On doubly warped product of complex finsler manifolds

Acta Mathematica Scientia 2016,36B(6):1747–1766 http://actams.wipm.ac.cn

ON DOUBLY WARPED PRODUCT OF COMPLEX FINSLER MANIFOLDS∗

Û℄)

Yong HE (

1,2

¨S²)

Chunping ZHONG (

1

1. School of Mathematical Sciences, Xiamen University, Xiamen 361005, China 2. School of Mathematical Sciences, Xinjiang Normal University, Urumqi 830053, China E-mail : [email protected]; [email protected] Abstract Let (M1 , F1 ) and (M2 , F2 ) be two strongly pseudoconvex complex Finsler manifolds. The doubly wraped product complex Finsler manifold (f2 M1 ×f1 M2 , F ) of (M1 , F1 ) and (M2 , F2 ) is the product manifold M1 × M2 endowed with the warped product complex Finsler metric F 2 = f22 F12 + f12 F22 , where f1 and f2 are positive smooth functions on M1 and M2 , respectively. In this paper, the most often used complex Finsler connections, holomorphic curvature, Ricci scalar curvature, and real geodesics of the DWP-complex Finsler manifold are derived in terms of the corresponding objects of its components. Necessary and sufficient conditions for the DWP-complex Finsler manifold to be K¨ ahler Finsler (resp., weakly K¨ ahler Finsler, complex Berwald, weakly complex Berwald, complex Landsberg) manifold are obtained, respectively. It is proved that if (M1 , F1 ) and (M2 , F2 ) are projectively flat, then the DWP-complex Finsler manifold is projectively flat if and only if f1 and f2 are positive constants. Key words

doubly warped products; complex Finsler metric; holomorphic curvature; geodesic

2010 MR Subject Classification

1

53C60; 53C40

Introduction

Singly warped product or simply warped product of Riemannian manifolds was first defined by O’Neill and Bishop in [12] to construct Riemannian manifolds with negative sectional curvature, then in [22], O’Neill obtained the curvature formulae of warped products in terms of curvatures of its components. The recent studies showed that warped product is useful in theoretical physics, particulary in general relativity. For instance, Beem, Ehrlich and Powell [11] pointed out that many exact solutions to Einstein’s field equation can be expressed in terms of Lorentzian warped products. Under the assumption that four-dimensional space-time to be a general warped product of two surfaces, Katanaev, Kl¨ osch and Kummer [16] explicitly constructed all global vacuum solutions to the four-dimensional Einstein equations. Doubly warped product of Riemannian manifolds was also of interesting and was studied by Unal [30]. ∗ Received

