Homogeneous Randers Research Articles

On Homogeneous Randers Metrics

In this paper, we study the curvature features of the class of homogeneous Randers metrics. For these metrics, we first find a reduction criterion to be a Berwald metric based on a mild restriction on their Ricci tensors. Then, we prove that every homogeneous Randers metric with relatively isotropic (or weak) Landsberg curvature must be Riemannian. This provides an extension of well-known Deng-Hu theorem that proves the same result for a homogeneous Berwald-Randers metric of non-zero flag curvature.

Geodesic graphs in Randers g.o. spaces

The concept of geodesic graph is generalized from Riemannian geometry to Finsler geometry, in particular to homogeneous Randers g.o. manifolds. On modified H-type groups which admit a Riemannian g.o. metric, invariant Randers g.o. metrics are determined and geodesic graphs in these Finsler g.o. manifolds are constructed. New structures of geodesic graphs are observed.

Naturally reductive homogeneous (α,β) spaces

In this paper, we study naturally reductive [Formula: see text]-metrics on homogeneous manifolds. We show that naturally reductive [Formula: see text]-metrics arise only when [Formula: see text] is naturally reductive and some conditions on [Formula: see text] is satisfied. We give an explicit formula for the flag curvature of naturally reductive [Formula: see text]metrics which improves the flag curvature formula of naturally reductive Randers metrics given in [D. Latifi, Naturally reductive homogeneous Randers spaces, J. Geom. Phys. 60 (2010) 1968–1973]. As a special case, we give an explicit formula for the flag curvature of bi-invariant [Formula: see text]-metrics on Lie groups.

Homogeneous Randers spaces admitting just two homogeneous geodesics

Homogeneous Randers spaces admitting just two homogeneous geodesics

Homogeneous Einstein–Randers metrics on symplectic groups

In this paper, we prove that Lie group Sp(n)(n≥3) admits at least one family of invariant Einstein–Randers metrics, which are not naturally reductive. We also give the classification of these metrics under isometries and determine their group of isometries. Moreover, we prove that if homogeneous Randers space (M,F) with navigation data (h,W) is Berwald, then (M,F) is naturally reductive if and only if the Riemannian metric (M,h) is naturally reductive.

Ricci-quadratic homogeneous Randers spaces

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 Berwald type.

Clifford-Wolf translations of homogeneous Randers spheres

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 local one-parameter groups of Clifford-Wolf translations for homogeneous Randers metrics on spheres. The most important point of this paper is a new phenomenon in Finsler geometry. Namely, we find that there are some CW-homogeneous Randers spaces which are essentially not symmetric. This is a great difference compared to Riemannian geometry, where any CW-homogeneous Riemannian manifold must be locally symmetric.

Curvatures of homogeneous Randers spaces

We study curvatures of homogeneous Randers spaces. After deducing the coordinate-free formulas of the flag curvature and Ricci scalar of homogeneous Randers spaces, we give several applications. We first present a direct proof of the fact that a homogeneous Randers space is Ricci quadratic if and only if it is a Berwald space. We then prove that any left invariant Randers metric on a non-commutative nilpotent Lie group must have three flags whose flag curvature is positive, negative and zero, respectively. This generalizes a result of J.A. Wolf on Riemannian metrics. We prove a conjecture of J. Milnor on the characterization of central elements of a real Lie algebra, in a more generalized sense. Finally, we study homogeneous Finsler spaces of positive flag curvature and particularly prove that the only compact connected simply connected Lie group admitting a left invariant Finsler metric with positive flag curvature is SU(2).

On Flag Curvature of Homogeneous Randers Spaces

AbstractIn this paper we give an explicit formula for the flag curvature of homogeneous Randers spaces of Douglas type and apply this formula to obtain some interesting results. We first deduce an explicit formula for the flag curvature of an arbitrary left invariant Randersmetric on a two-step nilpotent Lie group. Then we obtain a classification of negatively curved homogeneous Randers spaces of Douglas type. This results, in particular, in many examples of homogeneous non-Riemannian Finsler spaces with negative flag curvature. Finally, we prove a rigidity result that a homogeneous Randers space of Berwald type whose flag curvature is everywhere nonzero must be Riemannian.

