关于Finsler几何中的一类临界度量及Randers度量
摘要
Finsler度量的历史可以追溯到1854年黎曼的就职演说，然而黎曼很快将注意力集中于具有二次型表示的度量——黎曼度量．第一位系统探讨更一般的度量空间的是P．Finsler，在他的博士论文中([32])，成功地建立起了一般度量空间上的曲线、曲面理论．自此，“Finlser几何”的名称被广泛接受．之后，随着联络和曲率理论的发展，Finlser几何日臻成熟．另一方面，复Finsler度量概念的正确引述可能归功于G．Rizza([52])．作为复Finsler度量的重要例子，复流形上的内蕴度量在复几何研究中扮演着重要角色．
20世纪90年代以来，在陈省身先生的倡导下，Finsler几何获得了长足的发展．许多新的几何量被发现，涌现了众多优秀的工作([8]，[12]，[58]，[17]，[60]，[50]，[24]等等)．伴随着基础理论研究的发展，Finsler几何也被广泛应用于诸如物理学、生物学、信息与控制论和心理学等学科当中．
本文主要探讨Finsler度量的某些曲率性质．主要内容分为三部分，分别讨论Einstein-Hilbert泛函、实Randers度量和复Randers度量．
Einstein-Hilbert泛函
Einstein度量是黎曼几何中具有重要意义的一类度量([20])．就物理或变分角度而言，Einstein度量是Einstein-Hilbert泛函的临界点．设(M，g)为n-维黎曼流形，其数量曲率记为R_g，则Einstein-Hilbert泛函可以表示为其中dμ_M是(M，g)的体积元．
沿着这一思路，我们尝试在Finsler几何中作相应的推广．早在1995年，法国几何学家H.Akbar-Zadeh就从变分角度做了尝试([5])．然而不幸的是，由于他没有考虑到变分张量的可积性(参见[10])，因而得到的张量特征是有问题的．之后的十余年，未见有任何进展．在陈省身先生提出Finsler几何中Einstein度量存在性问题，以及David Bao建议了相应的Ricci流之后，研究Einstein-Hilbert泛函的临界点就显得尤为重要了．
在Finsler流形(M~n，F)上，通过Chern联络或者Berwald联络建立起的曲率理论，导出了包括旗曲率、Ricci曲率和Landsberg曲率在内的许多重要几何量[14]．其中旗曲率是截面曲率的推广，Ricci曲率是旗曲率的某种平均，而Landsberg曲率则是重要的非黎曼几何量．鉴于几乎所有的几何量都依赖于方向的选取，切丛TM和射影球丛SM取代了黎曼几何中底流形M的角色．利用SM上的Sasaki度量，我们将Einstein-Hilbert泛函推广为其中Ric是Finsler度量F的Ricci曲率，dμ_(SM)是SM上Sasaki度量的体积元、Vol(SM)是SM的体积．根据积分求迹公式，可以验证对于黎曼度量而言该泛函就是原来的Einstein-Hilbert泛函．
推导泛函(0．1．2)的Euler-Lagrange方程并研究临界度量的性质是第二章的主要工作．在临界方程推导中，除了计算的复杂性之外，关键在于散度公式以及一个Green-型积分公式．前者最早由Akbar-Zadeh在局部坐标下给出，后来莫小欢、沈一兵-张彦利用活动标架法简化了证明：后者的一种特殊情形最先被贺群-沈一兵发现，而我们则证明了一般情形：
引理0．1．1．设ψ和ω是SM上两个光滑函数，则其中(g~(ij))是基本张量(gij)：=(1/2[F~2]y~iy~j)的逆．
利用上述公式，可以避免文献[5]中变分张量的可积性问题，从而得到正确的临界点方程．
定理0．1．1．Einstein-Hilberg泛函(0．1．2)的Euler-Lagrange方程为其中r=1/Vol(SM)∫_(SM)Ric dμ_(SM)是Ric在SM上的平均值，J=J_idx~i是平均Lands-berg张量，“|”和“；”分别表示Chern联络的水平和竖直共变导数，而“·”则表示沿Hilbert方向求导．
定义0．1．1．满足方程(0．1．4)的Finsler度量称为ε-临界度量．
由定理0．1．1可以发现，Finsler几何中的临界度量不仅与Ricci曲率有关，也与非黎曼量Landsberg曲率有关，从而Einstein条件(Ricci曲率是常数)不足以使(0．1．4)成立，另一方面，Riemann-Einstein度量是临界的，于是黎曼几何中的Einstein度量是ε-临界度量．那么是否存在非黎曼的ε-临界度量呢?下面给出两个例子．
例0．1．1设α是Ricci平坦的黎曼度量，β是平行1-形式．则Randers度量F=α+β是ε-临界度量．
例0．1．2设(M，g)和(N，h)是两个Ricci平坦的黎曼流形，则在乘积空间M×N上，度量是ε-临界的．其中函数这里ε是非负实数，k是正整数．(参见[66，60])
利用某些Ricci恒等式以及球丛SM上的分析，我们得到下述刚性定理．
定理0．1．2．设M是紧致无边的光滑流形，F是M上具有常正旗曲率和常S-曲率的Finsler度量．则F是ε-临界度量的充要条件为F是常正曲率的黎曼度量．
这里所谓的S-曲率是沈忠民在研究体积比较定理时发现的，它是体积畸变沿着测地线的变化率．S-曲率在现代Finsler几何研究中有着重要意义．例如．如果F是可逆的，则[38]证明了常正旗曲率和零S-曲率已经蕴涵着F是黎曼度量．
由定理0．1．2可知，即使具有极好的曲率性质，也未必是ε-临界度量．因此．Einstein-Finsler度量与ε-临界度量的关系就是一个重要的问题．这里给出两者在曲面上的一个联系．
命题0．1．1．紧致无边曲面上，旗曲率非正的Einstein-Finsler度量是ε-临界的．
命题0．1．2．设F是曲面上的ε-临界度量．若其Landsberg曲率是水平常数，则F是Einstein-Landsberg度量．进一步，若F的旗曲率非零，则其必为黎曼度量．Randers度量
1941年，G．Randers在研究相对论时，为了讨论时空中类时方向的不对称性而引进了Randers度量([51])．它具有简洁的形式其中a_(ij)(x)dx~i(?)dx~j是黎曼度量，b_i(x)dx~i是1-形式．正是基于这种简单表达，Randers度量成为了Finsler几何里最令人感兴趣的一类度量．近年来，通过一系列的工作，如[17]、[27]、[16]、[35]以及[63]等等，人们对Randers度量有了更深入的理解．
第三章首先讨论了ε-临界的Randers度量．
利用Zermelo导航技术，文献[17]分类了常旗曲率的Randers度量．特别地，常旗曲率的Randers度量必然具有常S-曲率．从而，作为定理0．1．2的直接推论，有
命题0．1．3．紧致无边流形上具有常正旗曲率的ε-临界Randers度量必为黎曼度量，从而是常正曲率黎曼度量．
该命题可以视为球面上标准度量的某种刚性：在向量场的挠动下度量不再是临界的．
另一方面，例0．1．1给出了非平凡的临界点，它们属于Berwald-Randers度量．顺便指出，对于Randers度量而言，弱Landsberg、Landsberg和Berwald是等价的．那么，究竟怎样的Berwald-Randers度量是临界的呢?利用Randers度量的黎曼曲率公式([16])，我们得到
命题0．1．4．Berwald-Randers度量F=α+β是ε-临界度量，当且仅当它为下述情形之一：
(1)α是Einstein度量，β=0；
(2)α是Ricci平坦的黎曼度量．
其中，第一类是Riemann-Einstein度量，第二类正是例0．1．1所述之度量．注意到Calabi-Yau流形是Ricci平坦的，因此第二类有着丰富的例子．从另外一个角度来看，该命题给出了Calabi-Yau流形在Finsler几何中的一个重要意义：它们在特定挠动下仍是临界度量．
第三章还讨论了具有截面旗曲率的Randers度量．
如前所述，旗曲率是Finsler几何中最重要的几何量．它出现于弧长第二变分，是了解测地线结构的关键．点x处的“旗”包括“旗杆”y∈T_xM和包含y的“旗面(截面)”Π．一般地，若Π由{y，V}张成，则(y，Π)的旗曲率记为K(x，y，V)．因为这不依赖于V在截面Π中的选取，所以旗曲率也记为K(x，y，Π)．
Finsler度量称为具有数量旗曲率，如果其旗曲率不依赖于截面而只依赖于旗杆的选取，即K(x，y，Π)=K(x，y)．具有数量旗曲率的Finsler度量已经得到了较好的理解，特别是Randers度量．文献[48]证明了具有负数量旗曲率的Finsler度量必为Randers度量(dim≥3)．随后，文献[63]刻画了此类Randers度量的特征方程．利用Zermelo导航术，[28]完全解决了具有数量旗曲率和迷向S曲率的Randers度量分类问题(dim>3)．
那么一个自然的问题是，如果旗曲率只依赖于截面而与旗杆选取无关会怎样?即K(x，y，Π)=K(x，Π)，旗曲率可以约化为二平面Grassmann丛G_2(M)上的标量函数．我们称此种度量为具有截面旗曲率．显然，黎曼度量具有截面旗曲率，因此这是一个非黎曼条件．此外，常旗曲率度量显然也满足该条件．那么自然要问：
除了黎曼度量和常旗曲率度量之外，是否有非平凡的例子?