May 11, 2015; revised December 22, 2015. This work is supported by Program for New Century Excellent Talents in University (NCET-13-0510); National Natural Science Foundation of China (11271304, 11571288, 11461064); the Fujian Province Natural Science Funds for Distinguished Young Scholar (2013J06001); the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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The notion of warped product was recently extended to Finsler spaces. In [6, 7], Asanov obtained some models of relativity theory which were described by the warped product of Finsler metrics. In [18], Kozma, Peter and Varga studied the warped product of real Finsler manifolds. They obtained the relationship between the Cartan connection of the warped product Finsler metric and the Cartan connections of its components. More recently, Hushmandi and Rezaii [8] studied the warped product Finsler spaces of Landsberg type, and then in [9], Hushmandi, Rezaii and Morteza obtained the curvature of warped product Finsler spaces and the Laplacian of the Sasaki-Finsler metrics. In [23], Peyghan and Tayebi obtained the relationship between the Riemannian curvature of the doubly warped product Finsler manifold and the curvatures of its components. Let (M1 , F1 ) and (M2 , F2 ) be two complex Finsler manifolds. In [33], Wu and Zhong considered the product complex manifold M = M1 × M2 endowed with a complex Finsler metric F = f (K, H), where f (K, H) is a function of K = F12 and H = F22 . The possibility of F to be K¨ahler Finsler metric and complex Berwald metric were investigated. Recently in [35], Zhong showed that there are lots of strongly pseudoconvex (even strongly convex) unitary invariant complex Finsler metrics in domains in Cn . In this paper, we consider the warped product of strongly pseudoconvex complex Finsler manifolds. Our purpose of doing this is to study the possibility of constructing some special strongly pseudoconvex complex Finsler metrics such as K¨ahler Finsler metrics, weakly K¨ ahler Finsler metrics, complex Berwald metrics, complex Landsberg metrics and weakly complex Berwald metrics, among which to find possible way to obtain strongly pseudoconvex complex Finsler metrics which are of constant holomorphic curvatures. Note that it was prove in [33] that the Chern-Finsler nonlinear connection coefficients are independent of the choice of f , i.e., there is no difference between the case F 2 = f (K, H) and F 2 = F12 + F22 . In this paper, the warping product metric F on the product complex manifold M = M1 × M2 is F 2 = f22 F12 + f12 F22 , which generalizes [33] whenever f1 and f2 are not positive constants. This paper is organized as follows. In Section 2, we recall some basic concepts and notions in complex Finsler geometry. In Section 3, we derive the most often used complex Finsler connections (the Chern-Finsler connection, the complex Rund connection, the complex Berwald connection, and the complex Hashiguchi connection, etc.) of the DWP-complex Finsler manifold in terms of the corresponding connections of its components, respectively. In Section 4, we derive the formulae of the holomorphic curvature and Ricci scalar curvature of the DWPcomplex Finsler manifold in terms of the holomorphic curvatures and Ricci scalar curvatures of its components. In Section 5, we obtain the necessary and sufficient conditions for the DWP-complex Finsler manifold to be K¨ ahler Finsler (resp. weakly K¨ahler Finsler, complex Berwald, weakly complex Berwald, complex Landsberg, complex locally Minkowski) manifold. In Section 6, we derive the real geodesic equations of the DWP-complex Finsler manifold in terms of the geodesic equations of its components, and prove that if the warping function f1 (resp. f2 ) is a positive constant, then (M1 , F1 ) (resp. (M2 , F2 )) is a totally geodesic manifold of the DWP-complex Finsler manifold (f2 M1 ×f1 M2 , F ), and the projection of any geodesic of the DWP-complex Finsler manifold onto M1 (resp. M2 ) is a geodesic of (M1 , F1 ) (resp. (M2 , F2 )). In Section 7, we prove that if (M1 , F1 ) and (M2 , F2 ) are locally projectively flat manifolds, then the DWP-complex Finsler manifold (f2 M1 ×f1 M2 , F ) is projectively flat if and only if f1 and

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f2 are positive constants. The main results in this paper are as follows. Theorem 1.1 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of (M1 , F1 ) and (M2 , F2 ). Then the holomorphic curvature of (M, F ) along a holomorphic tangent vector ′ v = (v i , v i ) ∈ Tz1,0 M satisfying F1 (π1 (v)) = 1 and F2 (π2 (v)) = 1 is given by KF (v) =

∂ 2 ln f1 j k 4 ∂ 2 ln f2 j ′ k′ f12 h2 f22 g 2 4 KF1 (v1 ) − 2 f12 h v v − 2 f22 g v v + KF2 (v2 ). (1.1) 2 G G G G2 ∂z j ∂z k ∂z j ′ ∂z k′

