characterizations of complex finsler connections and weakly complex berwald metrics

Differential Geometry and its Applications 31 (2013) 648–671

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Differential Geometry and its Applications

www.elsevier.com/locate/difgeo

Characterizations of complex Finsler connections and weaklycomplex Berwald metrics

Liling Sun, Chunping Zhong ∗

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 19 September 2012Available online xxxxCommunicated by Z. Shen

MSC:53C6053C40

Keywords:Complex Berwald connectionHolomorphic sectional curvatureHolomorphic bisectional curvatureWeakly complex Berwald metric

Let M be a complex manifold endowed with a strongly pseudoconvex complex Finslermetric F . In this paper, characterizations of the complex Rund connection, complexBerwald connection and complex Hashiguchi connection that associated to F are given. Theprecise relationship of holomorphic sectional curvature, holomorphic bisectional curvatureand Ricci scalar curvature of F with respect to these connections are obtained. Moreover,it is proved that the conformal change F = eσ(z) F of F is a weakly complex Berwald metricon M if and only if F is a weakly complex Berwald metric on M .

© 2013 Elsevier B.V. All rights reserved.

1. Introduction and statement of results

As is well known, for a Riemannian manifold there is a unique connection, called the Levi-Civita connection, which istorsion free and metric compatible.

For a real Finsler manifold, there is no canonical connection. The most often used real Finsler connections are theCartan connection [1], the Chern–Rund connection [13], and the real Berwald connection [18]. The Cartan connection wascharacterized in [1], the Chern–Rund connection and the real Berwald connection were characterized in [2].

For a Hermitian manifold, or more general a Hermitian holomorphic vector bundle, there is a unique complex connection,called the Chern connection or the Hermitian connection, which is metric compatible.

For a strongly pseudoconvex complex Finsler manifold, there are also several well-known complex Finsler connectionsassociated to F . We refer to the Chern–Finsler connection [1], the complex Rund connection [21] and the complex Berwaldconnection [19]. The Chern–Finsler connection was characterized and comprehensively investigated in [1], and was widelyused in complex Finsler geometry [3–6,16,22–24].

In complex Finsler geometry, the Chern–Finsler connection is the most often used complex Finsler connection, butsometimes other complex Finsler connections are also useful and are more convenient. For example, the complex Rundconnection was systemically studied in [7–11], and was used to derive the formulas of complex horizontal and verticalLaplacians on Kähler–Finsler manifold [23]. It was also used to derive the Gauss, Codazzi and Ricci equations for holomor-phic Finsler submanifolds [24]. From a geometric point of view, the complex Rund connection has a simple law of verticalcovariant derivatives for complex Finsler tensors, which makes local calculations greatly simplified [23,24]. The complexBerwald connection was also used in [20] to investigate holomorphic subspaces in complex Finsler spaces.

* Corresponding author.E-mail address: [email protected] (C. Zhong).

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So it is important to investigate the relationship among these complex Finsler connections, as well as give characteri-zations of them. It is also important to clarify some geometric objects which are naturally associated to these connections,such as the holomorphic sectional curvature, the holomorphic bisectional curvature, and the Ricci scalar curvature withrespect to these connections from a differential geometric point of view.

To the authors knowledge, so far there are no characterizations of complex Rund connection, complex Berwald connectionand complex Hashiguchi connection. In this paper, we shall give characterizations of these complex Finsler connections.Essentially, these characterizations are global. We shall investigate the holomorphic sectional curvature, the holomorphicbisectional curvature and the Ricci scalar curvature of a given complex Finsler metric with respect to these connections.Their precise relationship among them are clarified. Moreover, the conformal transformation properties of complex Berwaldmetrics and weakly complex Berwald metrics are investigated.

We point it out here that our characterization of complex Berwald connection is different from the real case [2], sincethere are two complex non-linear connections (or equivalently complex horizontal bundles) associated to a strongly pseu-doconvex complex Finsler metric while there is only one real non-linear connection (or equivalently real horizontal bundle)canonically associated to a real Finsler metric. Nevertheless, the complex Rund connection and the complex Berwald con-nection share the same complex horizontal radial vector fields. This important observation is used successfully to give acharacterization of complex Berwald connection.

The main results are as follows:

Theorem 1.1. Let F be a strongly pseudoconvex complex Finsler metric on a complex manifold M, and let 〈·, ·〉 be the Hermitianstructure on V1,0 induced by F . Then there is a unique complex vertical connection D :X (V1,0) →X (T ∗

CM ⊗ V1,0) such that:

(i) D is good;(ii) for any H ∈ H1,0 and V , W ∈ V1,0 , one has

H〈V , W 〉 = 〈D H V , W 〉 + 〈V , D H W 〉, (1.1)

where H1,0 is the (1,0) part of the complex horizontal bundle HC associated to D;(iii) for any V ∈ V1,0 and H ∈ H1,0 , one has

θ (V , H) = 0, (1.2)

where θ is the (2,0)-torsion of D.

The complex Finsler connection characterized by Theorem 1.1 is called the complex Rund connection associated to F .

Theorem 1.2. Let F be a strongly pseudoconvex complex Finsler metric on a complex manifold M, and 〈·, ·〉 be the Hermitian structureon V1,0 induced by F . Then there is a unique complex vertical connection D such that:

(i) D is good;(ii) the complex radial horizontal vector field χ associated to D coincides with the complex radial horizontal vector field χ associated

to the Chern–Finsler connection D, i.e.,

χ = χ ; (1.3)

(iii) for any V , W ∈ V1,0 and H, U ∈ H1,0 , one has

θ (H, U ) ∈ V1,0, (1.4)

θ (V , H) = 0, (1.5)

θ (V , W ) = 0, (1.6)

where H1,0 is the (1,0) part of the complex horizontal bundle HC associated to D, and θ is the (2,0)-torsion of D.

The complex Finsler connection characterized by Theorem 1.2 is called the complex Berwald connection associated to F .




χ = χ ; (1.7)

650 L. Sun, C. Zhong / Differential Geometry and its Applications 31 (2013) 648–671

(iii) for U , V , W ∈X (V1,0),

U 〈V , W 〉 = 〈DU V , W 〉 + 〈V , DU W 〉; (1.8)

(iv) for V , W ∈ V1,0 and H, U ∈ H1,0 , one has

θ (H, U ) ∈ V1,0, (1.9)

θ (V , H) ∈ H1,0, (1.10)


The complex Finsler connection characterized by Theorem 1.3 is called the complex Hashiguchi connection associatedto F .

It follows from Theorems 1.1–1.3 that the complex Rund connection, complex Berwald connection and complexHashiguchi connection are different in the extent of metric compatibilities as well as the degree of vanishing of theirassociated (2,0)-torsions.

Next we shall investigate the holomorphic sectional curvature, the holomorphic bisectional curvature and Ricci scalarcurvature of F with respect to the above complex Finsler connections.

Denote D , D , D and D the Chern–Finsler connection, complex Rund connection, complex Berwald connection and com-plex Hashiguchi connection associated to F , respectively.

Denote K F (v), K F (v), K F (v) and K F (v) the holomorphic sectional curvatures of F with respect to D , D , D and D ,respectively.

Denote B F (v, w), B F (v, w), B F (v, w) and B F (v, w) the holomorphic bisectional curvatures of F with respect to D , D ,D and D , respectively.

Denote R F (v), R F (v), R F (v) and R F (v) the Ricci scalar curvatures of F with respect to D , D , D and D , respectively.One may ask whether these curvatures of F depend on the choice of complex Finsler connections associated to F . What’s

the relationship among them with respect to a given complex Finsler connection associated to F ? The following theoremgives an answer.

Theorem 1.4. Let M be a complex manifold endowed with a strongly pseudoconvex complex Finsler metric F . Then

(i) the holomorphic sectional curvature of F with respect to D, D, D and D are given by

K F (v) = K F (v) = K F (v) = K F (v); (1.11)

(ii) the holomorphic bisectional curvature of F with respect to D, D, D and D are given by

B F (v, w) = B F (v, w), B F (v, w) = B F (v, w), (1.12)

B F (v, w) = B F (v, w) + 1

G(v)G(w)Gαδν

(Γ α

β;μ − Γ αμ;β

)vβ wμwν

− 1

2

1

G(v)G(w)Gα

(Γ

γ

λ;ν − Γγ

ν;λ)

vλ∂γ

(Γ α

β;μ + Γ αμ;β

)vβ wμwν; (1.13)

(iii) the Ricci scalar curvature of F with respect to D, D, D and D are given by

R F (v) = R F (v), R F (v) = R F (v), R F (v) = R F (v) + 1

2χ

(Γ

μβ;μ − Γ

μμ;β

)vβ. (1.14)

As is well known, special complex Finsler metrics are important in investigating the differential geometry of complexFinsler metrics. In [7], the notion of complex Berwald metric was introduced and it was proved in [7] that if F is a complexBerwald manifold, then all complex Minkowski spaces T 1,0

p M with the corresponding norms F (p, ·) are isometric to eachother. Recently in [25], the second author introduced the notion of weakly complex Berwald metric. It was also shownin [25] that the complex Wrona metric [15]

F (z, v) = ‖v‖2√‖z‖2‖v‖2 − |〈z, v〉|2 (1.15)

is a weakly complex Berwald metric in Cn with the complex Berwald connection coefficients G

αβμ vanishing identically.

Moreover, the holomorphic sectional curvature and the Ricci scalar curvature of the complex Wrona metric vanish identi-cally.

Differs from real Finsler geometry, usually it is rather difficult to find explicit examples in complex Finsler geometrywhich contributes the right choice of some notions. The lack of explicit examples would raises the doubt that whether


there is a weakly complex Berwald metric with the complex Berwald connection coefficients Gαβμ not vanishing identically?

If there is, then to what extent its holomorphic sectional curvature is a constant? The following theorem gives an answer.

