arXiv research

Geodesics in Randers spaces of constant curvature are classified.

In this paper, we discuss anisotropic submanifolds and isoparametric hypersurfaces in a Randers space form (N,F) with the navigation datum (h,W). We find that (N, F) with respect to the BH-volume and (N,h) have the same isoparametric hypersurfaces although, in general, their isoparametric functions are different. This …

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.

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 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 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 paper explores almost Ricci solitons on Finsler spaces, proving conditions for their existence.

In this paper, we consider Randers change of some special $(α, β)-$ metrics. First we find the fundamental metric tensor and Cartan tensor of these Randers changed $(α, β)-$metrics. Next, we establish a general formula for inverse of fundamental metric tensors of these metrics. Finally, we find the necessary and su…

A characterization of the C-projective vector fields on a Randers spaces is presented in terms of a recently introduced non-Riemannian quantity defined by Z. Shen and denoted by ${\bfΞ}$; It is proved that the quantity ${\bfΞ}$ is invariant for C-projective vector fields. Therefore, the dimension of the algebra of the …

Study on isoparametric and constant-curvature hypersurfaces in Finsler spaces.

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.

We give a necessary and sufficient condition on a Randers space for the existence of a measure for which Shen's S-curvature vanishes everywhere. Moreover, such a measure coincides with the Busemann-Hausdorff measure up to a constant multiplication.

The paper explores conditions for Randers metrics to have compatible linear connections.

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…

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 certain circumstances tools of Riemannian geometry are sufficient to address questions arising in the more general Finslerian context. We show that one such instance presents itself in the characterisation of geodesics in Randers spaces of constant flag curvature. To achieve a simple, Riemannian derivation of this s…

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…

The collection of all projective vector fields on a Finsler space $(M, F)$ is a finite-dimensional Lie algebra with respect to the usual Lie bracket, called the projective algebra denoted by $p(M,F)$ and is the Lie algebra of the projective group $P(M,F)$. The projective algebra $p(M,F=α+β)$ of a Randers space is chara…

In some recent papers, the relations existing between the metric properties of Randers spaces and the conformal geometry of stationary Lorentzian manifolds were discovered and investigated. In this note, we focus on the equality between the index of a geodesic in a Randers space and that of its lightlike lift in the as…

The paper examines Randers metrics with isotropic scalar curvature properties.

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 $n\ge 3$, 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…

Paper finds flag curvature of submanifolds in Randers-Minkowski space using Zermelo data.

A Clifford-Wolf translation of a connected Finsler space is an isometry which moves each point the same distance. A Finsler space $(M, F)$ is called Clifford-Wolf homogeneous if for any two points $x_1, x_2\in M$ there is a Clifford-Wolf translation $ρ$ such that $ρ(x_1)=x_2$. In this paper, we give a complete classifi…

Study of Randers metrics on spheres with simple cut loci.

In this paper, we consider a Finsler space with a Randers change of Quartic metric F = $\sqrt[4]{α^4 + β^4} + β$. The conditions for this space to be with reversible geodesics are obtained. Further, we study some geometrical properties of F with reversible geodesics and prove that the Finsler metric F induces a general…

Paper classifies Randers metrics based on Ricci curvature properties.

A linear connection on a Finsler manifold is called compatible to the metric if its parallel transports preserve the Finslerian length of tangent vectors. Generalized Berwald manifolds are Finsler manifolds equipped with a compatible linear connection. Since the compatibility to the Finslerian metric does not imply the…

In this paper we consider simply connected Lie groups equipped with left invariant Randers metrics which arise from left invariant Riemannian metrics and left invariant vector fields. Then we study the intersection between automorphism and isometry groups of these spaces. Finally it has shown that for any left invarian…

In this paper, we first prove that homogeneous spaces $E_6/A_4$ and $E_6/A_1$ admit Einstein metrics which are $Ad(T\times A_1\times A_4)$-invariant, and then show that they admit Non-Riemannian Einstein-Randers metrics.

The study introduces new tensors for almost Finsler manifolds and analyzes their properties.

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.

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…

The paper studies Randers and $(α,β)$ equigeodesics on compact homogeneous manifolds.

Introduce sub-Randers metrics by adding a one-form to a sub-Riemannian metric

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 …

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 $x_1, x_2\in M$ there is a Clifford-Wolf translation $ρ$ such that $ρ(x_1)=x_2$. In this paper, we study Clifford-Wolf transl…

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…

Using the relativistic Fermat's principle, we establish a bridge between stationary-complete manifolds which satisfy the observer-manifold condition and pre-Randers metrics, namely, Randers metrics without any restriction on the one-form. As a consequence, we give a description of the causal ladder of such spacetimes i…

Study of Randers spacetimes and their Finsler gravity solutions.

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.

Integral formulae for foliated Riemannian manifolds provide obstructions for existence of foliations or compact leaves of them with given geometric properties. This paper continues our recent study and presents new integral formulae and their applications for codimension-one foliated Randers spaces. The goal is a gener…

We show that two-dimensional conformally related Douglas metrics are Randers

Study on isoperimetric problem in Randers planes achieving maximum area.

Let $G$ be a Lie group equipped with a left invariant Randers metric of Berward type $F$, with underlying left invariant Riemannian metric $g$. Suppose that $\widetilde{F}$ and $\widetilde{g}$ are lifted Randers and Riemannian metrics arising from $F$ and $g$ on the tangent Lie group $TG$ by vertical and complete lifts…

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…

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…