Theorem 1.1 implies that if KF1 (π1 (v)) = KF2 (π2 (v)) ≡ c, then KF (v) ≡ c if and only if the warping functions ln f1 and ln f2 are pluriharmonic functions on M1 and M2 , respectively. Theorem 1.2 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of (M1 , F1 ) and (M2 , F2 ). Then the Ricci scalar curvature of (f2 M1 ×f1 M2 , F ) associated to the Chern-Finsler connections is given by RicF = RicF1 + RicF2 , (1.2) where RicF1 and RicF2 are Ricci scalar curvature of (M1 , F1 ) and (M2 , F2 ), respectively. Theorem 1.2 implies that equality (1.2) is independent of the choice of the warping functions f1 and f2 . Theorem 1.3 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of two weakly K¨ahler Finsler manifolds (M1 , F1 ) and (M2 , F2 ). Then (f2 M1 ×f1 M2 , F ) is a weakly K¨ahler Finsler manifold if and only if the functions f1 and f2 are positive constants. Theorem 1.4 The DWP-complex Finsler manifold (f2 M1 ×f1 M2 , F ) is a complex Berwald manifold if and only if (M1 , F1 ) and (M2 , F2 ) are complex Berwald manifolds and the functions f1 and f2 are positive constants. Theorems 1.3 and 1.4 imply that unless the doubly warping function f1 and f2 are positive constants, the doubly warped product operation does not preserve weakly K¨ahler Finsler manifolds and complex Berwald manifolds. Theorem 1.5 The DWP-complex Finsler manifold (f2 M1 ×f1 M2 , F ) is a weakly complex Berwald manifold if and only if (M1 , F1 ) and (M2 , F2 ) are weakly complex Berwald manifolds. Theorem 1.5 implies that the doubly warped product operation preserves weakly complex Berwald metrics. Since there are lots of weakly complex Berwald metrics (see [35]), this theorem provides us an effective way to construct new weakly complex Berwald metrics. Theorem 1.6 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of two complex Finsler manifolds (M1 , F1 ) and (M2 , F2 ). (i) If the warping functions f1 (resp. f2 ) is a positive constant on M1 (resp. M2 ), then any geodesic of (M1 , F1 ) (resp. (M2 , F2 )) is a geodesic of (f2 M1 ×f1 M2 , F ), that is to say (M1 , F1 ) (resp. (M2 , F2 )) is a totally geodesic subspace of the doubly warped product complex Finsler space (f2 M1 ×f1 M2 , F ). (ii) If the warping functions f1 (resp. f2 ) is a positive constant on M1 (resp. M2 ), then the projection of any geodesic of the DWP-complex Finsler manifold (f2 M1 ×f1 M2 , F ) onto M1 (resp. M2 ) is a geodesic of (M1 , F1 ) (resp. (M2 , F2 )). Theorem 1.7 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of two strongly convex complex Finsler manifolds (M1 , F1 ) and (M2 , F2 ).

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(i) If (M1 , F1 ) and (M2 , F2 ) are locally projectively flat manifolds, then (f2 M1 ×f1 M2 , F ) is a locally projectively flat manifold if and only if the warping functions f1 and f2 are positive constants. (ii) If (M1 , F1 ) and (M2 , F2 ) are locally dually flat manifolds, then (f2 M1 ×f1 M2 , F ) is a ∂f2 j ′ 1 i is a real value function. locally dually flat manifold if and only if f1 h ∂f ∂z i v + f2 g ∂z j′ v

2

Preliminary

Let M be a complex manifold of complex dimension n. We denote z = (z 1 , · · · , z n ) the local holomorphic coordinates on M , and (z, v) = (z 1 , · · · , z n , v 1 , · · · , v n ) the induced local holomorphic coordinates on the holomorphic tangent bundle T 1,0 M of M . We shall assume that M is endowed with a strongly pseudoconvex complex Finsler metric F in the following sense. Definition 2.1 (see [1]) A strongly pseudoconvex complex Finsler metric F on a complex manifold M is a continuous function F : T 1,0 M → R+ satisfying ˜ = T 1,0 M − {zero section}; (i) G = F 2 is smooth on M ˜; (ii) F (p, v) > 0 for all (p, v) ∈ M (iii) F (p, ζv) = |ζ|F (p, v) for all (p, v) ∈ T 1,0 M and ζ ∈ C; (iv) the Levi matrix (or complex Hessian matrix) (Gαβ ) =

∂2G ∂v α ∂v β

!