我们研究了具有截面旗曲率的Randers度量．对于曲面情形，具有截面旗曲率等价于具有迷向旗曲率，即Einstein曲面．而Einstein-Randers曲面已经完全分类([16])，因此我们仅关注高维情形并得到如下刚性定理．
定理0．1．3．维数大于2的光滑流形上，非黎曼的Randers度量具有截面旗曲率的充要条件是它具有常旗曲率．
因此，就Randers度量而言，该问题就此解决．对于一般度量，最近[30]证明了一些负曲率刚性和弱Landsberg刚性结果．
复Randers度量
与实Randers度量的成熟研究相比，复Randers度量的研究现在只处于起步阶段．复Randers度量是形如的复Finsler度量，其中a_(ij)(z)dz~i(?)dz~j是Hermite度量,b_i(z)dz~i是(1，0)-形式．最早的基础工作应属文献[7]．本文第四章主要讨论复Randers度量的两个重要性质：联络的线性性和全纯曲率的刚性．因为黎曼面上的复Finsler度量必为Hermite度量，故本文所讨论之流形的复维数均大于一．
实Finsler几何中，如果一个度量的Berwald联络是线性的，则称该度量为Berwald度量．1981年，利用Berwald度量与Riemann度量具有相同和乐群，Z．Szabó对Berwald度量做了一个漂亮的分类定理([66])．
1996年，T．Aikou提出了复Finsler几何中Berwald度量的定义：复Finsler度量称为Berwald度量，如果它具有线性Berwald联络，并且是Kahler-Finsler度量([2])．这里所谓的Kahler度量的定义来自文献[8]．那里将Kahler性(联络的对称性)分为三个层次，即强Kahler、Kahler和弱Kahler．在本文第一章中，强Kahler和Kahler被证明是等价的，
定理0．1．4．Kahler-Finsler度量必为强Kahler度量．
因此，在复Finsler几何中仅有两种Kahler条件，即Kahler和弱Kahler．
如所知，实Randers度量是Berwald度量的充要条件是β关于α平行([45]).对于复Randers度量而言，我们得到下述定理．
定理0．1．5．复Randers度量F=α+|β|是Berwald度量，当且仅当α是Kahler度量，且β(?)β关于α平行．此时，F与α具有相同的和乐群．
例0．1．3设M为一Kahler流形，T为复一维的平环．在T上取平行向量场β，令α为T×M的直积度量．则F=α+|β|是T×M上的Berwald度量．
除和乐群之外，全纯曲率K_F(z，v)是极为重要的几何量．全纯曲率迷向是指K_F(z，v)=K_F(z)．Kahler几何中，迷向全纯曲率具有刚性．那么对于Kahler-Finsler度量，迷向全纯曲率是否仍然具有刚性?
本文讨论了一类特殊的Kahler-Finsler度量——复Berwald-Randers度量．其难点在于计算复Randers度量的全纯曲率K_F(z，v)．经过复杂的计算，在得到了全纯曲率公式之后，我们发现复Berwald-Randers度量也成立类似的刚性定理．
定理0．1．6．具有迷向全纯曲率的复Berwald-Randers度量必为Kahler-Hermite度量或局部Mincowski度量．整体地，有
定理0．1．7．设M为一单连通复流形，F=α+|β|是M上完备的复Berwald-Randers度量．若F具有迷向全纯曲率，则其必相似于下述度量之一：(1) M=CP~n,F是Fubini-Study度量；(2)M={z∈C~n：|z|<1}，F是Bergman度量;(3)M=C~n，α是欧氏度量且β(z，v)=β(0，v)e~(iθ)(z)，其中θ(z)是M上的实函数．
另一方面，根据曲率表达式，讨论由α为Kahler度量、β为全纯1-形式构成的复Randers度量很有意义．利用简单的消末定理，可得如下刚性定理．
命题0．1．5．设M~n是紧致无边复流形，F是由α为Kahler度量、β为全纯1-形式构成的复Randers度量．如果k_F(z，v)>0，则F是Kahler-Hermite度量．
最后，我们给出几个具有不同曲率性质的复Randers度量的例子．
例0．1．4 C~(2n)上，令α~2=δ_(AB)dz~A(?)dz~B，β=∑_(i=1)~nz~(i+n)dz~i-z~idz~(i+n)．则F(z，v)=α+|β|的全纯曲率为零．
例0．1．5令α为单位球△_1(?)C~n上的Bergman度量β是模长小于一的全纯形式，如β=∑z~idz~i．则F(z，v)=α+|β|的全纯曲率K_F≤-1/4．
例0．1．6在CP~n中取充分小的球△_ε(?)C~n，令α为Fubini-Study度量β=b_idz~i，其中b_i是常数．则在△+ε上有K_F>0．
The history of Finsler geometry can be traced back to the lecture given by B. Riemann in 1854, though he turned immediately to the quadratic case. P. Finsler was the first geometer who studied the general metrics systematically. In his thesis in 1918, Finsler developed a theory of curves and surfaces for such metrics. Therefrom, the name "Finlser space" was generally accepted. On the other hand, the correct notion of complex Finsler metrics was probably proposed firstly by G.B.Rizza ([52]). As important examples of complex Finsler metrics, the intrinsic metrics on complex manifolds play central roles in the studies of complex geometry.
From 90's of the 20th century, encouraged by S.-S. Chern([25]), many geometersentered the field of Finsler geometry. In the last decade, the study of Finsler geometry has taken on a new look ([12], [58], [17], [60], [50], [24] and etc.). On the other hand, as a powerful tool, Finsler metrics are also widely applied to physics, biology, control theory, psychology and etc.([13])
In this thesis, we will consider some properties of curvatures for Finsler metrics, both real and complex. The main text consists of three parts, which study Einstein-Hilbert functional, real Randers metrics and complex Randers metrics respectively.
