1) Berwald curvature
Berwald曲率
1.
In this article, we mainly discuss the relation between isotropic Berwald curvatures of projectively related Finsler metrics and properties of Finsler metrics with isotropic S-curvature.
本文主要讨论了射影相关下迷向Berwald曲率间的关系和在具有迷向S-曲率的条件下Finsler度量的某些性质。
2.
Here E denotes the mean Berwald curvature of F and H is the geometric quantity which characterizes the rate of the change of E along geodesics,“|”and“.
本文首先研究了完备的Douglas空间(M,F),证明了如果其Cartan张量是有界的,且满足H=0和Ejk?lm=0,则F为Berwald度量,其中E为F的平均Berwald曲率,H为刻划E沿测地线的变化率的几何量,“|”和“。
3.
Here E is the mean Berwald curvature of F,and H is the geometric quantity which characterizes the rate of the change of E along geodesics.
l|m=0,则F为Berwald度量,其中E为F的平均Berwald曲率,H为刻划E沿测地线的变化率的几何量。
2) mean Berwald curvature
平均Berwald曲率
1.
This paper studies two important classes of(α,β)-metrics in the form F=(α+β)m+1/αm and F= α +εβ+2β2/α-β4/(3α3) on an n-dimensional manifold and proves that these two kinds of(α,β)-metrics are of isotropic mean Berwald curvature if and only if they are of isotropic S-curvature,where α=aij(x)yiyj is a Riemannian metric and β=bi(x)yi is a 1-form and m is a real number with m≠1,0,-1/n.
研究了n-维流形上两类重要的(α,β)-度量——F=(α+β)m+1/αm和F=α+εβ+2β2/α-β4/(3α3),证明了这两类(α,β)-度量具有迷向S-曲率,当且仅当它们具有平均Berwald曲率,其中α=aij(x)yiyj是黎曼度量,β=bi(x)yi是非零1-形式,m为满足m≠-1,0,-1/n的非零实数。
2.
This paper studies two kinds of important geometric quantities the mean Berwald curvature and the mean Landsberg curvature for Randers metrics, describes these important curvatures and gives a sufficient and necessary condition for Randers metric satisfying E=0 or J=0 respectively.
研究了Randers度量的两类重要的几何量———平均Berwald曲率和平均Landsberg曲率,描述了这两类重要的曲率,且分别给出了Randers度量满足E=0或J=0的充分必要条
3.
Meanwhile, we show that the mean Berwald curvature S=0 implies that the Ricci curvature Ric is quadratic in projectively flat Finsler spaces.
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。
3) isotropic Berwald curvature
迷向Berwald曲率
1.
In this paper, we discuss the relation of the isotropic Berwald curvature for pojectively related Finsler metrics F and F, the necessary and sufficient condition that F is of isotropic S-curvature is obtained from the above result.
讨论了射影相关Finsler度量F与F的迷向Berwald曲率间的关系 ,并利用这种关系得到了一个射影相关下F具有迷向S 曲率的充分必要条
4) ispotropic mean Berwald curvature
迷向平均Berwald曲率
5) generalized isotropic Berwald curvature
广义迷向Berwald曲率
1.
In this article, some properties of generalized isotropic Berwald curvature are studied.
研究广义迷向Berwald曲率的性质, 得到: F是广义迷向Berwald曲率c(x, y)的当且仅当Dlijk =- c·kF-1hijyl,Eij =n+12c(x, y)F-1hij; 如果Lijk + c(x, y)FCijk =0, Dlijk =- c·kF-1hijyl, 则Eij =n+12c(x, y)F-1hij。
2.
In this paper, we study some properties of Finsler metric with generalized isotropic Berwald curvature and a class of special (α,β )-metric.
本文研究了一类特殊的(α,β)-度量以及具有广义迷向Berwald曲率的Finsler度量的性质。
6) weakly-Berwald metric
弱-Berwald度量
1.
Furthermore,this paper presents the conditions for them to be weakly-Berwald metrics.
同时给出了这两类(α,β)-度量为弱-Berwald度量的充要条件。
补充资料：Gauss曲率
Gauss曲率
Gausaan curvature
是曲面的第二基本形式(别x幻nd仙劝雀比正”tal form)，则Gau邓曲率能用公式 乙N一MZ K=共共一二鉴广 EG一F名来计算.Cau骆曲率恒等于球面映射(sPh汀i。习n.p)的J出刀bi行列式: S {K{尸。一J淤。于，这里P0是曲面上一点，s是包含P0的区域U的面积，S是U的球面象的面积，d是区域的直径.〔抽以弥曲率在椭画点(elliPtic Point)处是正的，在双曲点(hyPer加lic point)处是负的，在抛物点(para加licpoint)或平坦点(血t point)处为零，它可仅用第一基本形式的系数及其导数来表示(C明‘定理(CaJ骆th印rer。))，即 !EE云l {11}己F_一G K二，鑫夕}。。刀}十二节二‘飞二电-二石;一J‘+ 八一百丽矿}户’户。户。{’Zw!日。W }G民仅1 占F一E_〕 +—~-之址-一-一一二). 日v WJ’这里 WZ二EG一F2. 因为Ga璐曲率仅依赖于度量，即仅依赖于第一基本形式的系数，所以Gauss曲率在等距形变(士自m曰t幻n，ison犯山c)下是不变的.Ga口弱曲率在曲面论中起了特殊的作用，有许多关于它的计算公式(【21). 此概念由C.F.CaJ粥({11)引人，因而得名，【补注]全〔治毯骆曲率(to回Gauss枷curvat侧旧)(常简记为全曲率(to回cur呢lture))是指量 丁丁Kdo.(亦见Ga旧一D刀留峨定理(Ga理洛~B幻nnet小印n万n).) 对由x=x(s)所给出的光滑空间曲线C，C的总曲率K定义为C的球面象的长度(亦见球面标形(sPheri以1 indi口trix))，且能用沿C的关于Fr加以标架(见E滋.时三棱形(Fr乙nettri比过ron))(x，e.，e2，e3)的F滋.时公式(Fr‘netfomllllas)e，=‘，eZ，e;=一‘、e、+凡2e3，e3=一‘Ze:表示为 K一丁、lds.沈纯理译Ca.沼曲率【C.旧幽mo口，.to比;raycco皿Ic钾皿3.a〕，曲面的 正则曲面在一给定点的主曲率(prilldPal。印口.tl此)的乘积，若 I=dsZ=EduZ+2 Fdudy+GdvZ是曲面的第一基本形式(际tft田d旧lrntal forTn)及 11=侧“2+ZMdudy+Nd砂2
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