(2.1)

˜. is positive definite on M In this paper, we denote (Gνβ ) the inverse matrix of (Gαν ) such that Gνβ Gαν = δαβ . 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 , ∂z ν ∂v µ

Gα;ν =

∂2G . ∂z ν ∂v α

Let (M1 , F1 ) and (M2 , F2 ) be two strongly pseudoconvex complex Finsler manifolds with dimC M1 = m and dimC M2 = n, then M = M1 × M2 is a strongly pseudoconvex complex Finsler manifold with dimC M = m + n. Throughout this paper we use the natural product coordinate system on the product manifold M = M1 × M2 . Let (p0 , q0 ) be a point in M , then there are coordinate chart (U, z1 ) and (V, z2 ) on M1 and M2 , respectively, such that p0 ∈ U and q0 ∈ V . Thus we obtain a coordinate chart (W, z) on M such that W = U × V and (p0 , q0 ) ∈ W , and for all (p, q) ∈ W , z(p, q) = (z1 (p), z2 (q)), where π1 : M1 × M2 → M1 , π2 : M1 × M2 → M2 are natural projections. Let T 1,0 M1 , T 1,0 M2 and T 1,0 M be the holomorphic tangent bundles of M1 , M2 and M , respectively. Denote dπ1 : T 1,0 (M1 × M2 ) → T 1,0 M1 , dπ2 : T 1,0 (M1 × M2 ) → T 1,0 M2 the holomorphic tangent maps induced by π1 and π2 , respectively. Note that dπ1 (z, v) = (z1 , v1 ) and dπ2 (z, v) = (z2 , v2 ) for every v = (v1 , v2 ) ∈ Tz1,0 (M1 ×M2 ) with v1 = (v 1 , · · · , v m ) ∈ Tz1,0 M1 1 m+1 m+n 1,0 1,0 1,0 ˜ ˜ and v2 = (v , ··· ,v ) ∈ Tz2 M2 . Denote M1 = T M1 − {zero section}, M2 = T M2 − f=M f1 × M f2 ⊂ T 1,0 (M1 × M2 ) − {zero section}. {zero section}, M

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Definition 2.2 Let (M1 , F1 ) and (M2 , F2 ) be two strongly pseudoconvex complex Finsler manifolds and f1 : M1 → (0, +∞) and f2 : M2 → (0, +∞) be smooth functions. The doubly warped product (abbreviated as DWP) complex Finsler manifold of (M1 , F1 ) and (M2 , F2 ) is the product complex manifold M = M1 × M2 endowed with the complex Finsler metric f → R+ given by F :M F 2 (z, v) = (f2 ◦ π2 )2 (z)F12 (π1 (z), dπ1 (v)) + (f1 ◦ π1 )2 (z)F22 (π2 (z), dπ2 (v))

(2.2)

for z = (z1 , z2 ) ∈ M and v = (v1 , v2 ) ∈ Tz1,0 M − {zero section}. The functions f1 and f2 are called warped functions. The DWP-complex Finsler manifold of (M1 , F1 ) and (M2 , F2 ) is denoted by (f2 M1 ×f1 M2 , F ). It is obvious that the function F defined by (2.2) is a strongly pseudoconvex complex f rather than on Finsler metric on M . Note that it is natural to ask F to be defined on M 1,0 1,0 1,0 f1 × T M2 , or on T M1 × M f2 , since Fi is smooth T (M1 × M2 ) − {zero section}, or on M 1,0 on T Mi if and only if Fi comes from a Hermitian metric on Mi for i = 1, 2. If either f1 = 1 or f2 = 1, but not both of them, then (f2 M1 ×f1 M2 , F ) becomes a warped product of complex Finsler manifolds (M1 , F1 ) and (M2 , F2 ). If f1 ≡ 1 and f2 ≡ 1, then (f2 M1 ×f1 M2 , F ) becomes a product of complex Finsler manifolds (M1 , F1 ) and (M2 , F2 ). If neither f1 nor f2 is a constant, then we call (f2 M1 ×f1 M2 , F ) a nontrivial (proper) DWPcomplex Finsler manifold of (M1 , F1 ) and (M2 , F2 ). Notation Lowercase Greek indices such as α, β, γ, etc., will run from 1 to m + n, whereas lowercase Latin indices such as i, j, k, s, t, etc., will run from 1 to m, lowercase Latin indices with a prime, such as i′ , j ′ , k ′ , etc., will run from m + 1 to m + n. Quantities associated to 1