Theorem 1.5. Let F be a strongly pseudoconvex complex Finsler metric on M, and F = eσ(z) F be a conformal change of F . Then

(i) F is a weakly complex Berwald metric if and only if F is a weakly complex Berwald metric;(ii) the holomorphic sectional curvature K F (v) of F and the holomorphic sectional curvature K F (v) of F are related by

K F (v) = e−2σ

[K F (v) − 4

G

∂2σ

∂zμ∂zνvμvν

]; (1.16)

(iii) the Ricci scalar curvature R F (v) of F and the Ricci scalar curvature R F (v) of F are related by

R F (v) = R F (v) − (n + 1)∂2σ

∂zμ∂zνvμvν . (1.17)

In the last section, we shall use assertion (i) in Theorem 1.5 to construct more weakly complex Berwald metrics withG

αβμ not vanishing identically.

2. Complex non-linear connection and vertical connection

In this section, we shall recall some basic facts about complex non-linear connection and general theory of good complexvertical connections [1].

Let M be a complex manifold of complex dimension n, and π : T 1,0M → M be the holomorphic tangent bundle of M .We denote z = (z1, . . . , zn) the local holomorphic coordinates around p ∈ U ⊂ M , and (z, v) = (z1, . . . , zn, v1, . . . , vn) theinduced local holomorphic coordinates on π−1(U ) ⊂ T 1,0 M . Since most geometric objects associated to complex Finslermetrics live in T 1,0M . In this section we shall briefly recall some mechanisms to investigate general differential geometryof T 1,0M , which can be found in the excellent monograph [1]. We use Einstein sum convention throughout the paper.

The key point is to introduce a complex non-linear connection D on M to linearize the geometry of T 1,0M . Locally, D ischaracterized by its connection coefficients Γ α

μ , which are n × n functions defined on T 1,0M such that under change of

coordinates z → z on M and the induced change of coordinates (z, v) → (z, v) on T 1,0M ,

Γ αμ (z, v) = ∂ zα

∂zν

∂zβ

∂ zμΓ ν

β (z, v) − ∂2 zα

∂zν∂zβ

∂zν

∂ zμvβ. (2.1)

In the following we denote ∂μ the partial derivative with respect to the local coordinates zμ on M , and ∂μ the partialderivative with respect to fiber coordinates vμ .

Equal important objects in complex Finsler geometry are complex horizontal bundle HC and complex horizontal mapΘ : VC →HC , which are locally given by

Θ(∂μ) = δμ, Θ(∂μ) = δμ,

where δμ = ∂μ − Γ αμ ∂α and δμ denotes the complex conjugation of δμ . These conceptions are naturally considered in

complex Finsler geometry since they are well compatible with the canonical complex structure J on M and the canonicalcomplex structure J on T 1,0M , where we use the same symbol if it causes no confusion.

The relationship among the complex non-linear connection D , the complex horizontal bundle HC and the complexhorizontal map Θ is well clarified in [1].

Proposition 2.1. (See [1].) Let M be a complex manifold. Then the maps D → Θ D , Θ → HΘC

and HC → DHCdefine a one-to one

correspondence among complex horizontal bundles, complex non-linear connections and complex horizontal maps. More precisely,

D = DHΘ D

C

, Θ = ΘDHΘ

C , HC = HΘDHC

C. (2.2)

The first equality in (2.2) means that: for a given complex non-linear connection D , one can define a unique complex

horizontal map Θ D ; using Θ D , one can then define a unique complex horizontal bundle HΘ D

C, while using HΘ D

C, one can

then define a unique complex non-linear connection DHΘ D

C

, and DHΘ D

C

obtained in this way is in fact equal to D . The same

interpretations are applicable to other equalities in (2.2). Thus complex non-linear connection, complex horizontal map andcomplex horizontal bundle are essentially the same thing.

There is a natural way to obtain a complex non-linear connection from a good complex vertical connection [1]. In thefollowing we set M = T 1,0M − {0}, which is also called the slit holomorphic tangent bundle of M .


A complex vertical connection is a complex linear connection D :X (V1,0) →X (T ∗C

M ⊗ V1,0) such that

D X V = [X(

V α) + ωα

β (X)V β]∂α, (2.3)

D X V = D X V , D X V = X(

V α)∂α, D X V = D X V , (2.4)

for all X ∈ T 1,0M and V = V α∂α ∈X (V1,0), where throughout the paper over-lines denote complex conjugations. Note thatωα

β in (2.3) are connection 1-forms, which are of type (1,0). Define a bundle map Λ : T 1,0M → V1,0 by setting

Λ(X) =: D X ι, ∀X ∈ T 1,0M,

where ι = vα∂α is a global holomorphic vector field called complex radial vertical vector field. If Λ : V1,0 → V1,0 is a bundleisomorphism, then D is called a good complex vertical connection.

For a good complex vertical connection D , if we set H1,0 =: kerΛ, then HC =: H1,0 ⊕ H1,0 is a complex horizontalbundle such that T 1,0M =H1,0 ⊕ V1,0. Thus HC is completely determined by its (1,0)-part H1,0.

Denote Θ : VC → HC the complex horizontal map associated to HC . Then one can obtain complex linear connectionson the complex vector bundles T 1,0M or TCM just by setting

D X H =: Θ[D X

(Θ−1(H)

)], ∀X ∈ TCM, H ∈ X

(H1,0). (2.5)

For a good complex vertical connection D , we denote Γ αμ the associated complex non-linear connection, and set

δμ = ∂μ − Γ αμ ∂α, ψα = dvα + Γ α

μ dzμ. (2.6)

Then the globally defined vector field η = dzμ ⊗ ∂μ + dvα ⊗ ∂α on M can be expressed as

η = dzμ ⊗ δμ + ψα ⊗ ∂α. (2.7)

The exterior differential Dη is the sum of a T 1,0M-valued (2,0)-form θ = D ′η and a T 1,0 M-valued (1,1)-form τ = D ′′η. θ andτ are called the (2,0)-torsion and (1,1)-torsion of D , respectively. Locally, let ωα

β = Γ αβ;μ dzμ + Γ α

βγ ψγ be the connection1-forms of D . Then its (2,0)-torsion and (1,1)-torsion are given by

θ =[

1

2

(Γ

μν;σ − Γ

μσ ;ν

)dzσ ∧ dzν + Γ

μνγ ψγ ∧ dzν

]⊗ δμ +

{1

2

[δμ

(Γ α

ν

) − δν(Γ α

μ

)]dzμ ∧ dzν

+ [∂β

(Γ α

μ

) − Γ αβ;μ

]ψβ ∧ dzμ + 1

2

(Γ α

βγ − Γ αγ β

)ψβ ∧ ψγ

}⊗ ∂α (2.8)

and

τ = −[δν

(Γ α

μ

)dzμ ∧ dzν + ∂β

(Γ α

μ

)dzμ ∧ ψβ

] ⊗ ∂α. (2.9)

Proposition 2.2. Let D be a good complex vertical connection with the associated complex non-linear connection coefficients Γ αμ and

the connection 1-forms ωαβ = Γ α

β;μdzμ + Γ αβγ ψγ . Then

Γ αμ = Γ α

β;μvβ. (2.10)

Proof. Since H1,0 = ker D X ι,∀X ∈ T 1,0M . Take X = δμ then

0 = Dδμι = δμ(

vα)∂α + vαΓ

β

α;μ∂β = (−Γ αμ + vβΓ α

β;μ)∂α,

from which yields (2.10). �3. The Chern–Finsler connection

In this section, we shall briefly recall the Chern–Finsler connection. We shall assume from now that M is endowed witha strongly pseudoconvex complex Finsler metric F in the following sense.

Definition 3.1. [1] A strongly pseudoconvex complex Finsler metric F on a complex manifold M is a continuous functionF : T 1,0M → R

+ satisfying

(i) G = F 2 is smooth on M = T 1,0M − {zero section};(ii) F (p, v) > 0 for all (p, v) ∈ M;

(iii) F (p, ζ v) = |ζ |F (p, v) for all (p, v) ∈ T 1,0M and ζ ∈ C;


(iv) the Levi matrix (or complex Hessian matrix)

(Gαβ) =(

∂2G

∂vα∂vβ

)(3.1)

is positively definite on M .

In the following, we denote (G νβ ) the inverse matrix of (Gαν ) such that G νβ Gαν = δβα . We also use the notion in [1],

that is, the derivatives of G with respect to the v-coordinates and z-coordinates are separated by semicolon; for instance,

Gμ;ν = ∂2G

∂zν∂vμ, Gα;ν = ∂2G

∂zν∂vα.

Proposition 3.2. (See [1].) The derivatives of G satisfy the following contraction properties:

Gα vα = G, Gαβ vβ = Gα, Gαβ vβ = 0, vνGμν;β = 0, (3.2)

Gαβ vα vβ = G, Gα;μvα = G;μ, G βαGα = vβ, (3.3)

Gαβγ vγ = −Gαβ, Gαβγ vγ = Gαβ, Gαβγ vγ = 0. (3.4)

Note that condition (iv) in Definition 3.1 implies that F induces a Hermitian metric 〈·, ·〉 in the holomorphic verticalsub-bundles V1,0 over the 2n-dimensional complex manifold M . To see this, it suffice to define

〈∂α, ∂β〉v = Gαβ(z, v). (3.5)

There is a natural way to associate a complex non-linear connection to a given strongly pseudoconvex complex Finslermetric F . If we define

Γ α;μ =: G ναG ν;μ. (3.6)

Then it is easy to check that Γ α;μ satisfy (2.1).