Einstein-Hilbert functional
Let M be an n-dimensional compact manifold. As is well known, among Riemannian metrics on M there is an important cla.ss of metrics called Einstein metrics, which are the critical points of the normalized Einstein-Hilbert functional where R_g is the scalar curvature of the Riemannian metric g, and dμ_M is the volume element of g. This motivates us to consider the corresponding functional in Finsler geometry. An attempt in this direction was tried by H.Akbar-Zadeh ([5]). Unfortunately, it seems that one could not obtain the tensor characteristic on generalized Einstein metrics from the variation calculus in [5] (also cf. D.Bao's comment [10]). It encourages us to look for the Finslerian analogue of critical metrics from the point of view of differential geometry and variational calculus.
By virtue of the Chern connection on a Finsler manifold (M, F) with the Finsler metric F, we can define the flag curvature and the Ricci scalar, which are generalizations of the sectional curvature and the Ricci curvature in Riemanniangeometry, respectively ([14]). It is natural to define a similar functional in Finsler geometry by using the Ricci scalar and the volume form induced from the projective sphere bundle over (M,F). In fact, this functional can be defined bywhere Ric denotes the Ricci scalar and SM is the projective sphere bundle over M with volume element dμ_(SM).One can check easily (0.2.2) is just the previous (0.2.1) if F is Riemannian by means of the integral trace formula or Lemma1.4 in [34].
The purpose of the second chapter is to derive the Euler-Lagrange equation of the functional (0.2.2) and to study the properties of the critical metrics.
The main difficulties in deriving the critical equation are a divergence formulaand a Green type formula. The former one was firstly obtained by H.AkbarZadeh,and was reproved by Xiaohuan Mo and Qun He-Yibing Shen. A special case of the Green type formula was firstly proved by Qun He-Yibing Shen, and we discover the following generic form
Lemma 0.2.1. Letψandφbe two smooth functions on SM. Then it holds where (g~(ij)) is the inverse of the fundamental tensor (g_(ij)) := (1/2[F~2]_(y~iy~j).
With the help of this lemma, we get the correct equation.
Theorem 0.2.1. The Euler-Lagrange equation of the functional (0.2.2) iswherer=1/Vol(SM)∫_(SM)Ric dμ_(SM) is the average of Ric on SM, J = J_idx~i is the meanLandsberg tensor, "|" and ";" denote respectively the horizontal and the vertical covariant derivatives with respect to the Chern connection, and "·" denotes the covariant derivative along the Hilbert form.
Definition 0.2.1. A Finsler metric satisfying the equation (0.2.4) is called anε-critical metric.
It is easy to show that a Riemannian metric is anε-critical metric iff it is Einstein. On the other hand, we have the following non-Riemannian examples.
Example 0.2.1 Letαbe a Ricci-flat Riemannian metric andβbe parallel with respect toα, then the Randers metric F =α+βis anε-critical metric.
Example 0.2.2 Let (M,g) and (N,h) be two Ricci-flat Riemannian manifolds, then the metricisε-critical on the product manifold M×N, where the functionφ(s, t) can be defined asHereεis a nonnegative real number and k is a positive integer.
Applying some Ricci identities and the Hopf maximum principle, we obtain
Theorem 0.2.2. Let M be a closed manifold, and F be a Finsler metric on M with positive constant flag curvature and constant S-curvature. Then F is anε-critical metric if and only if it is a Riemannian metric with positive constant sectional curvature.
The S-curvature was discovered by Z.Shen in his study of volume comparison [58]. There are many rigidity results on the S-curvature. For instance, if F is reversible, then positive constant flag curvature and constant S-curvature will imply that F is Riemannian ([38]).
From Theorem 0.2.2, one can see that even Finsler space forms might not beε-critical. So it is interesting to study the relation betweenε-critical metrics and Einstein-Finsler metrics. In dimension two, we have
Proposition 0.2.1. On a closed surface, any nonpositively curved EinsteinFinslermetric isε-critical.
Proposition 0.2.2. Let F be anε-critical metric on a surface. If its Landsberg scalar is horizontal constant, then F is an Einstein-Landsberg metric. Moreover, if the Ricci curvature is nonzero, then F is Riemannian.
Real Randers metrics
Randers metrics were introduced by G.Randers in the context of general relativity,~([51]) and named by R.Ingarden in his paper on the electron microscope.~([36])
Being the sum of a Riemannian metricα=(?) and a 1-formβ= b_i(x)y~i,Randers metrics are the favorite metrics for Finsler geometers, and are well understood by some great works such as [27], [17], [16], [64], [63] and etc.
In Chapter 3. we first study theε-critical Randers metrics. By Zermelo navigation, [17] determined the Randers space forms. Particularly, any Randers metric with constant flag curvature must have constant S-curvature. So as a direct corollary of Theorem 0.2.2, we have
Proposition 0.2.3. Anyε-critical Randers metric with positive constant flag curvature on a closed manifold must be a Riemannian metric with positive constantsectional curvature.
In other words, the standard Riemannian metric on the sphere is isolated in the family of critical Randers metrics. On the other hand, Example 0.2.1 is nontrivial which belongs to the class of Berwald-Randers metrics. So it is interestingto understand what kind of Berwald-Randers metrics are indeed critical.
The following proposition will answer this question.
Proposition 0.2.4. A Berwald-Randers metric F =α+βisε-critical, if and only if it is one of the followings:(1)αis Einstein, andβ= 0;(2)αis Ricci-flat.
Notice that the second class contains the Calabi-Yau manifolds, so it is ample. From another point of view, it gives a new significance of Calabi-Yau manifolds in Finsler geometry which says that Calabi-Yau manifolds will remain critical under certain perturbations.
Besides the critical Randers metrics, Chapter 3 also considers Randers metricswith sectional flag curvature. The definition of sectional flag curvature is suggested in [62].
A flag planted at a base point x on the manifold M, consists of a flagpole y∈T_xM and a sectionΠcontaining y. Typically, forΠspanned by {y, V}, the flag curvature of (y,Π) is K(x, y, V). Since it is independent of the choice of V inΠ, we also write it as K(x, y,Π).
A Finsler metric is said to be of scalar flag curvature if the flag curvatureis independent of the section. In [48], it is verified that negatively curved Finsler metrics of scalar flag curvature on closed manifolds must be of Randers type(dim≥3). Later on, Randers metrics with scalar flag curvature were characterizedby [63]. Recently, [28] classified the Randers metrics with scalar curvature and constant S-curvature by Zermelo navigation(dim≥3).
Now, what happens if the flag curvature is independent of the flagpole? Such metrics are called of sectional flag curvature since the flag curvature dependsonly on the section. Under this condition, the flag curvature can be written as K(x,Π) just as in Riemannian geometry. Clearly, Riemannian metrics are of sectional flag curvature, so this condition is non-Riemannian. It is natural to ask:
Are there any nontrivial Finsler metrics with sectional flag curvature?
As a start, we first study the Randers metrics as usually do. In dimensiontwo, sectional flag curvature means isotropic flag curvature. Since EinsteinRandersmetrics have been determined by [16], we may only consider high dimensions.Finally, we find it is a rigid condition for Randers metrics. Precisely, it says
Theorem 0.2.3. In dim≥3, any non-Riemannian Randers metric with sectional flag curvature must have constant flag curvature.