2 ′

i (M1 , F1 ) and (M2 , F2 ) are denoted with upper indices 1 and 2, respectively, such as Γj;k , Γji′ ;k′ . f are transformed by the rules The local coordinates (z, v) on M ′

z˜i = z˜i (z 1 , · · · , z m ),

∂ z˜i v˜ = j v j , ∂z i

′

z˜i = z˜i (z m+1 , · · · , z m+n ), ′

∂ z˜i j ′ v . v˜ = ∂z j ′ i′

For ∂/∂v α , we have ′

∂ z˜j ∂ ∂ . ′ = i ∂v ∂z i′ ∂˜ vj′

∂ ∂ z˜j ∂ = , i ∂v ∂z i ∂˜ vj

(2.3)

Denote g = F12 , h = F22 , so that G = F 2 = f22 g + f12 h and gij =

∂2g ∂v i ∂v j

,

∂2h

hi′ j ′ =

∂v i′ ∂v j ′

.

The fundamental tensor matrix of F is given by (Gαβ ) =



2

∂ G ∂v α ∂v β



=

with its inverse matrix (Gβα ) given by 

(Gβα ) = 

f2−2 g ji 0



f 2g  2 ij

0 f12 hi′ j ′

0

0 f1−2 hj

′ i′



.

 

(2.4)

(2.5)

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f, which is locally spanned Let V 1,0 be the holomorphic vertical vector subbundle of T 1,0 M ∂ ∂ by the nature frame fields { ∂vi , ∂vi′ } and is called the doubly warped vertical distribution on f. Then, the complementary vector subbundle H1,0 to V 1,0 in T 1,0 M f is locally spanned by M {δi , δi′ }, where ′ ∂ ∂ ∂ − Γ;ij j − Γ;ij , i = 1, · · · , m, i ∂z ∂v ∂v j ′ ′ ∂ ∂ j ∂ − Γ;ij ′ j ′ , i′ = m + 1, · · · , m + n. δi′ = ′ − Γ;i′ i j ∂z ∂v ∂v 1,0 f. Thus the holomorphic tangent H is called the doubly warped horizontal distribution on M f admits the decomposition bundle T 1,0 M M f = H1,0 V 1,0 . (2.6) T 1,0 M

δi =

3

Connections of DWP-complex Finsler Manifold

In complex Finsler geometry, the Chern-Finsler connection is the most often used complex Finsler connection. There are also other complex Finsler connections, such as complex Rund connection, complex Berwald connection, complex Hashiguchi connection and Rund type complex Finsler connection used in various topics [27]. In this section we derive these connections of DWP-complex Finsler manifold, which are expressed in terms of connections of its components. α The Chern-Finsler complex nonlinear connection Γ;µ associate to a given strongly pseudoconvex complex Finsler metric F is given by α Γ;µ =: Gνα Gν;µ .

(3.1)

f ⊗ V 1,0 ) associated to a strongly The Chern-Finsler connection D : X (V 1,0 ) → X (TC∗ M pseudoconvex complex Finsler metric F was first introduced in [17] and systemically studied in [1]. Essentially, the Chern-Finsler connection associated to F is the Hermitian connection on the holomorphic vector bundle V 1,0 endowed with the Hermitian metric h·, ·i induced by F . The connection 1-forms ωβα of D is of type (1, 0) and are given by α α µ ωβα = Γβ;µ dz µ + Γβµ ψ ,

(3.2)

α = Gνα δµ (Gβν ), Γβ;µ Γ α = Gνα ∂˙µ (Gβν ),

(3.3)

α ˙ µ δµ = ∂µ − Γ;µ ∂α , ψ µ = dv µ + Γ;α dz α .