The Chern–Finsler connection D : X (V1,0) → X (T ∗C

M ⊗ V1,0) associated to F was first introduced in [17] and systemi-cally studied in [1]. Essentially, the Chern–Finsler connection associated to F is the Hermitian connection in V1,0 with theHermitian structure 〈·, ·〉. The connection 1-forms ωα

β of D is of type (1,0) and are given by

ωαβ = Γ α

β;μ dzμ + Γ αβμψμ, (3.7)

where

Γ αβ;μ = G ναδμ(Gβν), (3.8)

Γ αβμ = G να ∂μ(Gβν ), (3.9)

and

δμ = ∂μ − Γ α;μ∂α, ψμ = dvμ + Γ

μ;α dzα. (3.10)

Thus D is determined by the triple (Γ α;μ,Γ α

β;μ,Γ αβμ), where Γ α

;μ , Γ αβ;μ and Γ α

βμ are given by (3.6), (3.8) and (3.9),respectively.

Proposition 3.3. (See [1].) Let Γ α;μ,Γ α

β;μ and Γ αβμ be given by (3.6), (3.8) and (3.9), respectively. Then for every nonzero λ ∈ C,

Γ α;μ(z, λv) = λΓ α

;μ(z, v), Γ αβ;μ(z, λv) = Γ α

β;μ(z, v), (3.11)

Γ αβ;μ = ∂β

(Γ α

;μ), Γ α

β;μvβ = Γ α;μ, (3.12)

Γ αβμ = Γ α

μβ, Γ αβμvβ = 0. (3.13)

Proposition 3.4. Let Γ α;μ and Γ α

β;μ be given by (3.6) and (3.8), respectively. Then

ι(Γ α

;μ) = Γ α

;μ, ι(Γ α

;μ) = 0, (3.14)

ι(Γ α

β;μ) = 0, ι

(Γ α

β;μ) = 0. (3.15)


Proof. This follows from (3.11) and the Euler’s theorem of homogeneous functions. �Definition 3.5. (See [1].) Let F be a strongly pseudoconvex complex Finsler metric on a complex manifold M . F is called astrongly Kähler–Finsler metric iff Γ α

μ;β − Γ αβ;μ = 0; called a Kähler–Finsler metric iff (Γ α

μ;β − Γ αβ;μ)vμ = 0; called a weakly

Kähler–Finsler metric iff Gα(Γ αμ;β − Γ α

β;μ)vβ = 0.

Remark 3.6. A strongly Kähler–Finsler metric is obvious a Kähler–Finsler metric. The converse is also true [12].

Note that an important property of D is that it is both horizontal and vertical metrical. The Chern–Finsler connectionwas characterized by the following theorem.

Theorem 3.7. (See [1].) Let F be a strongly pseudoconvex Finsler metric on a complex manifold M, and let 〈·, ·〉 denote the Hermitianstructure on V1,0 induced by F . Then there is a unique complex vertical connection D :X (V1,0) →X (T ∗

CM ⊗ V1,0) such that

(i) D is good.

(ii)

X〈V , W 〉 = 〈D X V , W 〉 + 〈V , D X W 〉, (3.16)

for any X ∈ T 1,0M and V , W ∈X (V1,0).

4. The complex Rund connection

In this section, we shall introduce the complex Rund connection associated to F and give a characterization of it. Thecomplex Rund connection associated to F was first introduced in [21] and were systemically studied on complex Finslermanifolds [7,9] and more general complex Finsler vector bundles [10,11].

Throughout this section, we use D :X (V1,0) →X (T ∗C

M ⊗V1,0) to denote the complex Rund connection associated to F .

Note that D is a complex vertical connection satisfying (2.3) and (2.4), and its connection 1-forms are

ωαβ = Γ α

β;μ dzμ, (4.1)

where Γ αβ;μ are defined by (3.8). Obviously, ωα

β are the horizontal part of ωαβ , which are given by (3.7).

Proposition 4.1. Let D be the complex Rund connection associated to F . Then D is a good complex vertical connection.

Proof. Note that since D is a complex vertical connection, to prove that it is a good complex vertical connection, it issufficient to show that the following map

Λ(X) =: D X ι, ∀X ∈ T 1,0M

is regular for every X ∈ V1,0. In fact, it is easy to check that

Λ(V ) = V , ∀V ∈ V1,0,

which implies that D is a good complex vertical connection. �Proposition 4.2. Denote θ and τ the (2,0)-torsion and (1,1)-torsion of D, respectively. Denote

Ω = Ωαβ ⊗ (

dzβ ⊗ δα + ψβ ⊗ ∂α

)(4.2)

the components of the curvature Ω of D. Then

θ = 1

2

(Γ α

ν;μ − Γ αμ;ν

)dzμ ∧ dzν ⊗ δα, (4.3)

τ = −[δν

(Γ α

;μ)

dzμ ∧ dzν + ∂ν

(Γ α

;μ)

dzμ ∧ ψν] ⊗ ∂α, (4.4)

Ωαβ = −δν

(Γ α

β;μ)

dzμ ∧ dzν − ∂ν

(Γ α

β;μ)

dzμ ∧ ψν − ∂ν

(Γ α

β;μ)

dzμ ∧ ψν. (4.5)

Proof. Equalities (4.3) and (4.4) follows from (2.3.19) in [1] since the horizontal connection coefficients of the complex Rundconnection coincide with the horizontal connection coefficients of the Chern–Finsler connection, and the vertical connectioncoefficients of the complex Rund connection vanish identically.


Equality (4.5) follows from

Ωαβ = dωα

β − ωμβ ∧ ωα

μ

and the following formula:

δν(Γ α

β;μ) − δμ

(Γ α

β;ν) + Γ

γβ;μΓ α

γ ;ν − Γγβ;νΓ α

γ ;μ = 0, (4.6)

which was first proved in [21]. �Proposition 4.3. Let F be a strongly pseudoconvex complex Finsler metric on a complex manifold M, and D be the complex Rundconnection associated to F . Let Ω and τ be the curvature and torsion operators of D. Then for all X, Y ∈ X (H1,0) and V ∈ X (V1,0),we have

Ω(X, Y )ι = τ (X, Y ), Ω(X, V )ι = τ (X, V ), (4.7)

Ω(X, ι)V = 0, Ω(X, V )ι = 0, Ω(X, ι)V = 0, (4.8)⟨Ω(X, V )ι, ι

⟩ = 0. (4.9)

Proof. Let X, Y ∈X (H1,0) and V ∈X (V1,0), then it follows from (4.2) and (4.5) that

Ω(X, Y )ι = −δν(Γ α

β;μ)

XμY ν vβ ∂α, Ω(X, V )ι = −∂ν

(Γ α

β;μ)

XμV ν vβ ∂α. (4.10)

By the second equality in (3.12), we get

δν(Γ α

β;μ)

vβ = δν(Γ α

;μ), ∂ν

(Γ α

β;μ)

vβ = ∂ν

(Γ α

;μ). (4.11)

Equation (4.10) together with (4.11) yield

Ω(X, Y )ι = −δν(Γ α

;μ)

XμY ν ∂α, Ω(X, V )ι = −∂ν

(Γ α

;μ)

XμV ν ∂α.

Note that by (4.4), we have

τ (X, Y ) = −δν(Γ α

;μ)

XμY ν ∂α, τ (X, V ) = −∂ν

(Γ α

;μ)

XμV ν ∂α.

Thus we get (4.7). By the same reason we get (4.8).Next we prove (4.9). By (4.7), we have

⟨Ω(X, V )ι, ι

⟩ = ⟨−∂ν

(Γ α

;μ)

XμV ν ∂α, ι⟩ = −Gαγ ∂ν

(Γ α

;μ)

XμV ν vγ = −Gα∂ν

(Γ α

;μ)

XμV ν .

Since

∂ν

(Γ α

;μ) = ∂ν

(GμαGμ;β

) = −Gμγ G εαGγ εν Gμ;β + GμαGμν;β .

It follows that

Gα∂ν

(Γ α

;β) = Gα

[−Gμγ G εαGγ ενGμ;β + GμαGμν;β]

= −Gμγ vεGγ ενGμ;β + vμGμν;β = 0,

where we used (3.3) and (3.4). Thus we get (4.9) and this completes the proof. �Let 〈〈·, ·〉〉 :H1,0 ×H1,0 → C be the symmetric product defined by

〈〈H, K 〉〉 = Gαβ(v)Hα K β . (4.12)

It satisfies

〈〈H,χ〉〉 = 0, (4.13)

where χ = vμ(∂μ − Γ α;μ∂α) is the complex horizontal radial vector field associated to Γ α

;μ .

Note that the (h, h)-component of the curvature operator of the Chern–Finsler connection plays an important role inthe second variation of the arc length [1]. For the (h, h)-component of the curvature operator Ω of the complex Rundconnection, we have


Proposition 4.4. The (h, h)-component of the curvature operator Ω of D satisfy⟨Ω(X, Y )Z , W

⟩ = −⟨Ω(Y , X)Z , W

⟩, (4.14)⟨

Ω(X, Y )χ, W⟩ = ⟨

Ω(Y , X)W ,χ⟩ + ⟨⟨

Θ−1(τ (Y , X)), W

⟩⟩(4.15)

for any X, Y , Z , W ∈H1,0v .

If furthermore F is a Kähler–Finsler metric, then⟨Ω(χ, Y )X,χ

⟩ = ⟨Ω(X, Y )χ,χ

⟩ = ⟨Ω(X,χ)χ, Y

⟩ + ⟨⟨Θ−1

(τ (χ, X)

), Y

⟩⟩(4.16)

for any X, Y ∈H1,0v .

