Properties of Berwald scalar curvature

We prove that a Finsler manifold with vanishing Berwald scalar curvature has zero E-curvature. As a consequence, Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds. For (α,β)-metrics on manifold of dimension greater than 2, if the mean Landsberg curvature and the Berwald scalar curvature both vanish, then the Berwald curvature also vanishes.     

Minkowski空间的等价性定理及在Finsler几何的应用

首先利用中心仿射几何中的结果建立了Minkowski空间的等价性定理.作为在Finsler几何中的应用,我们证明满足一定条件的Landsberg空间为Berwald空间,这些条件可以是具有闭的Cartan型形式,S曲率为零或平均Berwald曲率为零.     

S-curvature of Doubly Warped Product of Finsler Manifolds

Doubly warped product of Finsler manifolds is useful in theoretical physics, particularly in general relativity. In this paper, we study doubly warped product of Finsler manifolds with isotropic mean Berwald curvature or weak isotropic S-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。     

The main invariants of a complex finsler space

In this paper we extend the results obtained in [3], where are investigated the general settings of the two-dimensional complex Finsler manifolds, with respect to a local complex Berwald frame. The geometry of such manifolds is controlled by three real invariants which live on T'M: two horizontal curvature invariants K and W and one vertical curvature invariant I. By means of these invariants are defined both the horizontal and the vertical holomorphic sectional curvatures. The complex Landsberg and Berwald spaces are of particular interest. Complex Berwald spaces coincide with Kähler spaces, in the two – dimensional case. We establish the necessary and sufficient condition under which K is a constant and we obtain a characterization for the Kähler purely Hermitian spaces by the fact K = W = constant and I = 0. For the class of complex Berwald spaces we have K = W = 0. Finally, a classification of two-dimensional complex Finsler spaces for which the horizontal curvature satisfies a special property is obtained.     

一类具有标量旗曲率的Berwald(α,β)-度量(英文)

本文研究了一类具有F=α+εβ+kα²/β形式的Finsler度量,其中α=（aijyⁱy^j）^1/2是Riemann度量,β=b_iyⁱ是非零1-形式,ε和k≠0是常数。得到了这个Finsler度量的S曲率消失和成为弱Berwald度量的充要条件。另外通过证明发现具有标量期曲率的Finsler度量成为弱Berwald度量的充要条件是它们成为Berwald度量,并且期曲率消失。在这种情况下,该Finsler度量就是局部Minkowski度量。     

A classification of unitary invariant weakly complex Berwald metrics of constant holomorphic curvature

In this paper, we give a characterization of strongly pseudoconvex complex Finsler metric F which is unitary invariant. A necessary and sufficient condition for F to be a weakly complex Berwald metric and a necessary and sufficient condition for F to be a weakly Kähler Finsler metric are given, respectively. We also give a classification of unitary invariant weakly complex Berwald metrics which are of constant holomorphic curvatures.     

芬斯勒射影几何中的Ricci曲率

本文研究了保持Ricci曲率不变的Finsler射影变换。给定一个紧致无边的n维可微流形M，证明了：对于一个从M上的Berwald度量到Riemann度量的C-射影变换，如果Berwald度量的Ricci曲率关于Riemann度量的迹不超过Riemann度量的标量曲率，则该射影变换是平凡的。     

A note on the fundamental group of Finsler manifolds with nonnegative flag curvature

In this note we prove that the fundamental group of any forward complete Finsler manifold with nonnegative flag curvature is finitely generated provided the line integral of T-curvature is small. In particular, the fundamental group of any forward complete Berwald manifold with nonnegative flag curvature is finitely generated.     

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.     

On Unitary Invariant Weakly Complex Berwald Metrics with Vanishing Holomorphic Curvature and Closed Geodesics

In this paper, the authors construct a class of unitary invariant strongly pseudoconvex complex Finsler metrics which are of the form F =√[ rf(s- t)[, where r = ||v||~ 2, s =| z,v |~2/r, t =|| z||~ 2, f(w) is a real-valued smooth positive function of w ∈ R,and z is in a unitary invariant domain M  C~n. Complex Finsler metrics of this form are unitary invariant. We prove that F is a class of weakly complex Berwald metrics whose holomorphic curvature and Ricci scalar curvature vanish identically and are independent of the choice of the function f. Under initial value conditions on f and its derivative f, we prove that all the real geodesics of F =√[rf(s- t)] on every Euclidean sphere S~(2n-1) M are great circles.     

On projective invariants of the complex Finsler spaces

In this paper we extend the results on projective changes of complex Finsler metrics obtained in Aldea and Munteanu (2012) [3], by the study of projective curvature invariants of a complex Finsler space. By means of these invariants, the notion of complex Douglas space is then defined. A special approach is devoted to the obtaining of equivalence conditions for a complex Finsler space to be a Douglas one. It is shown that any weakly Kähler Douglas space is a complex Berwald space. A projective curvature invariant of Weyl type characterizes complex Berwald spaces. These must be either purely Hermitian of constant holomorphic curvature, or non-purely Hermitian of vanishing holomorphic curvature. Locally projectively flat complex Finsler metrics are also studied.     

On Generalized Douglas–Weyl(α, β)-Metrics

In this paper, we study generalized Douglas–Weyl(α, β)-metrics. Suppose that a regular(α, β)-metric F is not of Randers type. We prove that F is a generalized Douglas–Weyl metric with vanishing S-curvature if and only if it is a Berwald metric. Moreover, by ignoring the regularity, if F is not a Berwald metric, then we find a family of almost regular Finsler metrics which is not Douglas nor Weyl. As its application, we show that generalized Douglas–Weyl square metric or Matsumoto metric with isotropic mean Berwald curvature are Berwald metrics.     

具有（α，β）度量的复Finsler流形

设M是复流形,具有复（α,β）度量F=αφ（｜β｜/α）,其中α为M上的Hermite度量,β为M上的（1,0）形式。本文得到与F相联系的复非线性联络系数Гiμ^i的表达式,且证明了：若β为M上的全纯（1,0）形式,并且关于α的Hermite联络γij^k（z）平行,则F是M上的复Berwald度量;若α是M上的Kaihler度量,则F是M上的强Kahler Finsler度量．     

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.     

Finsler Manifolds with Positive Constant Flag Curvature

It is shown that a Finsler metric with positive constant flag curvature and vanishing mean tangent curvature must be Riemannian. As applications, we also discuss the case of Cheng's maximal diameter theorem and Green's maximal conjugate radius theorem in Finsler manifolds.     

S-curvature of isotropic Berwald metrics

Isotropic Berwald metrics are as a generalization of Berwald metrics. Shen proved that every Berwald metric is of vanishing S-curvature. In this paper, we generalize this fact and prove that every isotropic Berwald metric is of isotropic S-curvature. Let F = α + β be a Randers metric of isotropic Berwald curvature. Then it corresponds to a conformal vector field through navigation representation.