A class of Finsler metrics of scalar flag curvature

It is known that every locally projectively flat Finsler metric is of scalar flag curvature. Conversely, it may not be true. In this paper, for a certain class of Finsler metrics, we prove that it is locally projectively flat if and only if it is of scalar flag curvature. Moreover, we establish a class of new non-trivial examples.     

A new class of projectively flat Finsler metrics with constant flag curvature K=1

In this paper, we consider a class of Finsler metrics which obtained by Kropina change of the class of generalized m-th root Finsler metrics. We classify projectively flat Finsler metrics in this class of metrics. Then under a condition, we show that every projectively flat Finsler metric in this class with constant flag curvature is locally Minkowskian. Finally, we find necessary and sufficient condition under which this class of metrics be locally dually flat.     

一类射影平坦的Finsler度量

本文主要研究由两个Riemann度量和一个1-形式构成的Finsler度量.首先,本文给出这类度量局部射影平坦的等价条件;其次,给出这类度量局部射影平坦且具有常旗曲率的分类情形;最后,构造这类度量局部射影平坦且具有常旗曲率K=-1的例子.     

Finsler warped product metrics with special Riemannian curvature properties

One of the fundamental problems in Finsler geometry is to study Finsler metrics of constant(or scalar) curvature. In this paper, we refine and improve Chen-Shen-Zhao equations characterizing Finsler warped product metrics of scalar flag curvature. In particular, we find equations that characterize Finsler warped product metrics of constant flag curvature. Then we improve the Chen-Shen-Zhao result on characterizing Einstein Finsler warped product metrics. As its application we construct explicitly many new warped product Douglas metrics of constant Ricci curvature by using known locally projectively flat spherically symmetric metrics of constant flag curvature.     

On a class of two-dimensional projectively flat finsler metrics with constant flag curvature

In this paper, we study a special class of two-dimensional Finsler metrics defined by a Riemannian metric and 1-form. We classify those which are locally projectively flat with constant flag curvature.     

General spherically symmetric Finsler metrics with constant Ricci and flag curvature

In this paper, we investigate the flag curvature of a special class of Finsler metrics called general spherically symmetric Finsler metrics, which are defined by a Euclidean metric and two related 1-forms. We find equations to characterize the class of metrics with constant Ricci curvature (tensor) and constant flag curvature. Moreover, we study general spherically symmetric Finsler metrics with the vanishing non-Riemannian quantity χ-curvature. In particular, we construct some new projectively flat Finsler metrics of constant flag curvature.     

Some projectively flat(α,β)-metrics

In this paper, we study a class of Finsler metrics in the form F = α + ∈β + 2k β2/α-k2β4/3α3 , where α= (√aijyiyj) is a Riemannian metric, β = biyi is a 1-form, and ∈ and k ≠ 0 are constants. We obtain a sufficient and necessary condition for F to be locally projectively flat and give the non-trivial special solutions. Moreover, it is proved that such projectively flat Finsler metrics with the constant flag curvature must be locally Minkowskian.     

On a class of weakly Einstein Finsler metrics

In this paper we study a special class of Finsler metrics—m-Kropina metrics which are defined by a Riemannian metric and a 1-form. We prove that a weakly Einstein m-Kropina metric must be Einsteinian. Further, we characterize Einstein m-Kropina metrics in very simple conditions under a suitable deformation, and obtain the local structures of m-Kropina metrics which are of constant flag curvature and locally projectively flat with constant flag curvature respectively.     

On some classes of projectively flat Finsler metrics with constant flag curvature

In this paper, we give the equivalent PDEs for projectively flat Finsler metrics with constant flag curvature defined by a Euclidean metric and two 1-forms. Furthermore, we construct some classes of new projectively flat Finsler metrics with constant flag curvature by solving these equivalent PDEs.     

On a class of projectively flat Finsler metrics with weakly isotropic flag curvature

In this paper, we study a class of Finsler metrics defined by a Riemannian metric and 1-form. We classify those metrics which are projectively flat with weakly isotropic flag curvature.     