Proof. Equality (4.14) is trivial. Now we prove (4.15). Let X, Y , Z , W ∈H1,0v , then we have⟨

Ω(X, Y )χ, W⟩ = −Gαγ δν

(Γ α

β;μ)

XμY ν vβ W γ . (4.17)

By (3.8), we have

Gαγ δν(Γ α

β;μ) = −G εαδν (Gαγ )δμ(Gβε) + δν ◦ δμ(Gβγ ) = −δμ(Gβε)Γ

εγ ;ν + δν ◦ δμ(Gβγ ). (4.18)

Conjugating (4.17) and using (4.18), we get⟨Ω(X, Y )χ, W

⟩ = [δμ(Gεβ )Γ ε

γ ;ν − δν ◦ δμ(Gγ β )]

XμY ν vβ W γ . (4.19)

Note that since

δν ◦ δμ − δμ ◦ δν = [δν, δμ] = −δν(Γ α

;μ)∂α + δμ

(Γ α

;ν)∂α,

it follows that

δν ◦ δμ(Gγ β ) = δμ ◦ δν(Gγ β ) − δν(Γ α

;μ)Gγ βα + δμ

(Γ α

;ν)Gγ βα. (4.20)

By (3.8), we have

δμ ◦ δν(Gγ β ) = δμ(GαβΓ α

γ ;ν) = δμ(Gεβ )Γ ε

γ ;ν + Gεβδμ(Γ ε

γ ;ν), (4.21)

this together with (4.20) and (3.4) yield

δν ◦ δμ(Gγ β )XμY ν vβ W γ = [δμ(Gεβ )Γ ε

γ ;ν + Gεβδμ(Γ ε

γ ;ν)]

XμY ν vβ W γ + δμ(Γ α

;ν)Gαγ XμY ν W γ .

Therefore⟨Ω(X, Y )χ, W

⟩ = −Gεβδμ(Γ ε

γ ;ν)

XμY ν vβ W γ − δμ(Γ α

;ν)Gαγ XμY ν W γ

= ⟨Ω(Y , X)W ,χ

⟩ + ⟨⟨Θ−1(τ (Y , X)

), W

⟩⟩.

Next we prove (4.16). Since 〈〈Θ−1(τ (Y , X)),χ 〉〉 = 0 for any X, Y ∈H1,0v , it follows from (4.15) that

⟨Ω(X, Y )χ,χ

⟩ = ⟨Ω(Y , X)χ,χ

⟩. (4.22)

Using the condition that F is a Kähler–Finsler metric, we get⟨Ω(Y , X)χ,χ

⟩ = −Gεβδμ(Γ ε

γ ;ν)

XμY ν vβ vγ

= −Gεβδμ(Γ ε

ν;γ)

XμY ν vβ vγ

= ⟨Ω(χ, X)Y ,χ

⟩, (4.23)

which implies the first equality in (4.16), that is,⟨Ω(χ, Y )X,χ

⟩ = ⟨Ω(X, Y )χ,χ

⟩. (4.24)

Using (4.15) again, we get⟨Ω(χ, X)Y ,χ

⟩ = ⟨Ω(X,χ)χ, Y

⟩ + ⟨⟨Θ−1(τ (X,χ), Y

⟩⟩. (4.25)

By (4.22),(4.23) and (4.25), we get⟨Ω(X, Y )χ,χ

⟩ = ⟨Ω(X,χ)χ, Y

⟩ + ⟨⟨Θ−1(τ (X,χ), Y

⟩⟩. (4.26)

Hence by Eqs. (4.23) and (4.26) we obtain (4.16). This completes the proof. �


Theorem 4.5. Let F be a strongly pseudoconvex complex Finsler metric on a complex manifold M, and let 〈·, ·〉 be the Hermitianstructure on V1,0 induced by F . Then there is a unique complex vertical connection D :X (V1,0) →X (T ∗

CM ⊗ V1,0) such that:

(i) D is good;(ii) for any H ∈ H1,0 and V , W ∈ V1,0 , one has

H〈V , W 〉 = 〈D H V , W 〉 + 〈V , D H W 〉, (4.27)

where H1,0 is the (1,0) part of the complex horizontal bundle HC associated to D;(iii) for any V ∈ V1,0 and H ∈ H1,0 , one has

θ (V , H) = 0, (4.28)

where θ is the (2,0)-torsion of D.

Proof. It is easy to check that the complex Rund connection D associated to F satisfies (i)–(iii).Conversely, we need to show that if a good complex vertical connection D satisfying (ii)–(iii), then it is the complex

Rund connection D associated to F .Define H1,0 =: ker D X ι for X ∈ T 1,0M . Then HC = H1,0 ⊕ H1,0 is the complex horizontal bundle associated to D , since

D is a good complex vertical connection. Now we denote Γ α;μ the complex non-linear connection coefficients associated

to H1,0, and define

δμ =: ∂μ − Γ α;μ∂α, ψμ =: dvμ + Γ

μ;α dzα.

Then locally there are functions Γ αβ;μ and Γ α

βμ defined on M such that the connection 1-forms ωαβ of D can be expressed as

ωαβ = Γ α


since D is a good complex vertical connection.Next we shall determine the connection coefficients Γ α


βμ .

By condition (4.27), if we take H = δμ ∈ H1,0, V = ∂β and W = ∂ν , then we have

δμ〈∂β , ∂ν〉 = 〈Dδμ

∂β , ∂ν〉 + 〈∂β , Dδμ

∂ν〉.Since D

δμ∂β = Γ α

β;μ∂α and Dδμ

∂ν = 0, it follows that

δμ(Gβν ) = Γ αβ;μGαν ,

or equivalently,

Γ αβ;μ = G να δμ(Gβν) = G να

(Gβν;μ − Γ

γ;μGβνγ

). (4.30)

By Proposition 2.2, we have

Γ α;μ = Γ α

β;μvβ,

since D is a good complex vertical connection. This together with (4.30) and (3.4) yield

Γ α;μ = G να

(Gβν;μ − Γ

γ;μGβνγ

)vβ = G ναG ν;μ = Γ α

;μ, (4.31)

where Γ α;μ is given by (3.6). Therefore

δμ = δμ, ψμ = ψμ,

where δμ and ψμ are given by (3.10). Substituting δμ = δμ in (4.30), we get

Γ αβ;μ = Γ α

β;μ, (4.32)

where Γ αβ;μ are given by (3.8).

Now we are in a position to determine Γ αβμ . Since by (4.31), (4.32) and the general formula (2.8) for the (2,0)-torsion

of a good complex vertical connection, we have


θ =[

1

2

(Γ α


)dzμ ∧ dzβ + Γ α

βμψμ ∧ dzβ

]⊗ δα + 1

2

(Γ α

βμ − Γ αμβ

)ψβ ∧ ψμ ⊗ ∂α,

where we used the first equality in (3.12), and the formula [δμ, δβ ] = 0 in Lemma 2.3.3 in [1], or equivalently, δβ(Γ α;μ) −

δμ(Γ α;β) = 0. Thus θ (V , H) = 0 for all V ∈ V1,0 and H ∈ H1,0 if and only if Γ α

βμ ≡ 0. So that ωαβ = Γ

μβ;μdzμ , which implies

that D is in deed the complex Rund D associated to F . �Corollary 4.6. Let H1,0 ,Γ α

;μ and Θ be the complex horizontal bundle, complex non-linear connection coefficients and complex hori-

zontal map associated to the Chern–Finsler connection D, respectively. Let H1,0 , Γ α;μ and Θ be the complex horizontal bundle, complex

non-linear connection coefficients and complex horizontal map associated to the complex Rund connection D, respectively. Then

H1,0 = H1,0, Γ α;μ = Γ α

;μ, Θ = Θ. (4.33)

Proof. This follows from Proposition 2.1, Theorem 3.7 and Theorem 4.5. �Using the Hermitian structure 〈·, ·〉 in V1,0, which is naturally induced by F , we can obtain a Hermitian structure 〈·, ·〉c in

H1,0, just by setting 〈H, K 〉c := 〈Θ−1(H),Θ−1(K )〉 for H, K ∈ H1,0, and asking H1,0 to be orthogonal to V1,0 with respectto 〈·, ·〉c .

Similarly, we can obtain a Hermitian structure 〈·, ·〉r in H1,0, just by setting 〈H, K 〉r := 〈Θ−1(H), Θ−1(K )〉 for H,

K ∈ H1,0, and asking H1,0 to be orthogonal to V1,0 with respect to 〈·, ·〉r .

Corollary 4.7. Let 〈·, ·〉c and 〈·, ·〉r be the Hermitian structures in H1,0 and H1,0 , respectively. Then

〈·, ·〉c = 〈·, ·〉r, 〈δα, δβ〉c = Gαβ . (4.34)

Proof. It follows from Corollary 4.6. �Formula (4.27) shows that the complex Rund connection is horizontal metrical. In addition to this, the complex Rund

connection enjoys some weak metric compatibility. Indeed it is easy to check that

Proposition 4.8. Let D be the complex Rund connection associated to F . Then

U 〈V , W 〉 = 〈DU V , W 〉 + 〈V , DU W 〉 + GαβμUα V β W μ, (4.35)

for any complex vertical vector fields U , V , W ∈ V1,0 . Especially, we have

ι〈V , W 〉 = 〈DιV , W 〉 + 〈V , D ιW 〉, ∀V , W ∈ V1,0

and

X〈ι, W 〉 = 〈D X ι, W 〉 + 〈ι, D X W 〉, ∀X ∈ T 1,0M, W ∈ V1,0.

Corollary 4.9. Let D be a complex Rund connection associated to F . Then D is vertical metrical if and only if F is a Hermitian metricon M.

Proof. This follows immediately from (4.35) since Gαβμ ≡ 0 if and only if F comes from a Hermitian metric on M . �5. Complex Berwald connection

In this section, we shall introduce the complex Berwald connection associated to F and give a characterization of it. Firstwe shall introduce another complex non-linear connection, which is obtained from Γ α

;μ . This complex non-linear connectionis different from Γ α , however, they share the same complex horizontal radial vector fields.
;μ


Define

Gα =: 1

2Γ α

;μvμ (5.1)

and

Gαμ =: ∂μ

(G

α), G

αβμ = ∂β

(G

αμ

). (5.2)

Then it is easy to check that Gαμ satisfy (2.1). So that Gα

μ are connection coefficients of a complex non-linear connectionassociated to F . In general, however, Γ α

;μ �=Gαμ .

