On a Class of Weakly-Berwald （α,β）-Metrics

In this paper, we study an important class of (α,β)-metrics in the form F = (α + β)m+1/αm on an n-dimensional manifold and get the conditions for such metrics to be weakly-Berwald metrics, 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. Furthermore, we also prove that this kind of (α,β)-metrics is of isotropic mean Berwald curvature if and only if it is of isotropic S-curvature. In this case, S-curvature vanishes and the metric is weakly-Berwald metric.

Minimal Surfaces and Gauss Curvature of Conoid in Fins-ler Spaces with (<i>α</i>, <i>β</i>)-Metrics

In this paper, minimal submanifolds in Finsler spaces with (α, β)-metrics are studied. Especially, helicoids are also minimal in (α, β)-Minkowski spaces. Then the minimal surfaces of conoid in Finsler spaces with (α, β)-metrics are given. Last, the Gauss curvature of the conoid in the 3-dimension Randers-Minkowski space is studied.

Projective Changes between Generalized (<i>α</i>, <i>β</i>)-Metric and Randers Metric

Projective change between two Finsler metrics arises from Information Geom-etry. Such metrics have special geometric properties and will play an important role in Finsler geometry. The purpose of the present paper is to find a relation to characterize the projective change between generalized (α, β) - metric ( μ1, μ2 and μ3 ≠ 0 are constants) and Randers metric , where α and are two Riemannian metrics, β and are 1-forms. Further, we study such projective change when generalized (α, β) -metric F has some curvature property.

对称的(α,β)度量的性质

对称的Finsler度量具有非常好的性质,有重要的研究价值.主要研究了对称的(α,β)度量的曲率性质,得到了对称的(α,β)度量的S-曲率,相对迷向平均Landsberg曲率之间的等价关系,并解决了沈忠民教授所提出的开放性问题的第4个问题中当F是(α,β)度量的情形:是否存在对称的(α,β)度量,当它是非Berwald度量时有S=0.

PROJECTIVELY FLAT MATSUMOTO METRIC AND ITS APPROXIMATION

Abstract In this article, the author studies the projectively flat Matsumoto metric F=α^2/（α -β）, where α=√αijy^iy^j is a Riemannian metric and β =biy^i is 1-form. Theyconclude that α is locally projectively fiat and β is paralled with respect to α. And get the same result for the higher order approximate Matsumoto metric.

关于一类特殊的(α，β)-度量的性质

研究一类β关于α是平行的并且Riemann度量α具有常曲率的(α,β)-度量F所具有的一些性质,证明了F要么是平坦平行度量,要么是与Riemann度量α共形相关的度量.

局部射影平坦且具有迷向S-曲率的两类重要的(α,β)-度量

研究了两类重要的分别形如F=α~2/（α-β）和F=α＋εβ＋kβ~2/α的（α,β）-度量,其中α=aij（x）yi yj为黎曼度量,β=bi（x）yi为流形上的1-形式,ε,k≠0为常数.得到了它们为局部射影平坦且具有迷向S-曲率的充要条件.

特殊的射影平坦(α,β)-度量(英文)

研究了近似指数度量并得到二阶近似指数度量射影平坦的充要条件是α射影平坦,β关于α平行.且对高阶指数度量也得到了相同的结果.这里,α=aijyiyj,β=biyi.

共形平坦的(α,β)-度量

本文主要研究共形平坦的(α,β)-度量.通过共形相关的Finsler度量间其测地系数间的关系,得到了(α,β)-度量是共形平坦的充分必要条件,并构造了若干共形平坦(α,β)-度量的例子.在此基础上,发现共形平坦且具有迷向S-曲率的(α,β)-度量一定是Minkowski度量或Riemann度量.

一类具有特殊曲率性质的(α,β)-度量

在一般的(α,β)-度量F=αφ(s)与Riemann度量α的Ricci曲率之间的关系基础上,证明了一类特殊的(α,β)-度量F=α2/α+β,在维数n≥3的流形上,如果F具有迷向的Ricci曲率,且β是闭的1-形式,则其Ricci曲率等于零.从而得到如果F=αα+2β具有常Ricci曲率,并且β是闭的1-形式,则其Ricci曲率等于零.