(3.5)

where

βµ

(3.4)

and In the following we denote ∂ , ∂z µ ∂ ∂˙µ = , ∂v µ ∂ ∂k = , ∂z k

∂µ =

µ = 1, · · · , m, m + 1, · · · , m + n, µ = 1, · · · , m, m + 1, · · · , m + n, k = 1, · · · , m,

∂k′ =

∂ , ∂z k′

k ′ = m + 1, · · · , m + n,

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∂ , ∂˙k = ∂v k

∂˙k′ =

k = 1, · · · , m,

∂ , ∂v k′

k ′ = m + 1, · · · , m + n.

Note that α α Γβ;µ = ∂˙β (Γ;µ ).

(3.6)

Lemma 3.1 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of (M1 , F1 ) and (M2 , F2 ). Then the Chern-Finsler complex nonlinear connection coefficients associated to F are given by   α (Γ;µ )=

where

1

′

i Γ;k

i Γ;k

i Γ;k ′

i Γ;k ′

′

∂f1 i′ v , ∂z k

′

i Γ;k = 2f1−1

i i Γ;k =Γ;k ,

2

∂f2 i v, ∂z k′ By using (2.5) and (3.1), we have

′

−1 i Γ;k ′ = 2f2

Proof

,

′

i i Γ;k ′ =Γ;k′ .

1

i i . Γ;k = Gji Gj;k + Gj i Gj ′ ;k = f2−2 g ji f22 gj;k =Γ;k ′

Similarly, we can obtain other equalities in Lemma 3.1.



Using (3.3), (3.4) and Lemma 3.1, after a straightfoward computation, we have Proposition 3.2 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of (M1 , F1 ) α α and (M2 , F2 ), Γβ;µ and Γβµ are the horizontal and vertical coefficients of the Chern-Finsler connection associated to F , respectively. Then 1 i i Γj;k = Γj;k ,

∂f2 i δ , ∂z k′ j ′ i = Γj;k ′ = 0

−1 i Γj;k ′ = 2f2 ′

i Γji′ ;k = Γji′ ;k′ = Γj;k

′

Γji′ ;k = 2f1−1

∂f1 i′ δ ′, ∂z k j

2 ′

′

Γji′ ;k′ =Γji′ ;k′ ,

and 1 i i Γjk = Γjk ,

2 ′

′

Γji′ k′ =Γji′ k′ , ′

′

′

i i i i i Γji′ k = Γjk ′ = Γj ′ k′ = Γjk = Γj ′ k = Γjk′ = 0.

ˆ : X (V 1,0 ) → X (T ∗ M ˜ ⊗ V 1,0 ) associated to a strongly The complex Rund connection D C pseudoconvex complex Finsler metric F was first introduced in [24] and were systemically ˆ are given by studied in [2, 3]. The connection 1-forms ω ˆ of D α dz µ , ω ˆ βα = Γβ;µ

(3.7)

α where Γβ;µ are defined by (3.3). Obviously, ω ˆ βα are the horizontal part of ωβα , which are given by (3.2). Using (3.3) and Lemma 3.1, we have

Proposition 3.3 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of (M1 , F1 ) α and (M2 , F2 ), Γβ;µ be the coefficients of the complex Rund connection associated to F . Then 1 i i , Γj;k = Γj;k

∂f2 i δ , ∂z k′ j i′ = Γj;k ′ = 0.

−1 i Γj;k ′ = 2f2 ′

i Γji′ ;k = Γji′ ;k′ = Γj;k

′

Γji′ ;k = 2f1−1

∂f1 i′ δ ′, ∂z k j

2 ′

′

Γji′ ;k′ =Γji′ ;k′ ,

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Next we shall derive the complex Berwald nonlinear connection coefficients Gα µ , which is α α obtained from Γ;µ . According to [21], the complex nonlinear connection Γ;µ always determine α µ a complex spray Gα =: 12 Γ;µ v . Conversely, the complex spray Gα induces another complex nonlinear connection denoted by Gα = ∂˙µ (Gα ), (3.8) µ

which are called the complex Berwald nonlinear coefficients associated to F . Note that we always have α µ µ α Γ;µ v = Gα µ v = 2G .