Proposition 5.1. Let Gαμ and G

αβμ be given by (5.2). Then for every nonzero λ ∈C,

Gαμ(z, λv) = λGα

μ(z, v), Gαβμ(z, λv) = G

αβμ(z, v), (5.3)

ι(G

αμ

) = Gαμ, ι

(G

αμ

) = 0, (5.4)

ι(G

αβμ

) = 0, ι(G

αβμ

) = 0. (5.5)

Proof. This follows from (5.1), (5.2), Proposition 3.3 and Proposition 3.4. �Setting

δμ = ∂μ −Gαμ∂α, ψμ = dvμ +G

μα dzα (5.6)

and

χ = vμδμ. (5.7)

Proposition 5.2. Let χ and χ be the complex radial horizontal vector fields associated to Γ α;μ and G

αμ , respectively. Then

χ = χ , or equivalently Γ α;μvμ = G

αμvμ. (5.8)

Complex Berwald connection was first introduced in [19], and was recently used to study the holomorphic subspace ofcomplex Finsler spaces [20]. Let D : X (V1,0) → X (T ∗

CM ⊗ V1,0) be the complex Berwald connection associated to F . It is a

complex vertical connection satisfies (2.3) and (2.4) with the connection 1-forms ωαβ given by

ωαβ = G

αβμ dzμ, (5.9)

where Gαβμ is given by (5.2).

In general, D is neither horizontal metrical nor vertical metrical. The horizontal connection coefficients Gαβμ of D , how-

ever, are symmetric with respect to the lower indexes.

Proposition 5.3. Let D be the complex Berwald connection associated to F . Then D is a good complex vertical connection.

Proof. The proof follows the same line as that of Proposition 4.1. �Definition 5.4. (See [7].) Let F be a strongly pseudoconvex complex Finsler metric on M . If locally the connection coefficientsΓ α

β;μ(z, v) of the associated Chern–Finsler connection are independent of fibre coordinates v , then F is called a complexBerwald metric.

Definition 5.5. (See [25].) Let F be a strongly pseudoconvex complex Finsler metric on M . If locally the connection coeffi-cients G

αβμ(z, v) of the associated complex Berwald connection are independent of fibre coordinates v , then F is called a

weakly complex Berwald metric.

Remark 5.6. Definition 5.4 and Definition 5.5 are independent of the choice of local holomorphic coordinates z on M andthe induced local holomorphic coordinates (z, v) on M .

It’s clear that if Γ αβ;μ are independent of v , then G

αβμ are also independent of v , the converse, however, is not true.

A fundamental example is the complex Wrona metric [25].


Proposition 5.7. Let θ and τ be the (2,0)-torsion and (1,1)-torsion of D, respectively. Let Ω = Ωαβ ⊗ (dzβ ⊗ δα + ψβ ⊗ ∂α) be the

components of the curvature Ω of D. Then

θ = 1

2

[δβ

(G

αμ

) − δμ(G

αβ

)]dzβ ∧ dzμ ⊗ ∂α, (5.10)

τ = −[δβ

(G

αμ

)dzμ ∧ dzβ + ∂β

(G

αμ

)dzμ ∧ ψβ

] ⊗ ∂α, (5.11)

and

Ωαβ = 1

2

[δν

(G

αβμ

) − δμ(G

αβν

) +GγβμG

αγ ν −G

γβνG

αγμ

]dzν ∧ dzμ

− δν(G

αβμ

)dzμ ∧ dzν − ∂ν

(G

αβμ

)dzμ ∧ ψν − ∂ν

(G

αβμ

)dzμ ∧ ψν . (5.12)

Proposition 5.8. Let F be a strongly pseudoconvex Finsler metric on a complex manifold M, and D be the complex Berwald connectionassociated to F . Let Ω and τ be the curvature and torsion operators of D. Then for all X, Y ∈X (H1,0) and U , V ∈X (V1,0), we have

Ω(·, ·)ι = τ , (5.13)

Ω(X, Y )ι = θ (X, Y ), (5.14)

Ω(χ, U )X = τ (X, U ), (5.15)

Ω(X, ι)U = Ω(X, U )ι = Ω(X, ι)U = Ω(χ, U )V = 0. (5.16)

Proposition 5.9. Let F be a strongly pseudoconvex Finsler metric on a complex manifold M, and D be the complex Berwald connectionassociated to F . Let Ω and τ be the curvature and torsion operators of D . Then for all X, Y , Z , W ∈ X (H1,0), we have the followingfirst and second Bianchi identities:

Ω(X, Y )Z + Ω(Y , Z)X + Ω(Z , X)Y = 0, (5.17)

(D XΩ)(Y , Z)W + (DY Ω)(Z , X)W + (D Z Ω)(X, Y )W

= −Ω(τ (X, Y ), Z

)W − Ω

(τ (Y , Z), X

)W − Ω

(τ (Z , X), Y

)W . (5.18)




χ = χ ; (5.19)

(iii) for any V , W ∈ V1,0 and H, U ∈ H1,0 , one has

θ (H, U ) ∈ V1,0, (5.20)

θ (V , H) = 0, (5.21)

θ (V , W ) = 0, (5.22)


Proof. It is easy to check that the complex Berwald connection D associated to F satisfies (i)–(iii).Conversely, we need to show that if a good complex vertical connection D satisfying (ii)–(iii), then it is the complex

Berwald connection associated to F .Define H1,0 =: ker D X ι for X ∈ T 1,0M . Then HC = H1,0 ⊕ H1,0 is a complex horizontal bundle associated to D , since D

is a good complex vertical connection. Now we denote Γ α;μ the connection coefficients of the complex non-linear connection

associated to H1,0, and define

δμ = ∂μ − Γ α;μ∂α, ψμ = dvμ + Γ

μ;α dzα. (5.23)

Then locally there are functions Γ α and Γ α defined on M such that the connection 1-forms ωα are given by
β;μ βμ β


ωαβ = Γ α

β;μ dzμ + Γ αβμψμ,

since D is a good complex vertical connection. Next we are need to determine the connection coefficients Γ α;μ , Γ α

β;μand Γ α

βμ .

Replacing θ , Γ αμ ,Γ α

β;μ , Γ αβμ with θ , Γ α

;μ , Γ αβ;μ , Γ α

βμ , respectively in (2.8), then condition (5.20) is equivalent to

Γ αβ;μ = Γ α

μ;β . (5.24)

Condition (5.21) is equivalent to

Γ αβμ = 0 (5.25)

and

Γ αβ;μ = ∂β

(Γ α

;μ). (5.26)

Condition (5.22) is equivalent to

Γ αβμ = Γ α

μβ. (5.27)

Since D is a good complex vertical connection, it follows from (2.10), (5.24) and (5.26) that

Γ α;μ = Γ α

β;μvβ = Γ αμ;β vβ = ∂μ

(Γ α

;β)

vβ = ∂μ

(Γ α

;β vβ) − Γ α

;μ,

from which we get

2Γ α;μ = ∂μ

(Γ α

;β vβ). (5.28)

Since by condition (ii), we have

Γ α;β vβ = Γ α

;β vβ, (5.29)

this together with (5.28) yield

Γ α;μ = 1

2∂μ

(Γ α

;β vβ) = ∂μ

(G

α) = G

αμ

and

Γ αβ;μ = ∂β

(Γ α

μ

) = ∂β

(G

αμ

) = Gαβμ. (5.30)

This completes the proof. �Using the Hermitian structure 〈·, ·〉 in V1,0, which is naturally induced by F , we can obtain a Hermitian structure 〈·, ·〉b in

H1,0, just by setting 〈H, K 〉b := 〈Θ−1(H), Θ−1(K )〉 for H, K ∈ H1,0, and asking H1,0 to be orthogonal to V1,0 with respectto 〈·, ·〉b .

In general the complex Berwald connection associated to F is neither horizontal metrical nor vertical metrical. Thefollowing proposition shows that it enjoys some weak metric compatibility.

Proposition 5.11. Let D be the complex Berwald connection associated to F . Then

ι〈ι, W 〉b = 〈∇ιι, W 〉b + 〈ι,∇ιW 〉b, ∀W ∈ V1,0.

If F is a weakly Kähler metric on M then

χ〈V , ι〉b = 〈∇χ V , ι〉b + 〈V ,∇χι〉b, ∀V ∈ V1,0.

Corollary 5.12. (See [25].) Let D be the complex Berwald connection associated to F . Then

H〈ι, ι〉 = 0, ∀H ∈ H1,0

if and only if F is a weakly Kähler–Finsler metric.


6. The complex Hashiguchi connection

In this section, we shall introduce the complex Hashiguchi connection associated to F and give a characterization ofit. The complex Hashiguchi connection is a complex analogue of the Hashiguchi connection in real Finsler geometry [18].We introduce it here for completeness.

The complex Hashiguchi connection D : X (V1,0) → X (T ∗C

M ⊗ V1,0) is a complex vertical connection satisfying (2.3)and (2.4). Its connection 1-forms ωα

β are given by

ωαβ = G

αβμ dzμ + Γ α

βμψμ, (6.1)

where Gαβμ are given by (5.2), Γ α

βμ are given by (3.9) and ψμ are given by (5.6).Note that both the horizontal and vertical connection coefficients of the complex Hashiguchi connection are symmetric

with respect to their lower indexes. It is also easy to check that D is vertical metrical.

Proposition 6.1. Let D be the complex Hushiguchi connection associated to F . Then D is a good complex vertical connection.