Finsler metrics with special curvature properties

In this paper, we discuss the relationship between the flag curvature and some non-Riemannian quantities of Finsler metrics of scalar curvature. In particular, we characterize projectively flat Finsler metrics with isotropic S-curvature.     

On dually flat Finsler metrics

In this paper, we study the locally dually flat Finsler metrics which arise from information geometry. An equivalent condition of locally dually flat Finsler metrics is given. We find a new method to construct locally dually flat Finsler metrics by using a projectively flat Finsler metric under the condition that the projective factor is also a Finsler metric. Finally, we find that many known Finsler metrics are locally dually flat Finsler metrics determined by some projectively flat Finsler metrics.     

Projectively Flat Singular Square Metrics with Constant Flag Curvature

In this paper, we study and characterize locally projectively flat singular square metrics with constant flag curvature. First, we obtain the sufficient and necessary conditions that singular square metrics are locally projectively flat. Furthermore, we classify locally projectively flat singular square metrics with constant flag curvature completely.     

A global classification result for Randers metrics of scalar curvature on closed manifolds

One important problem in Finsler geometry is that of classifying Finsler metrics of scalar curvature. By investigating the second-order differential equation for a class of Randers metrics with isotropic S$S$-curvature, we give a global classification of these metrics of scalar curvature, generalizing a theorem previously only known in the case of locally projectively flat Randers metrics.     

一类射影平坦的球对称的芬斯勒度量

本文研究了射影平坦芬斯勒度量的构造问题.通过分析射影平坦的球对称的芬斯勒度量的方程的解,构造了一类新的射影平坦的芬斯勒度量,并得到了射影平坦的球对称的芬斯勒度量的射影因子和旗曲率.     

The holonomy group of locally projectively flat Randers two-manifolds of constant curvature

In this paper, we investigate the holonomy structure of the most accessible and demonstrative 2-dimensional Finsler surfaces, the Randers surfaces. Randers metrics can be considered as the solutions of the Zermelo navigation problem. We give the classification of the holonomy groups of locally projectively flat Randers two-manifolds of constant curvature. In particular, we prove that the holonomy group of a simply connected non-Riemannian projectively flat Finsler two-manifold of constant non-zero flag curvature is maximal and isomorphic to the orientation preserving diffeomorphism group of the circle.     

On the classification of projectively flat Finsler metrics with constant flag curvature

In this paper, we study locally projectively flat Finsler metrics with constant flag curvature K. We prove those are totally determined by their behaviors at the origin by solving some nonlinear PDEs. The classifications when K=0$\mathbf{K}=0$, K=−1$\mathbf{K}=-1$ and K=1$\mathbf{K}=1$ are given respectively in an algebraic way. Further, we construct a new projectively flat Finsler metric with flag curvature K=1$\mathbf{K}=1$ determined by a Minkowski norm with double square roots at the origin. As an application of our main theorems, we give the classification of locally projectively flat spherical symmetric Finsler metrics much easier than before.     

Some projectively flat (α, β)-metrics

In this paper, we study a class of Finsler metrics in the form $F = \alpha + \varepsilon \beta + 2k\tfrac{{\beta ^2 }}{\alpha } - \tfrac{{k^2 \beta ^4 }}{{3\alpha ^3 }}$ , where $\alpha = \sqrt {\alpha _{ij} y^i y^j } $ is a Riemannian metric, β = b _i y ⁱ is a 1-form, and ε and k ≠ 0 are constants. We obtain a sufficient and necessary condition for F to be locally projectively flat and give the non-trivial special solutions. Moreover, it is proved that such projectively flat Finsler metrics with the constant flag curvature must be locally Minkowskian.     

Two-dimensional Finsler metrics with constant flag curvature

 We construct infinitely many two-dimensional Finsler metrics on 𝕊² and 𝔻² with non-zero constant flag curvature. They are all not locally projectively flat. Received: 10 October 2001 / Revised version: 15 July 2002