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

Hongchuan XIA Chunping ZHONG

Manuscript received December 24,2014.Revised June 12,2015.

1School of Mathematical Sciences,Xiamen University,Xiamen 361005,Fujian,China.

E-mail:cuttheknots@126.com

2Corresponding author.School of Mathematical Sciences,Xiamen University,Xiamen 361005,Fujian,China.E-mail:zcp@xmu.edu.cn

∗This work was supported by the National Natural Science Foundation of China(Nos.11271304,11171277),the Program for New Century Excellent Talents in University(No.NCET-13-0510),the Fujian Province Natural Science Funds for Distinguished Young Scholars(No.2013J06001)and the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry.

1 Introduction and Main Results

As is well known,complex Finsler metrics have become a very useful tool in geometric function theory of holomorphic mappings(see[2]).Two most important such metrics are the Carath´eodory and Kobayashi metrics,which share in higher dimensions the properties of the Poincar´e metric in the unit disc in C.In general,however,these two metrics do not have enough smoothness to allow a differential geometric study.In[8],Lempert proved a fundamental result which states that in smoothly bounded strictly convex domains in Cnthe Kobayashi and Carath´eodory metrics agree,and are strongly pseudoconvex complex Finsler metrics in the sense of Abate and Patrizio(see[2]).This fundamental result motivated several authors to investigate the Kobayashi metrics in strictly convex domains from a differential geometric point of view(see[2,11]).Even in strictly convex domains,however,we do not have the explicit formulae of the Kobayashi and Carath´eodory metrics.As was pointed out in[1], “the lack of consideration of explicit examples made the choice of the ‘right’notions in the complex setting difficult and sometimes rather artificial···,the lack of examples raised the doubt that perhaps metrics satisfying such strong conditions occur very infrequently.”So we need more explicit examples of complex Finsler metrics.

In the study of differential geometry of complex Finsler metrics,an important class of complex Finsler metrics comes from complex Berwald metrics,which includes Hermitian metrics and complex Minkowski metrics(see[3]).There are also lots of complex Berwald metrics which are neither Hermitian metrics nor complex Minkowski metrics(see[4]).

Let〈·,·〉be the canonical complex Euclidean inner product in Cn,and‖·‖be the norm induced by〈·,·〉,that is,forwhere in the following,bars and overlines denote conjugations of complex numbers.

In[14],the author introduced the notion of weakly complex Berwald metrics and proved that the complex Wrona metric(see[6,14]for more details)

in Cnis a weakly complex Berwald metric,but not a complex Berwald metric.It was proved in[13]that the conformal change of a weakly complex Berwald metric is also a weakly complex Berwald metric.More precisely,ifσ(z):M→R is a real smooth function onMandF:is a weakly complex Berwald metric in the sense of[14],thencalled a conformal change ofF,andis still a weakly complex Berwald metric.

Note that the complex Wrona metric(1.1)is only smooth on a subsetΩof the slit holomorphic tangent bundleof the holomorphic tangent bundleMore precisely(see[14]),

Our purpose in this paper is to construct a class of weakly complex Berwald metrics which are smooth on the whole slit holomorphic tangent bundlefor a unitary invariant domainM⊂Cn.More precisely,we shall introduce a class of unitary invariant complex Finsler metrics of the form

whereis a smooth positive function ofw∈R,andzis in a domainwhich is unitary invariant.Note that by Cauchy-Schwarz inequality we always haves≤t.

Our consideration of complex Finsler metrics of the form(1.2)is based on the following observation:The complex Wrona metric(1.1)can be rewritten as

while the later is equivalent to the whole slit holomorphic tangent bundleT1,0Cn-{zero section}.Thus in order to construct a weakly complex Berwald metric which is smooth on the whole slit holomorphic tangent bundleT1,0M-{zero section}for some domainM⊂Cn,it is natural to consider the class of complex Finsler metrics of the form(1.2),which is obtained by replacing the functionwith smooth positive functionsf(s−t)defined on the whole set{(t,s)∈[0,+∞)×[0,+∞):s≤t}.

We shall prove that wheneveris a strongly pseudoconvex complex Finsler metric,is necessarily a weakly complex Berwald metric with vanishing holomorphic curvature and Ricci scalar curvature,which are independent of the concrete choice of the functionf.We also prove that there are lots of functionsfsuch thatstrongly pseudoconvex complex Finsler metrics(see Proposition 3.2 and Corollary 3.1).Under some initial-value conditions onfand its derivativef′,we prove a surprising result,that is,the real geodesics of the weakly complex Berwald metricon every Euclidean sphereare great circles.

Our main results are as follows(see Theorems 4.1–4.3).

