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.     

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.     

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.     

一类射影平坦的球对称芬斯勒度量

In this paper, we study spherically symmetric Finsler metrics. Analyzing the solution of the projectively flat equation, we construct a new class of projectively flat Finsler metrics.     

On a New Class of Projectively Flat Finsler Metrics

A class of Finsler metrics with three parameters is constructed. Moreover, a sufficient and necessary condition for this Finsler metrics to be projectively flat was obtained.     

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.     

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.     

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 Square metrics of scalar flag curvature

In this paper, we study the problem whether a Finsler metric of scalar flag curvature is locally projectively flat. We consider a special class of Finsler metrics — square metrics which are defined by a Riemannian metric and a 1-form on a manifold. We show that in dimension n ≥ 3, any square metric of scalar flag curvature is locally projectively flat.     

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.     

Some constructions of projectively flat Finsler metrics

In this paper, we find some solutions to a system of partial differential equations that characterize the projectively flat Finsler metrics. Further, we discover that some of these metrics actually have the zero flag curvature.     

关于射影平坦Finsler空间

本文研究了射影平坦Finsler空间的几何量及其几何性质。证明了射影平坦Finsler空间的Ricci曲率可完全由射影因子简洁地刻画出来。同时还证明了，在射影平坦Finsler空间中，平均Berwald曲率S=0意味着Ricci曲率Ric是二次齐次的。此外，给出了一个射影平坦Finsler空间成为常曲率空间或局部Minkowski空间的充分条件。     

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.     

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.     

PROJECTIVELY FLAT FINSLER METRICS WITH ALMOST ISOTROPIC S-CURVATURE

This article characterizes projectively flat Finsler metrics with almost isotropic S-curvature.     

On generalized m-th root finsler metrics

In this paper, we characterize locally dually flat generalized m-th root Finsler metrics. Then we find a condition under which a generalized m-th root metric is projectively related to a m-th root metric. Finally, we prove that if a generalized m-th root metric is conformal to a m-th root metric, then both of them reduce to Riemannian metrics.     

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.     

Reversible Finsler metrics of constant flag curvature

Without the restriction of quadratic form as a Riemannian metric, a Finsler metric $F=F(x,y)$ on a smooth manifold M can be reversible (symmetric in y) or not. Reversible Finsler metrics have different properties from Riemannian metrics though it seems they are very close to Riemannian metrics. Hilbert metric is the famous reversible Finsler metric of negative constant flag curvature in the history, and it is projectively flat. Then it is natural to ask the question how to classify reversible projectively flat Finsler metrics of constant flag curvature and give more new examples? In this paper, we answer the above question by giving the classification when the flag curvature $\mathbf{K}=-1,0,1$ respectively. Especially, for the case when $\mathbf{K}=-1$, we show that the only reversible projectively flat Finsler metrics are just Hilbert metrics. For the case when $\mathbf{K}=1$, we give an algebraic way to construct explicit metric function by solving algebraic equations, such as by solving a quartic equation. When the flag curvature is zero, it is much easier to construct reversible projectively flat Finsler metrics than before.     

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.     

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.