关于(α,β)-度量的S-曲率

给出(α,β)-度量F=αφ(β／α)的S-曲率的计算公式．证得对一般的(α,β)-度量,当β为关于α长度恒定的Nilling1-形式时,S=0.研究了Matsumoto-度量F=α2／(α-β)和(α,β)-度量F=α+∈β+k(β2／α)的S-曲率,证得S=0)当且仅当β为关于α长度恒定的Killing1-形式．同时还得到这两类度量成为弱Berwald度量的充要条件．其中φ(s)为光滑函数,为黎曼度量,β(y)=bi(x)yi为非零1-形式且∈,k≠0为常数．

Projectively flat exponential Finsler metric

In this paper, we study a class of Finsler metric in the form F=αexp(β/α)+εβ, where α is a Riemannian metric and β is a 1-form, ε is a constant. We call F exponential Finsler metric. We proved that exponential Finsler metric F is locally projectively flat if and only if α is projectively flat and β is parallel with respect to α. Moreover, we proved that the Douglas tensor of expo-nential Finsler metric F vanishes if and only if β is parallel with respect to α. And from this fact, we get that if exponential Finsler metric F is the Douglas metric, then F is not only a Berwald metric, but also a Landsberg metric.

Projectively flat arctangent Finsler metric

In this work, we study a class of special Finsler metrics F called arctangent Finsler metric, which is a special （α, β）-metric, where a is a Riemannian metric and β is a 1-form, We obtain a sufficient and necessary condition that F is locally projectively fiat if and only if α and β satisfy two special equations. Furthermore we give the non-trivial solutions for F to be locally projectively fiat. Moreover, we prove that such projectively fiat Finsler metrics with constant flag curvature must be locally Minkowskian.

一类共形平坦的(α,β)-度量的研究

研究了共形平坦的(α,β)-度量F=α(β/α),这里α是一个黎曼度量,β是流形上的1-形式。证明了共形平坦的弱Landsberg的(α,β)-度量一定是黎曼度量或者闵可夫斯基度量。进一步,如果(s)是关于s的多项式,那么共形平坦且具有相对迷向平均Landsberg曲率的(α,β)-度量也一定是黎曼的或者闵可夫斯基度量的。

ON A CLASS OF DOUGLAS FINSLER METRICS

In this article, we study a class of Finsler metrics called general （α, β）-metrics, which are defined by a Riemannian metric α and a 1-form β. We determine all of Douglas general （α, β）-metrics on a manifold of dimension n 〉 2.

三江平原湿地小叶章群落沿土壤水分梯度物种组成及多样性变化

水分是影响湿地生态系统植物多样性的重要因素之一。以三江平原湿地小叶章群落为对象,研究了沿土壤水分梯度的典型草甸、沼泽化草甸和沼泽3种生境中群落物种组成及多样性特征。结果表明,3种生境均以小叶章占优势,但是伴生物种组成分化明显,其中,典型草甸指示种为二歧银莲花和垂梗繁缕,沼泽化草甸指示种为灰脉苔草和五脉山黧豆,沼泽指示种为漂筏苔草和毛果苔草;非度量多维尺度分析结果表明,草甸的物种组成差异更大,而沼泽和沼泽化草甸的物种组成差异较小。随土壤水分含量增加,小叶章的优势度逐渐降低,群落物种丰富度和Simpson优势度均呈逐渐降低的趋势,而Shannon-Wiener多样性与Pielou均匀度均呈逐渐上升的趋势;群落相似性系数(CJ、CS)呈现逐渐升高的趋势,二者的最大值分别出现在沼泽化草甸和沼泽,最小值分别出现在典型草甸和和沼泽化草甸,而Cody指数的变化格局呈现渐低的趋势,群落间共有物种数逐渐减少,物种替代速率降低。研究认为,三江平原小叶章湿地植物群落物种组成、Pielou均匀度、Simpson优势度和Shannon-Wiener多样性的变化格局与水分密切相关,β多样性的格局与水分联系紧密,物种丰富度的变化格局与水分无相关性,可能与物种自身的生物学特性以及其它环境因子有关,尚需进一步探讨。

具有迷向S-曲率的拟对称(α,β)-度量

本文主要讨论了具有迷向S-曲率的拟对称(α,β)-度量的等价条件,并研究了具有迷向S-曲率的拟对称(α,β)度量的一些好的性质.

对偶平坦和共形平坦的(α,β)-度量(英文)

本文主要研究了对偶平坦和共形平坦的(α,β)-度量.利用对偶平坦和共形平坦与其测地线的关系,得到了局部对偶平坦和共形平坦的Randers度量是Minkowskian度量的结论.进一步,推广到非Randers型的情形,我们证明了局部对偶平坦和共形平坦的非Randers型的(α,β)-度量在附加的条件下一定是Minkowskian度量.

一类对偶平坦且具有迷向S-曲率的(α,β)-度量

主要研究了一类特殊的(α,β)-度量F=αφ(s),s=β/α,其中φ(s)是关于s的k(k≥2)次多项式,α是一个Riemann度量,β是一个1-形式.得到了如下结果:F是对偶平坦的度量且具有迷向S-曲率的充分必要条件是F是Minkowski度量.

具有迷向S-曲率的两类(α,β)-度量

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.同时给出了这两类(α,β)-度量为弱-Berwald度量的充要条件.此时,它们的S-曲率为0.