Hence the characterization for Randers metrics is finished. Recently, some rigidity theorems for general Finsler metrics are verified in [30].
Complex Randers metrics
Contrast to the real case, there are few examples in complex Finsler geometry except the Kobayashi metric and the Caratheodory metric. Following the spirit of real Randers metrics, N. Aldea and G. Munteanu began the study of complex Randers metrics ([7]) which have the formwhere a_(ij)(z)dz~i(?)dz~j is a Hermitian metric and b_i(z)dz~i is a (1,0)-form. In Chapter4, we mainly study the connection and holomorphic curvature of complex Randers metrics. Since all the complex Finsler metrics on Riemann surfaces are Hermitian, we will always assume that the complex dimension of the manifolds considered in this part is greater than one.
In real Finsler geometry, a metric is said to be Berwaldian if it has linear Berwald connection. In 1981, Z.Szabo classified the Berwald metrics by noting that Berwald metrics share their holonomy groups with Riemannian metrics. In [2], the definition of complex Berwald space is given, and some conformal properties are verified.
A complex Finsler metric is said to be Berwaldian if it is a Kahler-Finsler metric with linear Berwald connection. Here the notion of Kahler-Finsler metric is given by [8] where the Kahlerness is divided into three levels, i.e. strongly Kahler, Kahler and weakly Kahler. In Chapter 1, we prove that Kahler and strongly Kahler are in fact equivalent.
Theorem 0.2.4. A Kahler-Finsler metric is actually strongly Kahler.
Hence, there are only two Kahler conditions with respect to the ChernFinslerconnection in the complex Finsler geometry.
As an analogue of real case, for complex Randers metrics we have
Theorem 0.2.5. A complex Randers metric F =α+ |β|is a Berwald metric if and only ifαis Kahlerian andβ(?)βis parallel with respect toα.
Example 0.2.3 Let M be a usual Kahler manifold, and T be a flat torus. Choose a parallel (1,0)-formβon T, and letαbe the Kahlerian product metric on T×M. Then F =α+ |β| is a complex Berwald metric on T×M.
Besides the connections, the holomorphic curvature is another important quantity in complex Finsler geometry. The holomorphic curvature K_F(z,v) is said to be isotropic, if it is independent of the direction v, i.e. K_F(z,v) = K_F(z). In Kahler geometry, isotropic holomorphic curvature implies certain rigidity. Then one may ask what happens in the complex Finsler realm? We study the complex Berwald-Randers metrics and partially answer this question. The main difficulty is to calculate the holomorphic curvature of Randers metrics. Fortunately,we obtain the formula and get a rigidity theorem.
Theorem 0.2.6. A Berwald-Randers metric with isotropic holomorphic curvaturemust be either usually Kdhlerian or locally Mincowskian.
Globally, we have
Theorem 0.2.7. Let M be a simply connected complex manifold, and F =α+|β| be a complete Berwald-Randers metric on M. If F has isotropic holomorphic curvature, then it must be one of the following:
(1) M = CP~n, F is the Fubini-Study metric;
(2) M = {z∈C~n : |z| < 1}, F is the Bergman metric;
(3) M = C~n,αis the Euclidean metric andβ(z,v) =β(0, v)e~(iθ(z)) whereθ(z) is a real function on M. Particularly, F is a Mincowskian metric.
On the other hand, we say a complex Randers metric is strong ifαis Kahlerian andβis holomorphic. Then it holds
Proposition 0.2.5. Let M~n be a closed complex manifold with a strongly complex Randers metric F =α+ |β|. If the holomorphic curvature of F is positive, then F is a Hermitian metric.
Here we give some examples which have different curvature properties.
Example 0.2.4 On C~(2n), letα~2=δ_(AB)dz~A(?)dz~B andβ=∑_(i=1)~nz~(i+n)dz~i-z~idz~(i+n).Then F(z,v)=α+|β| has zero holomorphic curvature.
Example 0.2.5 On the unit disc△_1(?)C~n,letαbe the Bergman metricandβbe a holomorphic one form with norm less than 1, for instanceβ=∑z~idz~i.Then the resulting Randers metric is complete and K_F≤-1/4.
Example 0.2.6 Taking a small disc△_ε(?)C~n in CP~n,letαbe the Fubini-Studymetric on itandβ= b_idz~i with constant b_i. Then K_F > 0 on△_εfor sufficiently smallε.
20世纪90年代以来，在陈省身先生的倡导下，Finsler几何获得了长足的发展．许多新的几何量被发现，涌现了众多优秀的工作([8]，[12]，[58]，[17]，[60]，[50]，[24]等等)．伴随着基础理论研究的发展，Finsler几何也被广泛应用于诸如物理学、生物学、信息与控制论和心理学等学科当中．
本文主要探讨Finsler度量的某些曲率性质．主要内容分为三部分，分别讨论Einstein-Hilbert泛函、实Randers度量和复Randers度量．
Einstein-Hilbert泛函
Einstein度量是黎曼几何中具有重要意义的一类度量([20])．就物理或变分角度而言，Einstein度量是Einstein-Hilbert泛函的临界点．设(M，g)为n-维黎曼流形，其数量曲率记为R_g，则Einstein-Hilbert泛函可以表示为其中dμ_M是(M，g)的体积元．