˘ : X (V 1,0 ) → X (T ∗ M ˜ ⊗ V 1,0 ) was first introduced in The complex Berwald connection D C [21], and its connection 1-form can be expressed as µ ω ˘ βα = Gα βµ dz ,

(3.9)

α ˙ Gα βµ = ∂β (Gµ ).

(3.10)

where

By Lemma 3.1 one may easily establish the following result. Lemma 3.4 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of (M1 , F1 ) and α (M2 , F2 ). Then the complex spray coefficients Gα associated to Γ;µ (or equivalently Gα µ ) are given by Gi = f2−1

1 ∂f2 i k′ i v v + G , ′ ∂z k

2

′

Gi = f1−1

′ ∂f1 i′ k v v + Gi . ∂z k

Using (3.8), (3.10) and Lemma 3.4, by a straight forward computation, we obtain the following result. Lemma 3.5 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of (M1 , F1 ) and (M2 , F2 ). Then the complex Berwald nonlinear connection coefficients Gα µ are given by   ′ Gik Gik ,  (Gα ) = µ ′ Gik′ Gik′ where

1 ∂f2 i k′ i δ v + G k, ∂z k′ k 2 ′ ∂f1 ′ = f1−1 k δki ′ v k + Gik′ , ∂z

Gik = f2−1 ′

Gik′

∂f2 i v, ∂z k′ ′ ∂f1 ′ Gik = f1−1 k v i . ∂z Gik′ = f2−1

Corollary 3.6 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of (M1 , F1 ) and (M2 , F2 ). Then the complex Berwald connection coefficients Gα βµ associated to F are given by 1

Gijk =Gijk , ′

′

Gij ′ k = Gikj ′ = f2−1

Gij ′ k = Gikj ′ = f1−1

∂f1 i′ δ ′, ∂z k j

∂f2 i δ , ∂z j ′ k

Gij ′ k′ = 0, 2

′

Gijk = 0,

′

′

Gij ′ k′ =Gij ′ k′ .

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ˇ : X (V 1,0 ) → X (T ∗ M ˜ ⊗ V 1,0 ) is a complex analogue The complex Hashiguchi connection D C of the Hashiguchi connection in real Finsler geometry [19]. Its connection 1-forms ω ˇ βα are given by ω ˇ α = Gα dz µ + Γ α ψ˘µ , (3.11) β

βµ

βµ

α µ µ α ˘µ where Gα βµ are given by (3.10), Γβµ are given by (3.4) and ψ = dv + G α dz .

Proposition 3.7 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of (M1 , F1 ) and (M2 , F2 ). Then the horizontal and vertical connection coefficients of the complex Hashiguchi α connection, respectively Gα βµ and Γβµ are given by Proposition 3.3 and Corollary 3.6. In [5], Aldea and Munteanu gave the definition of complex Landsberg space. A complex Finsler manifold (M, F ) is called a complex Landsberg manifold if Gγνµ = Lγνµ ,

(3.12)

where Lγνµ is the horizontal connection coefficients of the Rund type complex linear connection [21], i.e., h i 1 Lγνµ = Gτ γ δˇν (Gµτ ) + δˇµ (Gντ ) , (3.13) 2 where ˙ δˇν = ∂ν − Gα (3.14) ν ∂α . Proposition 3.8 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of (M1 , F1 ) and (M2 , F2 ). Then 1

Lijk = Lijk ,

2 ′

′

Lij ′ k′ =Lij ′ k′ , ′

′

′

Lij ′ k = Lijk′ = Lij ′ k′ = Lijk = Lij ′ k = Lijk′ = 0. Using (2.4), (2.5), (3.13), (3.14) and Lemma 3.5, we have i 1 ′h i 1 h Lijk = Gli δˇk (Gjl ) + δˇj (Gkl ) + Gl i δˇk (Gjl′ ) + δˇj (Gkl′ ) 2 2 i ′ ′ 1 li h = G (∂k − Gsk ∂˙s − Gsk ∂˙s′ )Gjl + (∂j − Gtj ∂˙t − Gtj ∂˙t′ )Gkl 2 i 1 1 1 h = Gli (∂k − Gsk ∂˙s )Gjl + (∂j − Gtj ∂˙t )Gkl 2 h 1 i 1 1 1 = f2−2 g li f22 δˇk (gjl )+ δˇj (gkl ) =Lijk . 2 Similarly, we can obtain other equalities of Proposition 3.8. Proof

