finsler metrics
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2021 ◽  
Vol 79 ◽  
pp. 101830
Author(s):  
M. Atashafrouz ◽  
B. Najafi ◽  
A. Tayebi

2021 ◽  
pp. 1-17
Author(s):  
Yong Fang

A Finsler manifold is said to be geodesically reversible if the reversed curve of any geodesic remains a geometrical geodesic. Well-known examples of geodesically reversible Finsler metrics are Randers metrics with closed [Formula: see text]-forms. Another family of well-known examples are projectively flat Finsler metrics on the [Formula: see text]-sphere that have constant positive curvature. In this paper, we prove some geometrical and dynamical characterizations of geodesically reversible Finsler metrics, and we prove several rigidity results about a family of the so-called Randers-type Finsler metrics. One of our results is as follows: let [Formula: see text] be a Riemannian–Finsler metric on a closed surface [Formula: see text], and [Formula: see text] be a small antisymmetric potential on [Formula: see text] that is a natural generalization of [Formula: see text]-form (see Sec. 1). If the Randers-type Finsler metric [Formula: see text] is geodesically reversible, and the geodesic flow of [Formula: see text] is topologically transitive, then we prove that [Formula: see text] must be a closed [Formula: see text]-form. We also prove that this rigidity result is not true for the family of projectively flat Finsler metrics on the [Formula: see text]-sphere of constant positive curvature.


Author(s):  
S. G. Elgendi

In this paper, as an application of the inverse problem of calculus of variations, we investigate two compatibility conditions on the spherically symmetric Finsler metrics. By making use of these conditions, we focus our attention on the Landsberg spherically symmetric Finsler metrics. We classify all spherically symmetric manifolds of Landsberg or Berwald types. For the higher dimensions [Formula: see text], we prove that all Landsberg spherically symmetric manifolds are either Riemannian or their geodesic sprays have a specific formula; all regular Landsberg spherically symmetric metrics are Riemannian; all (regular or non-regular) Berwald spherically symmetric metrics are Riemannian. Moreover, we establish new unicorns, i.e. new explicit examples of non-regular non-Berwaldian Landsberg metrics are obtained. For the two-dimensional case, we characterize all Berwald or Landsberg spherically symmetric surfaces.


Author(s):  
Hongjun Li ◽  
Chunhui Qiu ◽  
Hongchuan Xia ◽  
Guozhu Zhong

2021 ◽  
pp. 2150084
Author(s):  
Hongmei Zhu

In Finsler geometry, the projective Ricci curvature is an important projective invariant. In this paper, we investigate the projective Ricci curvature of a class of general [Formula: see text]-metrics satisfying a certain condition, which is invariant under the change of volume form. Moreover, we construct a class of new nontrivial examples on such Finsler metrics.


2021 ◽  
pp. 2150076
Author(s):  
Libing Huang ◽  
Xiaohuan Mo

In this paper, we discuss inverse problem in spray geometry. We find infinitely many sprays with non-diagonalizable Riemann curvature on a Lie group, these sprays are not induced by Finsler metrics. We also study left invariant sprays with non-vanishing spray vectors on Lie groups. We prove that if such a spray [Formula: see text] on a Lie group [Formula: see text] satisfies that [Formula: see text] is commutative or [Formula: see text] is projective, then [Formula: see text] is not induced by any (not necessary positive definite) left invariant Finsler metric. Finally, we construct an abundance of the left invariant sprays on Lie groups which satisfy the conditions in above result.


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