沿着这一思路，我们尝试在Finsler几何中作相应的推广．早在1995年，法国几何学家H.Akbar-Zadeh就从变分角度做了尝试([5])．然而不幸的是，由于他没有考虑到变分张量的可积性(参见[10])，因而得到的张量特征是有问题的．之后的十余年，未见有任何进展．在陈省身先生提出Finsler几何中Einstein度量存在性问题，以及David Bao建议了相应的Ricci流之后，研究Einstein-Hilbert泛函的临界点就显得尤为重要了．
在Finsler流形(M~n，F)上，通过Chern联络或者Berwald联络建立起的曲率理论，导出了包括旗曲率、Ricci曲率和Landsberg曲率在内的许多重要几何量[14]．其中旗曲率是截面曲率的推广，Ricci曲率是旗曲率的某种平均，而Landsberg曲率则是重要的非黎曼几何量．鉴于几乎所有的几何量都依赖于方向的选取，切丛TM和射影球丛SM取代了黎曼几何中底流形M的角色．利用SM上的Sasaki度量，我们将Einstein-Hilbert泛函推广为其中Ric是Finsler度量F的Ricci曲率，dμ_(SM)是SM上Sasaki度量的体积元、Vol(SM)是SM的体积．根据积分求迹公式，可以验证对于黎曼度量而言该泛函就是原来的Einstein-Hilbert泛函．
推导泛函(0．1．2)的Euler-Lagrange方程并研究临界度量的性质是第二章的主要工作．在临界方程推导中，除了计算的复杂性之外，关键在于散度公式以及一个Green-型积分公式．前者最早由Akbar-Zadeh在局部坐标下给出，后来莫小欢、沈一兵-张彦利用活动标架法简化了证明：后者的一种特殊情形最先被贺群-沈一兵发现，而我们则证明了一般情形：
引理0．1．1．设ψ和ω是SM上两个光滑函数，则其中(g~(ij))是基本张量(gij)：=(1/2[F~2]y~iy~j)的逆．
利用上述公式，可以避免文献[5]中变分张量的可积性问题，从而得到正确的临界点方程．
定理0．1．1．Einstein-Hilberg泛函(0．1．2)的Euler-Lagrange方程为其中r=1/Vol(SM)∫_(SM)Ric dμ_(SM)是Ric在SM上的平均值，J=J_idx~i是平均Lands-berg张量，“|”和“；”分别表示Chern联络的水平和竖直共变导数，而“·”则表示沿Hilbert方向求导．
定义0．1．1．满足方程(0．1．4)的Finsler度量称为ε-临界度量．
由定理0．1．1可以发现，Finsler几何中的临界度量不仅与Ricci曲率有关，也与非黎曼量Landsberg曲率有关，从而Einstein条件(Ricci曲率是常数)不足以使(0．1．4)成立，另一方面，Riemann-Einstein度量是临界的，于是黎曼几何中的Einstein度量是ε-临界度量．那么是否存在非黎曼的ε-临界度量呢?下面给出两个例子．
例0．1．1设α是Ricci平坦的黎曼度量，β是平行1-形式．则Randers度量F=α+β是ε-临界度量．
例0．1．2设(M，g)和(N，h)是两个Ricci平坦的黎曼流形，则在乘积空间M×N上，度量是ε-临界的．其中函数这里ε是非负实数，k是正整数．(参见[66，60])
利用某些Ricci恒等式以及球丛SM上的分析，我们得到下述刚性定理．
定理0．1．2．设M是紧致无边的光滑流形，F是M上具有常正旗曲率和常S-曲率的Finsler度量．则F是ε-临界度量的充要条件为F是常正曲率的黎曼度量．
这里所谓的S-曲率是沈忠民在研究体积比较定理时发现的，它是体积畸变沿着测地线的变化率．S-曲率在现代Finsler几何研究中有着重要意义．例如．如果F是可逆的，则[38]证明了常正旗曲率和零S-曲率已经蕴涵着F是黎曼度量．
由定理0．1．2可知，即使具有极好的曲率性质，也未必是ε-临界度量．因此．Einstein-Finsler度量与ε-临界度量的关系就是一个重要的问题．这里给出两者在曲面上的一个联系．
命题0．1．1．紧致无边曲面上，旗曲率非正的Einstein-Finsler度量是ε-临界的．
命题0．1．2．设F是曲面上的ε-临界度量．若其Landsberg曲率是水平常数，则F是Einstein-Landsberg度量．进一步，若F的旗曲率非零，则其必为黎曼度量．Randers度量
1941年，G．Randers在研究相对论时，为了讨论时空中类时方向的不对称性而引进了Randers度量([51])．它具有简洁的形式其中a_(ij)(x)dx~i(?)dx~j是黎曼度量，b_i(x)dx~i是1-形式．正是基于这种简单表达，Randers度量成为了Finsler几何里最令人感兴趣的一类度量．近年来，通过一系列的工作，如[17]、[27]、[16]、[35]以及[63]等等，人们对Randers度量有了更深入的理解．
第三章首先讨论了ε-临界的Randers度量．
利用Zermelo导航技术，文献[17]分类了常旗曲率的Randers度量．特别地，常旗曲率的Randers度量必然具有常S-曲率．从而，作为定理0．1．2的直接推论，有
命题0．1．3．紧致无边流形上具有常正旗曲率的ε-临界Randers度量必为黎曼度量，从而是常正曲率黎曼度量．
该命题可以视为球面上标准度量的某种刚性：在向量场的挠动下度量不再是临界的．
另一方面，例0．1．1给出了非平凡的临界点，它们属于Berwald-Randers度量．顺便指出，对于Randers度量而言，弱Landsberg、Landsberg和Berwald是等价的．那么，究竟怎样的Berwald-Randers度量是临界的呢?利用Randers度量的黎曼曲率公式([16])，我们得到
命题0．1．4．Berwald-Randers度量F=α+β是ε-临界度量，当且仅当它为下述情形之一：
(1)α是Einstein度量，β=0；
(2)α是Ricci平坦的黎曼度量．
其中，第一类是Riemann-Einstein度量，第二类正是例0．1．1所述之度量．注意到Calabi-Yau流形是Ricci平坦的，因此第二类有着丰富的例子．从另外一个角度来看，该命题给出了Calabi-Yau流形在Finsler几何中的一个重要意义：它们在特定挠动下仍是临界度量．
第三章还讨论了具有截面旗曲率的Randers度量．
如前所述，旗曲率是Finsler几何中最重要的几何量．它出现于弧长第二变分，是了解测地线结构的关键．点x处的“旗”包括“旗杆”y∈T_xM和包含y的“旗面(截面)”Π．一般地，若Π由{y，V}张成，则(y，Π)的旗曲率记为K(x，y，V)．因为这不依赖于V在截面Π中的选取，所以旗曲率也记为K(x，y，Π)．
Finsler度量称为具有数量旗曲率，如果其旗曲率不依赖于截面而只依赖于旗杆的选取，即K(x，y，Π)=K(x，y)．具有数量旗曲率的Finsler度量已经得到了较好的理解，特别是Randers度量．文献[48]证明了具有负数量旗曲率的Finsler度量必为Randers度量(dim≥3)．随后，文献[63]刻画了此类Randers度量的特征方程．利用Zermelo导航术，[28]完全解决了具有数量旗曲率和迷向S曲率的Randers度量分类问题(dim>3)．
那么一个自然的问题是，如果旗曲率只依赖于截面而与旗杆选取无关会怎样?即K(x，y，Π)=K(x，Π)，旗曲率可以约化为二平面Grassmann丛G_2(M)上的标量函数．我们称此种度量为具有截面旗曲率．显然，黎曼度量具有截面旗曲率，因此这是一个非黎曼条件．此外，常旗曲率度量显然也满足该条件．那么自然要问：
除了黎曼度量和常旗曲率度量之外，是否有非平凡的例子?
我们研究了具有截面旗曲率的Randers度量．对于曲面情形，具有截面旗曲率等价于具有迷向旗曲率，即Einstein曲面．而Einstein-Randers曲面已经完全分类([16])，因此我们仅关注高维情形并得到如下刚性定理．
定理0．1．3．维数大于2的光滑流形上，非黎曼的Randers度量具有截面旗曲率的充要条件是它具有常旗曲率．
因此，就Randers度量而言，该问题就此解决．对于一般度量，最近[30]证明了一些负曲率刚性和弱Landsberg刚性结果．
复Randers度量
与实Randers度量的成熟研究相比，复Randers度量的研究现在只处于起步阶段．复Randers度量是形如的复Finsler度量，其中a_(ij)(z)dz~i(?)dz~j是Hermite度量,b_i(z)dz~i是(1，0)-形式．最早的基础工作应属文献[7]．本文第四章主要讨论复Randers度量的两个重要性质：联络的线性性和全纯曲率的刚性．因为黎曼面上的复Finsler度量必为Hermite度量，故本文所讨论之流形的复维数均大于一．
实Finsler几何中，如果一个度量的Berwald联络是线性的，则称该度量为Berwald度量．1981年，利用Berwald度量与Riemann度量具有相同和乐群，Z．Szabó对Berwald度量做了一个漂亮的分类定理([66])．
1996年，T．Aikou提出了复Finsler几何中Berwald度量的定义：复Finsler度量称为Berwald度量，如果它具有线性Berwald联络，并且是Kahler-Finsler度量([2])．这里所谓的Kahler度量的定义来自文献[8]．那里将Kahler性(联络的对称性)分为三个层次，即强Kahler、Kahler和弱Kahler．在本文第一章中，强Kahler和Kahler被证明是等价的，
定理0．1．4．Kahler-Finsler度量必为强Kahler度量．
因此，在复Finsler几何中仅有两种Kahler条件，即Kahler和弱Kahler．
如所知，实Randers度量是Berwald度量的充要条件是β关于α平行([45]).对于复Randers度量而言，我们得到下述定理．
定理0．1．5．复Randers度量F=α+|β|是Berwald度量，当且仅当α是Kahler度量，且β(?)β关于α平行．此时，F与α具有相同的和乐群．
例0．1．3设M为一Kahler流形，T为复一维的平环．在T上取平行向量场β，令α为T×M的直积度量．则F=α+|β|是T×M上的Berwald度量．
除和乐群之外，全纯曲率K_F(z，v)是极为重要的几何量．全纯曲率迷向是指K_F(z，v)=K_F(z)．Kahler几何中，迷向全纯曲率具有刚性．那么对于Kahler-Finsler度量，迷向全纯曲率是否仍然具有刚性?
