F) is called Clifford-Wolf homogeneous (CW-homogeneous) if for any x,y∈M there is a CW-translation σ such that σ(x)=y. We prove that if F is a homogeneous Finsl…
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.
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 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.
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 M we mean a Finsler structure L∗=L+α, where L is a Riemannian structure and α is a 1-form on M. This structure was first introduced by Randers ~\cite{[8]} from the standpoint of general relativity. In this paper, we replace L 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 ildeF based on an existing one F. result Shows that F is of Berwald (Douglas) type if and only if ildeF is of Berwald (Douglas) type. 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.
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
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 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 ∥β∥α>1 and ∥β∥α≡1. 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.
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 n(n+2). result An n-dimensional Randers space has a C-projective algebra of maximum dimension n(n+2) 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.
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.
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…
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 M=R×S and Randers metrics on S. In particular, for stationary spacetimes, we give a simple characterization of when they are causally conti…
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.
The paper explores harmonic and asymptotically harmonic Finsler manifolds and their properties.
problem Characterizing and understanding harmonic Finsler manifolds.
method Investigation of various types of harmonic Finsler manifolds, characterizations via mean curvature and Laplacian, and construction techniques.
result Certain harmonic Finsler manifolds are of Einstein type and examples of non-Riemannian Finsler harmonic manifolds are provided.
In the year 1984 Shibata investigated the theory of a change which is called a β-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.