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.     

Some projectively flat (α,β)-metrics

In this paper, we study a class of Finsler metrics in the form , where is a Riemannian metric, 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.     

一类对偶平坦的球对称的芬斯勒度量

In this paper,we study spherically symmetric Finsler metrics.By analysing the solution of the spherically symmetric dually flat equation,we construct several new families of dually flat spherically symmetric Finsler metrics.     

PROJECTIVELY FLAT FINSLER METRICS WITH ALMOST ISOTROPIC S-CURVATURE

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

On Conformally Flat(α,β)-metrics with Special Curvature Properties

In this paper, we study a significant non-Riemannian quantity Ξ-curvature, which is defined by S-curvature. Firstly, we obtain the formula of Ξ-curvature for(α, β)-metrics. Based on it, we show that the Ξ-curvature vanishes for a class of(α, β)-metrics. In the end, we get the relation ofΞ-curvature for conformally related Finsler metrics, and classify conformally flat(α, β)-metrics with almost vanishing Ξ-curvature.     

On Einstein Matsumoto metrics

We study a special class of Finsler metrics,namely,Matsumoto metrics F=α2α-β,whereαis a Riemannian metric andβis a 1-form on a manifold M.We prove that F is a(weak)Einstein metric if and only ifαis Ricci flat andβis a parallel 1-form with respect toα.In this case,F is Ricci flat and Berwaldian.As an application,we determine the local structure and prove the 3-dimensional rigidity theorem for a(weak)Einstein Matsumoto metric.     

ON DOUBLY WARPED PRODUCT OF COMPLEX FINSLER MANIFOLDS

Let(M_1,F_1) and(M_2,F_2) be two strongly pseudoconvex complex Finsler manifolds.The doubly wraped product complex Finsler manifold(f_2M_1×f_1 M_2,F) of(M_1,F_1)and(M_2,F_2) is the product manifold M_1×M_2 endowed with the warped product complex Finsler metric F~2 =f_2~2 F_1~2 + f_1~2F_2~2,where f_1 and f_2 are positive smooth functions on M_1 and M2,respectively.In this paper,the most often used complex Finsler connections,holomorphic curvature,Ricci scalar curvature,and real geodesics of the DWP-complex Finsler manifold are derived in terms of the corresponding objects of its components.Necessary and sufficient conditions for the DWP-complex Finsler manifold to be Kahler Finsler(resp.,weakly Kahler Finsler,complex Berwald,weakly complex Berwald,complex Landsberg) manifold are obtained,respectively.It is proved that if(M_1,F_1) and(M_2,F_2) are projectively flat,then the DWP-complex Finsler manifold is projectively flat if and only if f_1 and f_2 are positive constants.     

On Douglas General(α,β)-metrics

Douglas metrics are metrics with vanishing Douglas curvature which is an important projective invariant in Finsler geometry. To find more Douglas metrics, in this paper we consider a class of Finsler metrics called general(α,β)-metrics, which are defined by a Riemannian metricα=(a_(ij)(x)y~iy~j)~(1/2) and a 1-form β= b_i(x)y~i. We obtain the differential equations that characterizes these metrics with vanishing Douglas curvature. By solving the equivalent PDEs, the metrics in this class are totally determined. Then many new Douglas metrics are constructed.     

Complete Kaehler Submanifolds in CP~n

In this paper we mainly investigate projectively flat complete Kaehler sub-manifolds, in CPn. We give the pinching constants and the local structure.     

本刊英文版Vol.36(2020),No.6论文摘要（英文）

正Moduli of Polarized Calabi-Yau Pairs Janos KOLLAR Chen Yang XU Abstract We prove that the irreducible components of the moduli space of polarized CalabiYau pairs are projective.Projectively Flat Singular Square Metrics with Constant Flag Curvature Guang Zu CHEN Xin Yue CHENG Abstract In this paper,we study and characterize locally projectively flat singular square     

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 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.     

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.     

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.     

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.     

On some projectively flat finsler metrics in terms of hypergeometric functions

This paper gives an explicit construction of a family of projectively flat Finsler metrics by using hypergeometric functions and a special class of projectively flat Randers metrics.     

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 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.     

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.