Proof. Note that the vertical connection coefficients Γ αβμ of the complex Hashiguchi connection satisfy (3.13). Thus the

proof follows the same line as that of Proposition 4.1. �Proposition 6.2. Let θ and τ be the (2,0)-torsion and (1,1)-torsion of D, respectively. Let Ω = Ωα

β ⊗ (dzβ ⊗ δα + ψβ ⊗ ∂α) be the

components of the curvature Ω of D. Then

θ = Γ αβμψμ ∧ dzβ ⊗ δα + 1

2

[δβ

(G

αμ

) − δμ(G

αβ

)]dzβ ∧ dzμ ⊗ ∂α, (6.2)

τ = −[δβ

(G

αμ

)dzμ ∧ dzβ + ∂β

(G

αμ

)dzμ ∧ ψβ

] ⊗ ∂α, (6.3)

and

Ωαβ = 1

2

[δν

(G

αβμ

) − δμ(G

αβν

) +GγβμG

αγ ν −G

γβνG

αγμ

]dzν ∧ dzμ − δν

(G

αβμ

)dzμ ∧ dzν

− [∂ν

(G

αβμ

) − δμ(Γ α

βν

) +GγβμΓ α

γ ν −GαγμΓ

γβν

]dzμ ∧ ψν + [

∂ν

(Γ α

βμ

) − ΓγβνΓ α

γμ

]ψν ∧ ψμ

− δν(Γ α

βμ

)ψμ ∧ dzν − ∂ν

(G

αβμ

)dzμ ∧ ψν − ∂ν

(Γ α

βμ

)ψμ ∧ ψν . (6.4)

Note that the complex Berwald connection D and the complex Hashiguchi connection D share the same complex hori-zontal bundle with the associated complex non-linear connection coefficients G

αμ given by (5.2). Since

Gαμ = 1

2∂μ

(Γ α

;β vβ) = 1

2

(Γ α

μ;β + Γ αβ;μ

)vβ.

It follows that Gαμ = Γ α

;μ if and only if Γ αβ;μvβ = Γ α

μ;β vβ , that is, F is a Kähler–Finsler metric [12]. In this case, the complexBerwald connection coincides with the complex Rund connection, and the complex Hashiguchi connection coincides withthe Chern–Finsler connection.




χ = χ ; (6.5)

(iii) for U , V , W ∈X (V1,0),

U 〈V , W 〉 = 〈DU V , W 〉 + 〈V , DU W 〉; (6.6)

(iv) for V , W ∈ V1,0 and H, U ∈ H1,0 , one has

θ (H, U ) ∈ V1,0, (6.7)

θ (V , H) ∈ H1,0, (6.8)



Proof. It is easy to check that the complex Hashiguchi connection D associated to F satisfies (i)–(iv).Conversely, we need to show that if D is a good complex vertical connection satisfying (ii)–(iv), then it is the complex

Hashiguchi connection associated to F .

Define H1,0 =: ker D X ι for X ∈ T 1,0M . Then HC = H1,0 ⊕ H1,0 is a complex horizontal bundle associated to D , since Dis a good complex vertical connection.

Denote Γ α;μ the connection coefficients of the complex non-linear connection associated to H1,0, and define

δk = ∂k − Γ α;μ∂α, ψμ = dvμ + Γ

μ;α dzα. (6.9)

Then locally there are functions Γ αβ;μ and Γ α

βμ such that the connection 1-forms ωαβ of D are given by

ωαβ = Γ α


since D is a good complex vertical connection.Next we are need to determined the connection coefficients Γ α


βμ .

Substituting U = ∂μ, V = ∂β , W = ∂ν in (6.6), we get D V W = Γ αβν ∂α, D V W = 0. Thus (6.6) implies that

∂μ(Gβν ) = Γ αβμGαν , (6.11)

from which we get

Γ αβμ = G να ∂μ(Gβν ) = Γ α

βμ, (6.12)

where Γ αβμ are given by (3.9).

Replacing θ,Γ αμ ,Γ α

β;μ,Γ αβμ with θ , Γ α

β;μ, Γ αβμ , respectively in (2.8), then condition (6.7) implies that

Γ αβ;μ = Γ α

μ;β . (6.13)

Condition (6.8) implies that

Γ αβ;μ = ∂β

(Γ α

;μ). (6.14)

Now it follows from Proposition 2.2, (6.13) and (6.14) that

Γ α;μ = Γ α

β;μvβ = Γ αμ;β vβ = ∂μ

(Γ α

;β)

vβ = ∂μ

(Γ α

;β vβ) − Γ α

;μ,

from which we get

2Γ α;μ = ∂μ

(Γ α

;β vβ). (6.15)

Since by condition (ii), we have

Γ α;β vβ = Γ α

;β vβ, (6.16)

this together with (6.15) yield

Γ α;μ = 1

2∂μ

(Γ α

;β vβ) = ∂μ

(G

α) = G

αμ, (6.17)

where Gαμ are given by (5.2). So that

Γ αβ;μ = ∂β

(Γ α

;μ) = ∂β

(G

αμ

) = Gαβμ, (6.18)

where Gαβμ are given by (5.2). This completes the proof. �

Corollary 6.4. Let H1,0 ,Γ α;μ and Θ be the complex horizontal bundle, complex non-linear connection coefficients and complex horizon-

tal map associated to the complex Berwald connection D, respectively. Let H1,0 , Γ α;μ and Θ be the complex horizontal bundle, complex

non-linear connection coefficients and complex horizontal map associated to the complex Hashiguchi connection D, respectively. Then

H1,0 = H1,0, Γ α;μ = Γ α

;μ, Θ = Θ. (6.19)


Proof. This follows from Proposition 2.1, Theorem 5.10 and Theorem 6.3. �Now we can obtain a Hermitian structure 〈·, ·〉h in H1,0, just by setting 〈H, K 〉h := 〈Θ−1(H), Θ−1(K )〉 for H, K ∈ H1,0,

and asking H1,0 to be orthogonal to V1,0 with respect to 〈·, ·〉h .

Corollary 6.5. Let 〈·, ·〉b and 〈·, ·〉h be the Hermitian structures in H1,0 and H1,0 , respectively. Then

〈·, ·〉b = 〈·, ·〉h, 〈δα, δβ〉b = Gαβ . (6.20)

Proof. It follows from Corollary 6.4. �7. Holomorphic curvatures of complex Finsler connections

In this section, we shall introduce (from differential geometric point of view) a new definition of holomorphic sectionalcurvature, holomorphic bisectional curvature and Ricci scalar curvature with respect to a general good complex verticalconnection associated to a strongly pseudoconvex complex Finsler metric. We obtain the precise relationship among thesecurvatures with respect to the above mentioned complex Finsler connections.

Let M be a complex manifold endowed with a strongly pseudoconvex complex Finsler metric F . In literature, the holo-morphic sectional curvature, holomorphic bisectional curvature and Ricci scalar curvature of F were defined in terms of theChern–Finsler connection D associated to F . For example, the definition of holomorphic sectional curvature [1], holomorphicsectional and bisectional curvatures in [14] and Ricci scalar curvature in [19] were all defined in terms of the Chern–Finslerconnection associated to F .

Since there are several complex Finsler connections associated to F and they are all good complex vertical connections.One may ask what will happen if those curvatures are defined in terms of other complex Finsler connections associatedto F .

In this section, we shall introduce a definition of holomorphic sectional curvature, holomorphic bisectional curvatureand Ricci scalar curvature with respect to an arbitrary good complex vertical connection D associated to F . This definitioncoincides with those given in [1,14] and [19] when D is the Chern–Finsler connection associated to F .

Let H1,0 be the complex horizontal bundle associated to D, or equivalently, Γ αμ be the complex non-linear connection

coefficients associated to D. Let χDv : T 1,0

p M →H1,0v be a horizontal lift map such that

χDv (w) = wμ

[∂μ − Γ α

μ (v)∂α

], (7.1)

for every nonzero vector w ∈ T 1,0p M . The vector field χ =: χD

v (v) = vμ[∂μ −Γ αμ (v)∂α] is called the complex radial horizon-

tal vector field associated to D. Denote 〈·, ·〉 the Hermitian inner product in V1,0 which is naturally induced by F , we alsouse the same symbol 〈·, ·〉 to denote the Hermitian inner product in H1,0, which is transferred from V1,0 via the complexhorizontal map Θ : V1,0 →H1,0 associated to H1,0. In addition, we denote Ω the curvature operator associated to D.

Definition 7.1. Let M be a complex manifold endowed with a strongly pseudoconvex complex Finsler metric F , and D is anarbitrary good complex vertical connection associated to F .

(i) The holomorphic sectional curvature K F (v) of F along a nonzero vector v ∈ T 1,0p M is given by

K F (v) = K F(χ(v)

) = 2

G2(v)

⟨Ω(χ,χ)χ,χ

⟩v . (7.2)

(ii) The holomorphic bisectional curvature B F (v, w) of F along nonzero vectors v, w ∈ T 1,0p M is given by

B F (v, w) = 2

G(v)G(w)

⟨Ω

(χD

v (w),χDv (w)

)χD

v (v),χDv (v)

⟩v . (7.3)

(iii) The Ricci scalar curvature R F (v) of F along a nonzero vector v ∈ T 1,0p M is given by

R F (v) = −χ(Γ

μμ

). (7.4)

Denote D , D , D and D the Chern–Finsler connection, complex Rund connection, complex Berwald connection and com-plex Hashiguchi connection associated to F , respectively.

Denote K F (v), K F (v), K F (v) and K F (v) the holomorphic sectional curvatures of F along a nonzero vector v ∈ T 1,0p M

with respect to D , D , D and D , respectively.Denote B F (v, w), B F (v, w), B F (v, w) and B F (v, w) the holomorphic bisectional curvatures of F along nonzero vectors

v, w ∈ T 1,0p M with respect to D , D , D and D , respectively.


Denote R F (v), R F (v), R F (v) and R F (v) the Ricci scalar curvatures of F along a nonzero vector v ∈ T 1,0p M with respect

to D , D , D and D , respectively.