本文讨论了一类特殊的Kahler-Finsler度量——复Berwald-Randers度量．其难点在于计算复Randers度量的全纯曲率K_F(z，v)．经过复杂的计算，在得到了全纯曲率公式之后，我们发现复Berwald-Randers度量也成立类似的刚性定理．
定理0．1．6．具有迷向全纯曲率的复Berwald-Randers度量必为Kahler-Hermite度量或局部Mincowski度量．整体地，有
定理0．1．7．设M为一单连通复流形，F=α+|β|是M上完备的复Berwald-Randers度量．若F具有迷向全纯曲率，则其必相似于下述度量之一：(1) M=CP~n,F是Fubini-Study度量；(2)M={z∈C~n：|z|<1}，F是Bergman度量;(3)M=C~n，α是欧氏度量且β(z，v)=β(0，v)e~(iθ)(z)，其中θ(z)是M上的实函数．
另一方面，根据曲率表达式，讨论由α为Kahler度量、β为全纯1-形式构成的复Randers度量很有意义．利用简单的消末定理，可得如下刚性定理．
命题0．1．5．设M~n是紧致无边复流形，F是由α为Kahler度量、β为全纯1-形式构成的复Randers度量．如果k_F(z，v)>0，则F是Kahler-Hermite度量．
最后，我们给出几个具有不同曲率性质的复Randers度量的例子．
例0．1．4 C~(2n)上，令α~2=δ_(AB)dz~A(?)dz~B，β=∑_(i=1)~nz~(i+n)dz~i-z~idz~(i+n)．则F(z，v)=α+|β|的全纯曲率为零．
例0．1．5令α为单位球△_1(?)C~n上的Bergman度量β是模长小于一的全纯形式，如β=∑z~idz~i．则F(z，v)=α+|β|的全纯曲率K_F≤-1/4．
例0．1．6在CP~n中取充分小的球△_ε(?)C~n，令α为Fubini-Study度量β=b_idz~i，其中b_i是常数．则在△+ε上有K_F>0．
The history of Finsler geometry can be traced back to the lecture given by B. Riemann in 1854, though he turned immediately to the quadratic case. P. Finsler was the first geometer who studied the general metrics systematically. In his thesis in 1918, Finsler developed a theory of curves and surfaces for such metrics. Therefrom, the name "Finlser space" was generally accepted. On the other hand, the correct notion of complex Finsler metrics was probably proposed firstly by G.B.Rizza ([52]). As important examples of complex Finsler metrics, the intrinsic metrics on complex manifolds play central roles in the studies of complex geometry.
From 90's of the 20th century, encouraged by S.-S. Chern([25]), many geometersentered the field of Finsler geometry. In the last decade, the study of Finsler geometry has taken on a new look ([12], [58], [17], [60], [50], [24] and etc.). On the other hand, as a powerful tool, Finsler metrics are also widely applied to physics, biology, control theory, psychology and etc.([13])
In this thesis, we will consider some properties of curvatures for Finsler metrics, both real and complex. The main text consists of three parts, which study Einstein-Hilbert functional, real Randers metrics and complex Randers metrics respectively.
Einstein-Hilbert functional
Let M be an n-dimensional compact manifold. As is well known, among Riemannian metrics on M there is an important cla.ss of metrics called Einstein metrics, which are the critical points of the normalized Einstein-Hilbert functional where R_g is the scalar curvature of the Riemannian metric g, and dμ_M is the volume element of g. This motivates us to consider the corresponding functional in Finsler geometry. An attempt in this direction was tried by H.Akbar-Zadeh ([5]). Unfortunately, it seems that one could not obtain the tensor characteristic on generalized Einstein metrics from the variation calculus in [5] (also cf. D.Bao's comment [10]). It encourages us to look for the Finslerian analogue of critical metrics from the point of view of differential geometry and variational calculus.
By virtue of the Chern connection on a Finsler manifold (M, F) with the Finsler metric F, we can define the flag curvature and the Ricci scalar, which are generalizations of the sectional curvature and the Ricci curvature in Riemanniangeometry, respectively ([14]). It is natural to define a similar functional in Finsler geometry by using the Ricci scalar and the volume form induced from the projective sphere bundle over (M,F). In fact, this functional can be defined bywhere Ric denotes the Ricci scalar and SM is the projective sphere bundle over M with volume element dμ_(SM).One can check easily (0.2.2) is just the previous (0.2.1) if F is Riemannian by means of the integral trace formula or Lemma1.4 in [34].
The purpose of the second chapter is to derive the Euler-Lagrange equation of the functional (0.2.2) and to study the properties of the critical metrics.
The main difficulties in deriving the critical equation are a divergence formulaand a Green type formula. The former one was firstly obtained by H.AkbarZadeh,and was reproved by Xiaohuan Mo and Qun He-Yibing Shen. A special case of the Green type formula was firstly proved by Qun He-Yibing Shen, and we discover the following generic form
Lemma 0.2.1. Letψandφbe two smooth functions on SM. Then it holds where (g~(ij)) is the inverse of the fundamental tensor (g_(ij)) := (1/2[F~2]_(y~iy~j).
With the help of this lemma, we get the correct equation.
Theorem 0.2.1. The Euler-Lagrange equation of the functional (0.2.2) iswherer=1/Vol(SM)∫_(SM)Ric dμ_(SM) is the average of Ric on SM, J = J_idx~i is the meanLandsberg tensor, "|" and ";" denote respectively the horizontal and the vertical covariant derivatives with respect to the Chern connection, and "·" denotes the covariant derivative along the Hilbert form.
Definition 0.2.1. A Finsler metric satisfying the equation (0.2.4) is called anε-critical metric.
It is easy to show that a Riemannian metric is anε-critical metric iff it is Einstein. On the other hand, we have the following non-Riemannian examples.
Example 0.2.1 Letαbe a Ricci-flat Riemannian metric andβbe parallel with respect toα, then the Randers metric F =α+βis anε-critical metric.
Example 0.2.2 Let (M,g) and (N,h) be two Ricci-flat Riemannian manifolds, then the metricisε-critical on the product manifold M×N, where the functionφ(s, t) can be defined asHereεis a nonnegative real number and k is a positive integer.
Applying some Ricci identities and the Hopf maximum principle, we obtain
Theorem 0.2.2. Let M be a closed manifold, and F be a Finsler metric on M with positive constant flag curvature and constant S-curvature. Then F is anε-critical metric if and only if it is a Riemannian metric with positive constant sectional curvature.
The S-curvature was discovered by Z.Shen in his study of volume comparison [58]. There are many rigidity results on the S-curvature. For instance, if F is reversible, then positive constant flag curvature and constant S-curvature will imply that F is Riemannian ([38]).
