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.     

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.     

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 metrics with constant （or scalar） flag curvature

A Finsler metric on a manifold M with its flag curvature K is said to be almost isotropic flag curvature if K =3c + σ where σ and c are scalar functions on M.In this paper,we establish the intrinsic re...     

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.     

On a class of projectively flat Finsler metrics

In this paper, we classify locally projectively flat general $(\alpha ,\beta )$-metrics $F=\alpha \varphi (b^2,\frac{\beta }{\alpha })$ on an $n(\ge 3)$-dimensional manifold if α is of constant sectional curvature and ${\varphi }_1\ne 0$. Furthermore, we find equations to characterize this class of metrics with constant flag curvature and determine their local structures.     

A new characterization of Finsler metrics with constant flag curvature 1

The purpose of this article is to derive an integral inequality of Ricci curvature with respect to Reeb field in a Finsler space and give a new geometric characterization of Finsler metrics with constant flag curvature 1.     

On spherically symmetric Finsler metrics with vanishing Douglas curvature

We obtain the differential equation that characterizes the spherically symmetric Finsler metrics with vanishing Douglas curvature. By solving this equation, we obtain all the spherically symmetric Douglas metrics. Many explicit examples are included.     

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.     

On a class of projectively Ricci-flat Finsler metrics

Every Finsler metric induces a spray on a manifold. With a volume form on a manifold, every spray can be deformed to a projective spray. The Ricci curvature of a projective spray is called the projective Ricci curvature. The projective Ricci curvature is an important projective invariant in Finsler geometry. In this paper, we study and characterize projectively Ricci-flat square metrics. Moreover, we construct some nontrivial examples on such Finsler metrics.     

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.     

Finsler metrics of scalar flag curvature with special non-Riemannian curvature properties

In this paper, we study Finsler metrics of scalar flag curvature. We find that a non-Riemannian quantity is closely related to the flag curvature. We show that the flag curvature is weakly isotropic if and only if this non-Riemannian quantity takes a special form. This will lead to a better understanding on Finsler metrics of scalar flag curvature.      

Characterization and structure of Finsler spaces with constant flag curvature

The geometric characterization and structure of Finsler manifolds with constant flag curvature (CFC) are studied. It is proved that a Finsler space has constant flag curvature 1 (resp. 0) if and only if the Ricci curvature along the Hilbert form on the projective sphere bundle attains identically its maximum (resp. Ricci scalar). The horizontal distributionH of this bundle is integrable if and only ifM has zero flag curvature. When a Finsler space has CFC, Hilbert form’s orthogonal complement in the horizontal distribution is also integrable. Moreover, the minimality of its foliations is equivalent to given Finsler space being Riemannian, and its first normal space is vertical Project supported by Wang KC Fundation of Hong Kong and the National Natural Science Foundation of China (Grant No. 19571005).     

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.     

Finsler metrics with special Riemannian curvature properties

The Weyl curvature is one of the fundamental quantities in Finsler geometry because it is a projective invariant. By determining the Weyl curvature of a class of Finsler metrics, we find a lot of Finsler metrics of quadratic Weyl curvature which are non-trivial in the sense that they are not of quadratic Riemann curvature.