Theorem 7.2. Let M be a complex manifold endowed with a strongly pseudoconvex complex Finsler metric F . Then

(i) the holomorphic sectional curvature of F with respect to D, D, D and D are given by

K F (v) = K F (v) = K F (v) = K F (v); (7.5)

(ii) the holomorphic bisectional curvature of F with respect to D, D, D and D are given by

B F (v, w) = B F (v, w), B F (v, w) = B F (v, w), (7.6)

B F (v, w) = B F (v, w) + 1

G(v)G(w)Gαδν

(Γ α


)vβ wμwν

− 1

2

1

G(v)G(w)Gα

(Γ

γ

λ;ν − Γγ

ν;λ)

vλ∂γ

(Γ α

β;μ + Γ αμ;β

)vβ wμwν; (7.7)

(iii) the Ricci scalar curvature of F with respect to D, D, D and D are given by

R F (v) = R F (v), R F (v) = R F (v), R F (v) = R F (v) + 1

2χ

(Γ

μβ;μ − Γ

μμ;β

)vβ. (7.8)

Proof.

(i) the holomorphic sectional curvature of F .Denote by Ω , Ω , Ω and Ω the curvature operators of D , D , D and D , respectively.First note that [1]

⟨Ω(χ,χ)χ,χ

⟩c = −Gαδν

(Γ α

;μ)

vμvν = −Gαχ(Γ α

;μ)

vμ. (7.9)

By (4.2),(4.5), and the second equality in (3.12), we get

Ω(χ,χ)χ = −δν(Γ α

;μ)

vμ vνδα. (7.10)

By Corollary 4.6 and Corollary 4.7, we have

⟨Ω(χ,χ)χ,χ

⟩r = −Gαβδν

(Γ α

;μ)

vμvν vβ = −Gαχ(Γ α

;μ)

vμ.

Thus K F (v) = K F (v).By Theorem 5.10 and Proposition 5.7, we have

Ω(χ , χ )χ = −χ(Γ α

;μ)

vμδα. (7.11)

By Theorem 6.3 and Proposition 6.2, we have

Ω(χ , χ )χ = −χ(Γ α

;μ)

vμδα. (7.12)

Eqs. (7.11) and (7.12) together with Corollary 6.4 and Corollary 6.5 yield

⟨Ω(χ , χ )χ , χ

⟩b = ⟨

Ω(χ,χ)χ,χ⟩h = −Gαχ

(Γ α

;μ)

vμ,

which implies (7.5);(ii) the holomorphic bisectional curvature of F .

Since χ Dv (w), χ D

v (v) are horizontal vectors, it follows from Corollary 2.3.7 in [1] that

Ω(χ D

v (w),χ Dv (w)

)χ D

v (v) = Rαβ;μν wμwν vβδα,

where

Rαβ;μν = −δν

(Γ α

β;μ) − Γ α

βσ δν(Γ σ

;μ),

so that

⟨Ω

(χ D

v (w),χ Dv (w)

)χ D

v (v),χ Dv (v)

⟩ = Gαγ Rα wμwν vβ vγ ,
c β;μν


where we used the equality

〈δα, δγ 〉c =: ⟨Θ−1(δα),Θ−1(δγ )⟩ = 〈∂α, ∂γ 〉 = Gαγ .

Since

Γ αβσ vβ = 0, Gαγ vγ = Gα, Γ α

β;μvβ = Γ α;μ,

it follows that⟨Ω

(χ D

v (w),χ Dv (w)

)χ D

v (v),χ Dv (v)

⟩c = −Gαδν

(Γ α

;μ)

wμwν .

Next by Proposition 4.2, we have

Ω(χ D

v (w),χ Dv (w)

)χ D

v (v) = −δν(Γ α

β;μ)

wμwν vβδα = −δν(Γ α

;μ)

wμwνδα.

Thus it follows from Corollary 4.6 and Corollary 4.7 that

⟨Ω

(χ D

v (w),χ Dv (w)

)χ D

v (v),χ Dv (v)

⟩r(v) = −Gαδν

(Γ α

;μ)

wμwν .

Therefore B F (v, w) = B F (v, w) for nonzero vectors v, w ∈ T 1,0p M .

Note that

χ Dv (v) = vμ

[∂μ −G

αμ(v)∂α

] = χ Dv (v), χ D

v (w) = wμ[∂μ −G

αμ(v)∂α

] = χ Dv (w).

By Proposition 5.7 and Proposition 6.2, the horizontal (1,1)-part of Ω and Ω are the same, i.e.,

Ω(h,h) = Ω(h,h) = −δν(G

αβμ

)dzμ ∧ dzν ⊗ dzβ ⊗ δα.

Thus Corollary 6.4 and Corollary 6.5 imply B F (v, w) = B F (v, w) for all nonzero vectors v, w ∈ T 1,0p M . More precisely,

it is easy to check that

Ω(χ D

v (w),χ Dv (w)

)χ D

v (v) = −δν(G

αβμ

)wμwν vβ δα.

Since

〈δα, δβ〉b =: ⟨Θ−1(∂α), Θ−1(∂β)⟩ = ⟨

Θ−1(∂α), Θ−1(∂β)⟩ = Gαβ .

It follows that⟨Ω

(χ D

v (w),χ Dv (w)

)χ D

v (v),χ Dv (v)

⟩b = −Gαδν

(G

αμ

)wμwν .

Next, we derive the relationship between B F (v, w) and B F (v, w).Since G

α = 12 Γ α

;μvμ , it follows that

Gαμ = 1

2

(Γ α

;μ + Γ αμ;β vβ

). (7.13)

Thus

δν = δν + 1

2

(Γ

γ

λ;ν − Γγ

ν;λ)

vλ∂γ .

So that

−Gαδν(G

αμ

)wμwν = −1

2Gαδν

(Γ α

;μ)

wμwν − 1

2Gαδν

(Γ α

μ;β vβ)

wμwν

− 1

4Gα

(Γ

γ

λ;ν − Γγ

ν;λ)

vλ∂γ

(Γ α

β;μ + Γ αμ;β

)vβ wμwν

= −Gαδν(Γ α

;μ)

wμwν + 1

2Gαδν

(Γ α


)vβ wμwν

− 1

4Gα

(Γ

γ

λ;ν − Γγ

ν;λ)

vλ∂γ

(Γ α

β;μ + Γ αμ;β

)vβ wμwν .

Therefore,

B F (v, w) = B F (v, w) + 1

G(v)G(w)Gαδν

(Γ α


)vβ wμwν

− 1

2

1

G(v)G(w)Gα

(Γ

γ

λ;ν − Γγ

ν;λ)

vλ∂γ

(Γ α

β;μ + Γ αμ;β

)vβ wμwν;


(iii) the Ricci scalar curvature of F .Since the complex horizontal radial vector field associated to D , D , D and D are the same. By Definition 7.1, we have

R F (v) = R F (v) = −χ(Γ

μ;μ

), R F (v) = R F (v) = −χ

(G

μμ

),

where μ is summed from 1 to n.Now by (7.13), we get

R F (v) = −1

2χ

(Γ

μ;μ + Γ

μμ;β vβ

) = R F (v) + 1

2χ

(Γ

μβ;μ − Γ

μμ;β

)vβ. �

Corollary 7.3. Let M be a complex manifold endowed with a Kähler–Finsler metric F . Then

K F (v) = K F (v) = K F (v) = K F (v), (7.14)

B F (v, w) = B F (v, w) = B F (v, w) = B F (v, w), (7.15)

R F (v) = R F (v) = R F (v) = R F (v). (7.16)

8. Examples of weakly complex Berwald metrics

In this section, we shall investigate some conformal properties of complex Berwald metric and weakly complex Berwaldmetric, and show that actually there are lots of weakly complex Berwald metrics in the sense of [25]. Moreover, the preciserelationship of the holomorphic sectional curvature and Ricci scalar curvature of F and its conformal change F = eσ F areobtained.

It was also shown in [25] that the complex Wrona metric is a weakly complex Berwald metric. This assertion was provedby showing that G

α = 12 Γ α

;μvμ ≡ 0 while Γ α;μ do not vanish identically. It was also prove that the holomorphic sectional

curvature and the Ricci scalar curvature of the complex Wrona metric vanishing identically.One may wonder whether there exists a weakly complex Berwald metric with G

αβμ not vanishing identically? As is

well known, complex Berwald metrics can be obtained from complex (α,β)-metrics [7]. They can also be obtained via theconformal change of a complex Minkowski metric [8]. In the following, we shall show that this method is also effective toconstruct weakly complex Berwald metrics.

Definition 8.1. (See [8].) Let F be a strongly pseudoconvex complex Finsler metric on M . A conformal change of F is thechange F → F = eσ(z) F for a smooth real valued function σ on M .

Theorem 8.2. Let F be a strongly pseudoconvex complex metric on M, and F = eσ(z) F be a conformal change of F . Then

(i) F is a complex Berwald metric if and only if F is a complex Berwald metric;(ii) the holomorphic sectional curvature K F (v) of F and the holomorphic sectional curvature K F (v) of F are related by

K F (v) = e−2σ

[K F (v) − 4

G

∂2σ

∂zμ∂zνvμvν

]; (8.1)


R F (v) = R F (v) − (n + 1)∂2σ

∂zμ∂vνvμvν . (8.2)

Proof.

(i) This assertion is due to [8].(ii) Denote Γ α

;μ and Γ α;μ the complex non-linear connection coefficients of the Chern–Finsler connections associated to F

and F , respectively. Then it is easy to check that

Γ α;μ = 2

∂σ

∂zμvα + Γ α

;μ. (8.3)

Note that the holomorphic sectional curvature K F (v) of F at v ∈ M is given by

K F (v) = − 2

G2(v)Gαχ

(Γ α

;μ)

vμ.

Let χ be the complex radial horizontal vector field associated to F . Then


χ = vμ(∂μ − Γ α

;μ∂α

) = χ − 2∂σ

∂zμvμι,

where χ = vμ(∂μ − Γ α;μ) and ι = vα∂α . Thus the holomorphic sectional curvature K F (v) of F at v ∈ M is given by

K F (v) = − 2

G2(v)Gαχ

(Γ α

;μ)

vμ = − 2

G2(v)Gα

(χ − 2

∂σ

∂zμvμι

)(2

∂σ

∂zμvα + Γ α

;μ)

vμ.