Clifford–Wolf translations of Finsler spaces

Abstract In this paper, we study Clifford–Wolf translations of Finsler spaces. We give a characterization of those Clifford–Wolf translations generated by Killing vector fields. In particular, we show that there is a natural interrelation between the local one-parameter groups of Clifford–Wolf translations and the Killing vector fields of constant length. In the special case of homogeneous Randers spaces, we give some explicit sufficient and necessary conditions for a Killing vector field to have a constant length, in which case the local one-parameter group of isometries generated by the Killing field consist of Clifford–Wolf translations. Finally, we construct explicit examples to explain some of the results of this paper.

Three dimensional homogeneous Finsler manifolds

AbstractIn this paper we study three dimensional homogeneous Finsler manifolds. We first obtain a complete list of the three‐dimensional homogeneous manifolds which admit invariant Finsler metrics. Then we consider invariant Randers metrics and present the classification of three dimensional homogeneous Randers spaces under isometrics.

Invariant naturally reductive Randers metrics on homogeneous spaces

Purpose: The purpose of this paper is to study the geometric properties of naturally reductive homogeneous Randers spaces. Methods: We use Lie theory methods in the study of Finsler geometry. Results: We first prove that if a Randers metric is naturally reductive, then the underlying Riemannian metric is naturally reductive. Then, we show that, for Berwald type Randers metric, if the underlying Riemannian metric is naturally reductive, then the Randers metric is naturally reductive. Finally, we give a geometric criterion of homogeneous naturally reductive Randers spaces. Conclusions: This paper provides a convenient method to construct naturally reductive Randers metrics on homogeneous Riemannian manifolds.

Homogeneous Randers spaces with isotropic S-curvature and positive flag curvature

In this paper, we will give a complete classification of homogeneous Randers spaces with isotropic S-curvature and positive flag curvature. This results in a large class of Finsler spaces with non-constant positive flag curvature. At the final part of the paper, we prove a rigidity result asserting that a homogeneous Randers space with almost isotropic S-curvature and negative Ricci scalar must be Riemannian.

Naturally reductive homogeneous Randers spaces

In this paper, we study naturally reductive Randers metrics on homogeneous manifolds. We first prove that naturally reductive Randers metrics are of Berwald type. We then give an explicit formula for the flag curvature of naturally reductive Randers metrics. Finally a necessary and sufficient condition for invariant Randers metrics on homogeneous manifolds being naturally reductive is given.

Invariant Finsler metrics on polar homogeneous spaces

We study invariant Finsler metrics on polar homogeneous manifolds. After establishing existence results, we prove that an invariant Finsler metric on a nonsymmetric polar homogeneous manifold of a simply connected compact simple Lie group is Berwaldian if and only if it is Riemannian. As an application, we prove that on each such manifold with generalized rank of at least 2, there exist infinitely many invariant Finsler metrics that are reversible, non-Berwaldian and of vanishing S-curvature; this kind of space is sought after in an open problem of Shen. Finally, using one type of polar homogeneous manifold, we give a classification of homogeneous Randers spaces with positive constant flag curvature.

The S-curvature of homogeneous Randers spaces

In this paper, we give an explicit formula of the S-curvature of homogeneous Randers spaces and prove that a homogeneous Randers space with almost isotropic S-curvature must have vanishing S-curvature. As an application, we obtain a classification of homogeneous Randers space with almost isotropic S-curvature in some special cases. Some examples are also given.

Homogeneous geodesics in homogeneous Finsler spaces

In this paper, we study homogeneous geodesics in homogeneous Finsler spaces. We first give a simple criterion that characterizes geodesic vectors. We show that the geodesics on a Lie group, relative to a bi-invariant Finsler metric, are the cosets of the one-parameter subgroups. The existence of infinitely many homogeneous geodesics on the compact semi-simple Lie group is established. We introduce the notion of a naturally reductive homogeneous Finsler space. As a special case, we study homogeneous geodesics in homogeneous Randers spaces. Finally, we study some curvature properties of homogeneous geodesics. In particular, we prove that the S-curvature vanishes along the homogeneous geodesics.

On homogeneous Finsler spaces

In this paper we study homogeneous Finsler spaces and show that they are forward complete. As a special case we consider homogeneous Randers spaces and show that if these spaces have constant flag curvature then the underlying Riemannian space is locally symmetric. Also we extend some of classical results in Riemannian homogeneous spaces to those in homogeneous Finsler spaces.