2 Preliminaries

In this section,we shall recall some necessary notations and definitions,which can be found in[2].

LetMbe a complex manifold of complex dimensionn.Letz=(z1,···,zn)be a local coordinate system inM,andv=(v1,···,vn)be the local fibre coordinate system defined by the local holomorphic frame fieldon the holomorphic tangent bundleT1,0MofM.Then(z,v)=(z1,···,zn,v1,···,vn)is a local coordinate system forT1,0M.In the following we denote bythe complement of the zero section inT1,0M,i.e.,

Note that ifFis a unitary invariant complex Finsler metric onM,then locallyMis necessarily unitary invariant in the sense thatAz∈Mwheneverz∈MandA∈U(n).

Letbe the inverse matrix ofIn the following,for functionsG,r,tandsdefined onwe denote by indexes likeα,and so on the derivatives with respect to thev-coordinates;the derivative with respect to thez-coordinates will be indexed after a semicolon;for instance,

In complex Finsler geometry,there are several well-known complex Finsler connections,for example,the Chern-Finsler connection(see[2]),the complex Rund connection(see[12])and the complex Berwald connection(see[10]).These connections are suitable for considering different problems in complex Finsler geometry.As we know,given a real Finsler metricF,there is only one nonlinear connection associated toF(see[9]).Given a strongly pseudoconvex complex Finsler metricF,there are two complex nonlinear connections associated toF:The Chern-Finsler nonlinear connection(see[2])and the complex Berwald nonlinear connection which is also called the Cartan complex nonlinear connection in[10].Their corresponding complex nonlinear connection coefficients are denoted byand,respectively.

LetFbe a strongly pseudoconvex complex Finsler metric.Then(see[2,10])

3 Fundamental Tensor and Nonlinear Connection

In this section,we shall derive the fundamental tensorthe Chern-Finsler nonlinear connection coefficientsand the complex Berwald nonlinear connection coefficientsthat are associated to a strongly pseudoconvex complex Finsler metricrespectively.

and(3.3)follows.Note that the fundamental matrixHis a nonsingular matrix sinceFis strongly pseudoconvex,so by the formula of the inverse of a small-rank adjustment(see[7]p.19),we can safely suppose that

wherekis given by the second equality of(3.5).So

whereYis given by the second equality of(3.6).Substituting(3.10)into(3.8),we get(3.4).

Denote byMn×m(C)the set of alln×mmatrices over the complex number field C.

Since ann×nHermitian matrixHhas exactlynreal eigenvalues,counting multiplicities,we immediately obtain thatFis a strongly pseudoconvex complex Finsler metric if and only ifHis a positive de finite matrix on?M;if and only if

Note that sincec0=f−sf′,it is easy to check that

by which the third inequality of(3.17)implies the second one,soFis strongly pseudoconvex if and only if(3.15)–(3.16)hold.

Remark 3.1It follows from the proof of Proposition 3.2 that ifn=2,thenFis a strongly pseudoconvex complex Finsler metric if and only if(3.16)holds.

Corollary 3.1Suppose that f(w)is a real-valued smooth positive function of w=s−t∈Rsatisfying

DifferentiatingG=rf(s−t)with respect tovβandzμin turn gives

Substituting(3.30)into the second equality of(3.25)yields(3.21).

4 Proofs of Main Results

Note that the holomorphic curvature(see[2])and the Ricci scalar curvature(see[10])of a strongly pseudoconvex complex Finsler metricFwith respect to the Chern-Finsler connection are defined respectively by

Proof By Proposition 3.3,

So by(4.9),it is necessary thatf′=0,i.e.,f(s−t)=a(s−t)+bfor some constantsa,b∈R.By Proposition 3.2,it follows thatb＞0 andb−at＞0.In this case,it is easy to check that the equalities in(4.8)hold identically.Conversely,iff(s−t)=a(s−t)+bfor some

it follows thatGαβdepends only onz=(z1,···,zn)andfsatisfies the condition in Proposition 3.2,so thatFis actually a Hermitian metric.

Substituting(4.14)and(4.17)into(4.13),we obtain the geodesic equations

On the other hand,by Proposition 3.1,we have

which implies(4.11).Differentiating(4.11)with respect toτ,we get

By the assumption we have

Thus it follows from(4.11)and(4.26)–(4.27)that the geodesicσgiven by(4.11)actually satisfies

By the explicit formula(4.11)for the geodesicσ(τ)ofF,we see thatσ(τ)is actually a periodic function with period 2π.It is clear thatσis a closed geodesic andσ(0)=σ(2π)=p.Moreover,by(4.28)we have

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