By Euler’s theorem of homogeneous function, we have

ι

(2

∂σ

∂zμvα + Γ α

;μ)

= 0. (8.4)

So that

K F (v) = − 2

G2(v)Gαχ

(2

∂σ

∂zμvα + Γ α

;μ)

vμ

= − 2

G2(v)Gαχ

(2

∂σ

∂zμ

)vα vμ − 2

G2(v)Gαχ

(Γ α

;μ)

vμ

= − 4

G

∂2σ

∂zμ∂zνvμvν + e−2σ K F (v).

(iii) By (8.3) we get

Gαμ = ∂σ

∂zμvα + ∂σ

∂zνvνδα

μ +Gαμ,

where Gαμ and G

αμ are complex non-linear connection coefficients associated to the complex Berwald connections of

F and F , respectively. Thus the Ricci scalar curvature R F (v) of F is given by

R F (v) = −χ(G

μ;μ

)

= −(χ − 2

∂σ

∂zμvμι

)[(n + 1)

∂σ

∂zμvμ +G

μμ

]

= −(n + 1)∂2σ

∂zμ∂vνvμvν + R F (v). �

Remark 8.3. In Theorem 8.2, F is not necessary a complex Minkowski metric on M .

Corollary 8.4. Let F be a complex Minkowski metric on M, and F = eσ(z) F be a conformal change of F . Then

(i) F is a complex Berwald metric on M;(ii) the holomorphic sectional curvature K F (v) of F is given by

K F (v) = − 4

G

∂2σ

∂zμ∂zνvμvν; (8.5)

(iii) the Ricci scalar curvature R F (v) of F is given by

R F (v) = −(n + 1)∂2σ

∂zμ∂vνvμvν . (8.6)

Proof. This follows immediately from Theorem 8.2. In deed, for a complex Minkowski metric F on M , we have Γ α;μ ≡ 0,

Γ αβ;μ ≡ 0 and K F (v) ≡ 0. �

Corollary 8.5. Let F be a complex Minkowski metric on M, and F = eσ(z) F be a conformal change of F . Then

(i) K F (v) = c = 0 if and only if σ(z) is pluriharmonic function on M;(ii) K F (v) = c > 0 if and only if

F 2 = −4

c

∂2σ

∂zμ∂zνvμvν

for some strictly plurisubharmonic function −σ(z) on M;


(iii) K F (v) = c < 0 if and only if

F 2 = −4

c

∂2σ

∂zμ∂zνvμvν

for some strictly plurisubharmonic function σ(z) on M.

Proof. This follows from Corollary 8.4. �Example 8.6. Let z = (z1, z2) ∈ C

2, v = (v1, v2) ∈ T 1,0z C

2, and σ(z) is a real valued smooth function in C2. Then it is easy to

check that

Fε(z, v) = e12 σ (z)

√|v1|2 + |v2|2 + ε

√|v1|4 + |v2|4 (8.7)

is a complex Berwald metric in C2 for every ε > 0.

Example 8.7. (See [19].) Let z = (z1, z2) ∈ C2, v = (v1, v2) ∈ T 1,0

z C2, and σ(z) is a real valued smooth function in C

2. Then

L A S(z, v) = e2σ (z)√

|v1|4 + |v2|4 (8.8)

is a complex Berwald metric in C2.

Theorem 8.8. Let F be a strongly pseudoconvex complex Finsler metric on M, and F = eσ(z) F be a conformal change of F . Then

(i) F is a weakly complex Berwald metric if and only if F is a weakly complex Berwald metric;(ii) the holomorphic sectional curvature K F (v) of F and the holomorphic sectional curvature K F (v) of F are related by

K F (v) = e−2σ

[K F (v) − 4

G

∂2σ

∂zμ∂zνvμvν

]; (8.9)


R F (v) = R F (v) − (n + 1)∂2σ


Proof. It is easy to check that the complex non-linear connection coefficients Γ α;μ of the Chern–Finsler connection associated

to (M, F ) are given by

Γ α;μ = 2

∂σ

∂zμvα + Γ α

;μ, (8.11)

where Γ α;μ are the complex non-linear connection coefficients of the Chern–Finsler connection associated to F . So that

Gα = 1

2Γ α

;μvμ = ∂σ

∂zμvα vμ +G

α, (8.12)

where Gα = 1

2 Γ α;μvμ . Thus the complex non-linear connection coefficients G

αμ of the complex Berwald connection associ-

ated to (M, F ) are given by

Gαμ = ∂σ

∂zμvα + ∂σ

∂zνδαμvν +G

αμ, (8.13)

where Gαμ are the complex non-linear connection coefficients of the complex Berwald connection associated to F . So that

the complex Berwald connection coefficients Gαβμ of the complex Berwald connection associated to (M, F ) are given by

Gαβμ = ∂σ

∂zμδαβ + ∂σ

∂zβδαμ +G

αβμ, (8.14)

where Gαβμ are the complex Berwald connection coefficients of the complex Berwald connection associated to F . Therefore

Gαβμ are independent of fiber coordinates v if and only if Gα

βμ are independent of fiber coordinates v . This completes theassertion (i).

Next note that the holomorphic sectional curvature K F (v) of F is


K F (v) = − 2

G2(v)Gαχ

(Γ α

;μ)

vμ.

If we denote χ the complex radial horizontal vector field of F . Then

χ = ∂μ − Γ α;μ∂α = χ − ∂σ

∂zμvμι,

where χ = vμ(∂μ − Γ α;μ∂α) and ι = vα∂α . Thus the holomorphic sectional curvature K F (v) of F is

K F (v) = − 2

G2(v)Gαχ

(Γ α

;μ)

vμ = − 2

G2(v)Gα

(χ − 2

∂σ

∂zμvμι

)(2

∂σ

∂zμvα + Γ α

;μ)

vμ

= − 4

G(v)

∂2σ

∂zμ∂vνvμvν − e−2σ 2

G2(v)Gαχ

(Γ α

;μ)

vμ = e−2σ

[K F (v) − 4

G

∂2σ

∂zμ∂zνvμvν

].

(iii) The Ricci scalar curvature R F (v) of F is

R F (v) = −χ(G

μμ

) = −(χ − 2

∂σ

∂zμvμι

)[(n + 1)

∂σ

∂zμvμ +G

μμ

]

= −χ

[(n + 1)

∂σ

∂zμvμ +G

μμ

]= −(n + 1)

∂2σ

∂zμ∂zνvμvν − χ

(G

μμ

)

= R F (v) − (n + 1)∂2σ

∂zμ∂zνvμvν . �

Note that it is easy to check that Γ αβ;μ and G

αβμ are related by the following equality:

∂γ

(Γ α

β;μ)

vμ + Γ αβ;γ + Γ α

γ ;β = 2Gμγ β. (8.15)

It follows from (8.15) that a complex Berwald metric is necessary a weakly complex Berwald metric, the converse, however,is not true. We have the special case when these two notions coincide.

Proposition 8.9. If F is a Kähler–Finsler metric on M, then F is a complex Berwald metric if and only if F is a weakly complex Berwaldmetric.

Proof. Since we always have Γ αβ;μ =G

αβμ if F is a Kähler–Finsler metric. �

As an application of Theorem 8.8, the following Proposition shows that there are many weakly complex Berwald metrics.

Corollary 8.10. Let F be a weakly complex Berwald metric in Cn such that Gα

βμ ≡ 0. Let F = eσ(z) F be a conformal change of F such

that the complex gradient ∇σ = ( ∂σ∂z1 , . . . , ∂σ

∂zn ) �= 0. Then

(i) F is a weakly complex Berwald metric with Gαβμ not vanishing identically;

(ii) the holomorphic sectional curvature K F (v) of F is

K F (v) = − 4

G

∂2σ

∂zμ∂zνvμvν; (8.16)

(iii) the Ricci scalar curvature R F (v) of F is given by

R F (v) = −(n + 1)∂2σ


Proof. This follows immediately from (8.14) and Theorem 8.8, since Gαβμ ≡ 0 implies

Gα ≡ 0, G

αμ ≡ 0, K F (v) ≡ 0, R F (v) ≡ 0. �

Corollary 8.11. Let F be the complex Wrona metric in Cn and F = eσ(z) F be the conformal change of F . Then

(i) F is a weakly complex Berwald metric with Gα not vanishing identically;
βμ


(ii) if the holomorphic sectional curvature K F (v) of F is a constant c, then it is necessary that c = 0;(iii) if K F (v) ≡ 0 and R F (v) ≡ 0, then it is necessary that σ(z) is a strictly pluriharmonic function.

Proof. (i) It is trivial. (ii) By Theorem 6.7 in [25] we have K F (v) = R F (v) ≡ 0. Thus if K F = c �= 0, then Theorem 8.10 impliesthat F 2 is quadratic with respect to v . This is a contradiction since F is a complex Wrona metric, which is not quadraticwith respect to v . Thus c ≡ 0.

(iii) By (8.16) and (8.17), K F (v) ≡ 0 and R F (v) ≡ 0 if and only if

∂2σ

∂zμ∂zνvμvν = 0

for every nonzero vector v ∈ T 1,0p C

n , which implies that σ(z) is a strictly pluriharmonic function. �Acknowledgements

This work is sponsored by the National Natural Science Foundation of China (Grant Nos. 11271304, 10971170, 11171277);the Natural Science Foundation of Fujian Province of China for Distinguished Young Scholars (Grant No. 2013J06001); theScientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry; the Fundamental Re-search Funds for the Central Universities (Grant No. 2012121006). The authors would like to thank the referee for his/hervaluable suggestions and comments which make this paper greatly improved.

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characterizations of complex finsler connections and weakly complex berwald metrics

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