From Theorem 0.2.2, one can see that even Finsler space forms might not beε-critical. So it is interesting to study the relation betweenε-critical metrics and Einstein-Finsler metrics. In dimension two, we have
Proposition 0.2.1. On a closed surface, any nonpositively curved EinsteinFinslermetric isε-critical.
Proposition 0.2.2. Let F be anε-critical metric on a surface. If its Landsberg scalar is horizontal constant, then F is an Einstein-Landsberg metric. Moreover, if the Ricci curvature is nonzero, then F is Riemannian.
Real Randers metrics
Randers metrics were introduced by G.Randers in the context of general relativity,~([51]) and named by R.Ingarden in his paper on the electron microscope.~([36])
Being the sum of a Riemannian metricα=(?) and a 1-formβ= b_i(x)y~i,Randers metrics are the favorite metrics for Finsler geometers, and are well understood by some great works such as [27], [17], [16], [64], [63] and etc.
In Chapter 3. we first study theε-critical Randers metrics. By Zermelo navigation, [17] determined the Randers space forms. Particularly, any Randers metric with constant flag curvature must have constant S-curvature. So as a direct corollary of Theorem 0.2.2, we have
Proposition 0.2.3. Anyε-critical Randers metric with positive constant flag curvature on a closed manifold must be a Riemannian metric with positive constantsectional curvature.
In other words, the standard Riemannian metric on the sphere is isolated in the family of critical Randers metrics. On the other hand, Example 0.2.1 is nontrivial which belongs to the class of Berwald-Randers metrics. So it is interestingto understand what kind of Berwald-Randers metrics are indeed critical.
The following proposition will answer this question.
Proposition 0.2.4. A Berwald-Randers metric F =α+βisε-critical, if and only if it is one of the followings:(1)αis Einstein, andβ= 0;(2)αis Ricci-flat.
Notice that the second class contains the Calabi-Yau manifolds, so it is ample. From another point of view, it gives a new significance of Calabi-Yau manifolds in Finsler geometry which says that Calabi-Yau manifolds will remain critical under certain perturbations.
Besides the critical Randers metrics, Chapter 3 also considers Randers metricswith sectional flag curvature. The definition of sectional flag curvature is suggested in [62].
A flag planted at a base point x on the manifold M, consists of a flagpole y∈T_xM and a sectionΠcontaining y. Typically, forΠspanned by {y, V}, the flag curvature of (y,Π) is K(x, y, V). Since it is independent of the choice of V inΠ, we also write it as K(x, y,Π).
A Finsler metric is said to be of scalar flag curvature if the flag curvatureis independent of the section. In [48], it is verified that negatively curved Finsler metrics of scalar flag curvature on closed manifolds must be of Randers type(dim≥3). Later on, Randers metrics with scalar flag curvature were characterizedby [63]. Recently, [28] classified the Randers metrics with scalar curvature and constant S-curvature by Zermelo navigation(dim≥3).
Now, what happens if the flag curvature is independent of the flagpole? Such metrics are called of sectional flag curvature since the flag curvature dependsonly on the section. Under this condition, the flag curvature can be written as K(x,Π) just as in Riemannian geometry. Clearly, Riemannian metrics are of sectional flag curvature, so this condition is non-Riemannian. It is natural to ask:
Are there any nontrivial Finsler metrics with sectional flag curvature?
As a start, we first study the Randers metrics as usually do. In dimensiontwo, sectional flag curvature means isotropic flag curvature. Since EinsteinRandersmetrics have been determined by [16], we may only consider high dimensions.Finally, we find it is a rigid condition for Randers metrics. Precisely, it says
Theorem 0.2.3. In dim≥3, any non-Riemannian Randers metric with sectional flag curvature must have constant flag curvature.
Hence the characterization for Randers metrics is finished. Recently, some rigidity theorems for general Finsler metrics are verified in [30].
Complex Randers metrics
Contrast to the real case, there are few examples in complex Finsler geometry except the Kobayashi metric and the Caratheodory metric. Following the spirit of real Randers metrics, N. Aldea and G. Munteanu began the study of complex Randers metrics ([7]) which have the formwhere a_(ij)(z)dz~i(?)dz~j is a Hermitian metric and b_i(z)dz~i is a (1,0)-form. In Chapter4, we mainly study the connection and holomorphic curvature of complex Randers metrics. Since all the complex Finsler metrics on Riemann surfaces are Hermitian, we will always assume that the complex dimension of the manifolds considered in this part is greater than one.
In real Finsler geometry, a metric is said to be Berwaldian if it has linear Berwald connection. In 1981, Z.Szabo classified the Berwald metrics by noting that Berwald metrics share their holonomy groups with Riemannian metrics. In [2], the definition of complex Berwald space is given, and some conformal properties are verified.
A complex Finsler metric is said to be Berwaldian if it is a Kahler-Finsler metric with linear Berwald connection. Here the notion of Kahler-Finsler metric is given by [8] where the Kahlerness is divided into three levels, i.e. strongly Kahler, Kahler and weakly Kahler. In Chapter 1, we prove that Kahler and strongly Kahler are in fact equivalent.
Theorem 0.2.4. A Kahler-Finsler metric is actually strongly Kahler.
Hence, there are only two Kahler conditions with respect to the ChernFinslerconnection in the complex Finsler geometry.
As an analogue of real case, for complex Randers metrics we have
Theorem 0.2.5. A complex Randers metric F =α+ |β|is a Berwald metric if and only ifαis Kahlerian andβ(?)βis parallel with respect toα.
Example 0.2.3 Let M be a usual Kahler manifold, and T be a flat torus. Choose a parallel (1,0)-formβon T, and letαbe the Kahlerian product metric on T×M. Then F =α+ |β| is a complex Berwald metric on T×M.
Besides the connections, the holomorphic curvature is another important quantity in complex Finsler geometry. The holomorphic curvature K_F(z,v) is said to be isotropic, if it is independent of the direction v, i.e. K_F(z,v) = K_F(z). In Kahler geometry, isotropic holomorphic curvature implies certain rigidity. Then one may ask what happens in the complex Finsler realm? We study the complex Berwald-Randers metrics and partially answer this question. The main difficulty is to calculate the holomorphic curvature of Randers metrics. Fortunately,we obtain the formula and get a rigidity theorem.
Theorem 0.2.6. A Berwald-Randers metric with isotropic holomorphic curvaturemust be either usually Kdhlerian or locally Mincowskian.
Globally, we have
Theorem 0.2.7. Let M be a simply connected complex manifold, and F =α+|β| be a complete Berwald-Randers metric on M. If F has isotropic holomorphic curvature, then it must be one of the following:
(1) M = CP~n, F is the Fubini-Study metric;
(2) M = {z∈C~n : |z| < 1}, F is the Bergman metric;
(3) M = C~n,αis the Euclidean metric andβ(z,v) =β(0, v)e~(iθ(z)) whereθ(z) is a real function on M. Particularly, F is a Mincowskian metric.
On the other hand, we say a complex Randers metric is strong ifαis Kahlerian andβis holomorphic. Then it holds
Proposition 0.2.5. Let M~n be a closed complex manifold with a strongly complex Randers metric F =α+ |β|. If the holomorphic curvature of F is positive, then F is a Hermitian metric.
Here we give some examples which have different curvature properties.
Example 0.2.4 On C~(2n), letα~2=δ_(AB)dz~A(?)dz~B andβ=∑_(i=1)~nz~(i+n)dz~i-z~idz~(i+n).Then F(z,v)=α+|β| has zero holomorphic curvature.
Example 0.2.5 On the unit disc△_1(?)C~n,letαbe the Bergman metricandβbe a holomorphic one form with norm less than 1, for instanceβ=∑z~idz~i.Then the resulting Randers metric is complete and K_F≤-1/4.
