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.
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.
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.
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. 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.
Optimizes search paths in river environments using Finslerian geometry.
problem Optimizing search paths in river environments.
method Using Finslerian geometry and time-optimal paths based on Randers metric.
result Time-optimal paths in river environments.
Paper finds new minimal surfaces in hyperbolic Randers space.
problem Finding minimal surfaces in hyperbolic Randers spaces.
method Deriving formulas for mean curvature and finding explicit expressions for BH-minimal surfaces.
result First examples of BH-minimal surfaces in hyperbolic Randers space.
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 the present paper we study Randers metics of Berwald type on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. On these spaces, the Randers metrics arising from invariant hyper-Hermitian metrics are considered. Then we give explicit formulas for computing flag curvature of th…
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…
Study on flag curvatures of specific metrics on Lie groups.
problem Investigate flag curvatures of left invariant (α,β)-metrics and Randers metrics. method Explicit formulas and necessary/sufficient conditions for Berwald and Douglas types.
result Flag curvatures of (α,β)-metrics and Randers metrics on Lie groups can be zero, positive, or negative. In this paper we identify all simply connected 3-dimensional real Lie groups which admit Randers or Matsumoto metrics of Berwald type with a certain underlying left invariant Riemannian metric. Then we give their flag curvatures formulas explicitly.
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 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…
Generalizes navigation problem on curved spaces with speed constraints.
problem Navigating on curved spaces with speed limitations.
method Applied Finsler metric of Randers type to solve the problem.
result Solved the Zermelo navigation problem on Riemannian manifolds with speed constraints.
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.
A Finsler space is called Ricci-quadratic if its Ricci curvature Ric(x,y) is quadratic in y. It is called a Berwald space if its Chern connection defines a linear connection directly on the underlying manifold M. In this article, we prove that a homogeneous Randers space is Ricci-quadratic if and only if it is of…
In this paper, it is proved that any conformal vector field is homothetic on a locally projectively flat (α,β)-space of non-Randers type in dimension n≥3, and the local solutions of such a vector field are determined. While on a locally projectively flat Randers space, examples showthat the conformal vector fiel…
Study proves existence of multiple geodesics in a specific metric space.
problem Existence of multiple geodesics in a manifold with a Randers-Kropina metric.
method Lusternik-Schnirelman theory applied to a homotopy type of solutions of an affine control system.
result Proves existence of infinitely many geodesics between two points in a non-contractible manifold.
Optimal ship paths on perturbed spheroids found using control theory.
problem Finding time-optimal ship paths on an ellipsoid with a perturbation.
method Optimal control theory with Finsler metric of Randers type.
result Solutions to the problem on an oblate ellipsoid.
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. 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…
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.
The paper studies special metrics on tangent Lie groups.
problem Investigating lifted (α,β)-metrics of Douglas type on tangent Lie groups. method Constructing and analyzing vertical and complete lifted (α,β)-metrics on tangent Lie groups. result Necessary and sufficient conditions for these metrics to be of Douglas type are derived.
Introduces Finslerian convolution metrics and their properties.
problem No specific problem stated; focuses on new metric concept.
method Definition and study of Finslerian convolution metrics.
result Characterization of Finslerian convolution metrics of Riemannian, Minkowskian, and Randers types.
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…
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 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 of navigation on Hermitian manifolds with complex wind effects.
problem Zermelo navigation problem on Hermitian manifolds with complex wind.
method Admits space-dependent relative speed, discusses projectively related metrics, geodesics, and necessary conditions for locally projectively flat solutions.
result Presented necessary and sufficient conditions for locally projectively flat solutions.
The paper explores generalized Douglas-Weyl metrics and their properties.
problem Characterizing and understanding generalized Douglas-Weyl metrics.
method Analyzing properties of (α,β)-metrics and proving conditions for being generalized Douglas-Weyl metrics. result Generalized Douglas-Weyl metrics are Berwald metrics under certain conditions.
In this paper we study the geometry of simply connected two-step nilpotent Lie groups of dimension five. We give the Levi-Civita connection, curvature tensor, sectional and scalar curvatures of these spaces and show that they have constant negative scalar curvature. Also we show that the only space which admits left in…
Study on curvatures of specific metric on Heisenberg group.
problem Analyzing curvatures of a specific type of metric on a Heisenberg group.
method Examined left invariant Z-Randers metrics on the five-dimensional Heisenberg group.
result Existence of flags with strictly negative and strictly positive curvatures.
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.
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 …
Navigation in Lorentz Finsler geometry induces isoparametric hypersurfaces.
problem Defining and analyzing isoparametric hypersurfaces in Lorentz Finsler geometry.
method Using a navigation process with a Finsler metric and a tangent vector field, isoparametric functions and hypersurfaces are defined and analyzed.
result Local correspondences between isoparametric functions and hypersurfaces are established.
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…
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.
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.
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.
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.
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 …
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.