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.     

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.     

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.     

A BLOW-UP CRITERION OF SPHERICALLY SYMMETRIC STRONG SOLUTIONS TO 3D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE BOUNDARY

In this article, we consider the free boundary value problem of 3D isentropic compressible Navier-Stokes equations. A blow-up criterion in terms of the maximum norm of gradients of velocity is obtained for the spherically symmetric strong solution in terms of the regularity estimates near the symmetric center and the free boundary respectively.     

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.     

R^n空间上具有正曲率和负曲率的度量一些例子

From the class of conformally flat metric of gij=f(|x|^2)δij defined on R^n,where f:[0,∞]→（0,∞）is a C^2 function,we find one class of metrics with positive curvature and two classes of complete metrics with negative curvature.     

Finsler空间上的Weyl曲率

The Weyl curvature of a Finsler metric is investigated. This curvature constructed from Riemannain curvature. It is an important projective invariant of Finsler metrics. The author gives the necessary conditions on Weyl curvature for a Finsler metric to be Randers metric and presents examples of Randers metrics with non-scalar curvature. A global rigidity theorem for compact Finsler manifolds satisfying such conditions is proved. It is showed that for such a Finsler manifold,if Ricci scalar is negative,then Finsler metric is of Randers type.     

A priori Estimates and Existence for a Class of Fully Nonlinear Elliptic Equations in Conformal Geometry

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive.     

Dually flat general spherically symmetric Finsler metrics

Dually flat Finsler metrics arise from information geometry which has attracted some geometers and statisticians. In this paper, we study dually flat general spherically symmetric Finsler metrics which are defined by a Euclidean metric and two related 1-forms. We give the equivalent conditions for those metrics to be locally dually flat. By solving the equivalent equations, a group of new locally dually flat Finsler metrics is constructed.     

一类对偶平坦的球对称的芬斯勒度量

本文研究了对偶平坦的芬斯勒度量的构造问题.通过分析球对称的对偶平坦的芬斯勒度量的方程的解,我们构造了一类新的对偶平坦的芬斯勒度量,并得到了球对称的芬斯勒度量成为对偶平坦的充分必要条件.     

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.     

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 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.     

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.     

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 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.     

On dually flat square metrics

Square metrics arise from several classification problems in Finsler geometry. They are the rare Finsler metrics to be of excellent geometry properties. It is proved that every non-Riemannian dually flat square metric must be Minkowskian if the dimension ≥3. We also obtain a rigidity result in dually flat Matsumoto metrics.