Example 0.2.6 Taking a small disc△_ε(?)C~n in CP~n,letαbe the Fubini-Studymetric on itandβ= b_idz~i with constant b_i. Then K_F > 0 on△_εfor sufficiently smallε.
引文
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[6] H.Akbar-Zadeh, Geometry of generalized Einstein Manifolds, C. R. Acad. Sci. Paris, Ser.I 339(2004), 125-130.
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[2] T.Aikou, Some Remarks on Locally conformal complex Berwald spaces, Contemporary Mathematics, Amer. Math. Soc, Vol. 196(1996), 109-120.
[3] T.Aikou, Complex manifolds modeled on a complex Minkowski space, Jour. Math. Kyoto Univ., 35(1996), 85-103.
[4] H.Akbar-Zadeh, Les espaces de Finsler et certaines de leurs generalisations, Ann. Ec. Norm. Sup. 3~e serie 80(1963), 1-79.
[5] H.Akbar-Zadeh, Generalized Einstein Manifolds, Jour. of Geom. and Phys. 17(1995), 342-380. (#MR 1365208(99m:53130).)
[6] H.Akbar-Zadeh, Geometry of generalized Einstein Manifolds, C. R. Acad. Sci. Paris, Ser.I 339(2004), 125-130.
[7] N.Aldea, G.Munteanu, On complex Finsler spaces with Randers metric, manuscript, (2006).
[8] M.Abate, G.Patrizio, Finsler metrics-A Global Approach, with applications to geometric function theory, Springer, LNM1591, (1994).
[9] G.S.Asanov, Finsleroid-Finsler spaces of positive-definite and relativistic types, Reports on Math. Physics, 58(2006), No.2, 275-300.
[10] D.Bao, review, AMS MathSciNet (Mathematical Reviews on the web), http://www.ams.org/msnpdf/al365208.pdf
[11] D.Bao, On two curvature-driven problems in Ricmann-Finlser geometry, manuscript, (2007).
[12] D.Bao, S.-S.Chern, A note on the Gauss-Bonnet theorem for Finsler spaces, Ann. Math. 143(1996),233-252.
[13] D.Bao, S.-S.Chern, Z.Shen editors, Finsler Geometry, Contemporary Mathematics 196, Amer. Math. Soc, 1995.
[14] D.Bao, S.-S.Chern, Z.Shen, An Introduction to Riemann-Finsler Geometry, Grad. Texts in Math. 200, Springer, New York, (2000).
[15] D.Bao, C.Robles. On Randers metrics of constant curvature, Rep. on Math.Phys. 51(2003), 9-42.
[16] D.Bao, C.Robles, Ricci and Flag Curvatures in Finsler Geometry, Riemann-Finlser Geometry, MSRI Publications, 50(2004), 198-256.
[17] D.Bao, C.Robles, Z.Shen, Zermelo Navigation on Riemannian Manifolds, Jour. of Diff.Geom. 66(2004), 391-449.
[18] L.Berwald, Untersuchung der Krummung allgemeiner metrischer Raume auf Grund des in ihnen herrschenden Parallelismus, Math. Z. 25(1926), 40-73.
[19] L.Berwald, Parallelubertragung in allgemeinen Raurnen, Atti Congr. Inter. Mat. Bologna 4(1928), 263-270.
[20] A.L.Besses, Einstein manifolds, Ergebnisse der Math, (3)10, Springer, Berlin, (1987).
[21] F.Brickell, A nes proof of Deicke's theorem on homogeneous functions, Proc. of Amer. Math. Soc, 16(1965),190-191.
[22] H.Busemann, The Geometry of Finsler Spaces, Bull, of Amer. Math. Soc, (1950), 5-16.
[23] C.Caratheodory, Uber erne spezielle Metric, die in der Theorie der analitis-chen Functionen antritt, Atti Pontificia Acad. Sc. Nuovi Lincei 80(1927), 135-141.
[24] J.-G.Cao, P.Wong, Finsler Geometry of Projectivized vector bundles, manuscript, (2003).
[25] S.-S.Chern, Riemannian Geometry as a Special case of Finsler geometry, Contemporary Mathematics, 196, Amer. Math. Soc, (1996), 51-58.
[26] X.Cheng, X.Mo, Z.Shen, On the Flag Curvature of Finsler Metrics of Scalar Curvature, Jour.London Math. Soc. (2)68:3(2003), 762-780.
[27] X.Cheng, Z.Shen, Randers metrics with special curvature properties, Osaka J. Math. 40(2003), 87-101.
[28] X.Cheng, Z.Shen, Randers metrics of scalar flag curvature, manuscript, (2007).
[29] S.-S.Chern, Z.Shen, Riemann-Finsler Geometry, World Scientific Singapore,(2005).
[30] B.Chen, L.Zhao, Finsler metrics with nonzero sectional flag curvature, manuscript, (2008).
[31] A.Deicke, Uber die Finsler-Raume mit A_i = 0, Arch. Math. 4(1953),45-51.
[32] P.Finsler, Ueber Kurven und Flachen in allgeneinen Raumen, Dissertation, (1918).
[33] M.Hashiguchi, Y.Ichijyo, On some special (α,β)-metrics, Rep. Fac. Sci. Kagoshima Univ. 8(1975), 39-46.
[34] Q.He, Y.-B.Shen, Some Results on Harmonic Maps for Finsler Manifolds, Inter. Jour, of Math., 16, No.9(2005), 1017-1031.
[35] Q.He, Y.-B.Shen, On Berstein type theorems in Finsler spaces with the volume form induced form the projective sphere bundle, Proc. of Amer. Math. Soc, 134(2006), No.3, 871-880.
[36]R.S.Ingarden, On the geometrically absolute optical represectation in the electron microscope, Trav. Soc. Sci. Lettr. Wroclaw B45( 1957),3-60.
[37] S.Kikuchi, On the condition that a space with (α,β)-metrics be locally Minkowskian, Tensor (N.S.) 33(1979), 242-246.
[38] C.Kim, J.Yim, Finsler manifold with positive constant flag curvature, Geom. Dedicata 98(2003), 47-56.
[39] S.Kobayashi, Negative vector bundles and complex Finsler structures, Nagoya Math. Jour. 57(1975), 153-166.
[40] S.Kobayashi, Invariant distances on complex manifolds and holomorhpic mappings, Jour. Math. Soc. Japan, 19(1976), 460-480.
[41] S.Kobayashi, K.Nomizu, Foundations of Differential Geometry, Volume 11(1969), 369-370.
[42] L.Lempert, La metrique de Kobayashi et la representation des domaines sur la boule, Bull. Soc. Math. Prance 109(1981), 427-474.
[43] B.Li, Z.Shen, On a Class of Weak Landsberg Metrics, Sci. in China(A), 50(2007), No.1,75-85.
[44] K.Liu. X.Sun, S.T.Yau, Canonical metrics on the moduli space of Riemann surface I, Jour. Diff. Geom. 68, 3(2004), 571-637.
[45] M.Matsumoto, On Finsler spaces with Randers metric and special forms of important tensors, Jour. Math. Kyoto Univ. (Kyoto Daigaku J. Math.). 14(1974), 477-498.
[46] X.Mo, Characterization and structure of Finsler spaces with constant flag curvature, Sci. in China Ser. A, 41(1997), 900-917.
[47] X.Mo, Harmonic maps from Finsler manifolds, Illinois Jour. Math., 45(2001), 1331-1345.
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