4



Holomorphic Curvature and Ricci Scalar Curvature of Doubly Warped Product Complex Finsler Manifold

Our purpose in this section is to derive formulae of the holomorphic curvature and Ricci scalar curvature of DWP-complex Finsler manifold in terms of the holomorphic curvature and Ricci scalar curvature of its components . In a complex Finsler space (M, F ), there are two ways of defining the holomorphic curvature of F .

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The first method gives the holomorphic curvature on (M, F ) from the Gauss curvature on the unit disc △ ⊂ C via a holomorphic map ϕ : △ → M . More precisely, the holomorphic curvature is defined by KF (v) = sup{K(ϕ∗ G)(0)} [31]. The second method to define the holomorphic curvature on (M, F ) from the curvature tensor Ω of a complex Finsler connection and this was considered by Kobayashi [17], and locally expression of the holomorphic curvature of a strongly pseudoconvex Finsler metric F along v ∈ Tz1,0 M with respect to Chern-Finsler 0 connection D is given by Abate and Patrizio [1]: 2 α µ ν KF (v) = − 2 Gα δν (Γ;µ )v v . (4.1) G In [3], Aikou proved that the above two definitions is equivalent to each other, and if F comes from a Hermitian metric on M , then the holomorphic curvature is just the holomorphic sectional curvature in the usual sense [32]. In [27], Sun and Zhong pointed out that the holomorphic curvature of a strongly pseudoconvex Finsler metric F along a nonzero vector v ∈ Tz1,0 M with respect to the Chern-Finsler 0 ˆ ˘ or connection D, or the complex Rund connection D, or the complex Berwald connection D, ˇ are coincide with each other. In the following, we use the complex Hashiguchi connection D, the Chern-Finsler connection to derive the holomorphic curvature of the DWP-complex Finsler manifold. Theorem 4.1 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of (M1 , F1 ) and (M2 , F2 ). Then the holomorphic curvature of (M, F ) along a holomorphic tangent vector ′ v = (v i , v i ) ∈ Tz1,0 M satisfying F1 (π1 (v)) = 1 and F2 (π2 (v)) = 1 is given by f22 g 2 4 2 ∂ 2 ln f2 j ′ k′ f12 h2 4 2 ∂ 2 ln f1 j k v v − v v + K (v ) − f f g KF2 (v2 ). (4.2) h F 1 1 1 G2 G2 G2 2 ∂z j ′ z k′ G2 ∂z j z k Proof The holomorphic curvature of the DWP-complex Finsler manifold (f2 M1 ×f1 M2 , F ) with respect to the Chern-Finsler connection is given by i ′ ′ ′ ′ 2 h KF (v) = − 2 Gi δν (Γ;ji )v j v ν + Gi δν (Γ;ji ′ )v j v ν + Gi′ δν (Γ;ji )v j v ν + Gi′ δν (Γ;ji ′ )v j v ν . G ′ ′ i′ Since Γ;j depend only on z i , v i and are holomorphic with respect to v i , we have ′ 2 ∂  −1 ∂f1 i′  j k 2 2f v v v − 2 Gi′ δν (Γ;ji )v j v ν = − 2 f12 hi′ G G ∂z j ∂z k   2 2 ∂f1 ∂f1 −1 ∂ f1 = − 2 f12 h − 2f1−2 + 2f vj vk G ∂z k ∂z j ∂z j z k 4 ∂ 2 ln f1 j k = − 2 f12 h v v . G ∂z j z k KF (v) =

′

′

′

Since Γ;ji ′ depend only on z i , v i , thus ′ ′ ′ ′ ′ ′ 2 2 2 Gi′ δν (Γ;ji ′ )v j v ν = − 2 f12 hi′ δk (Γ;ji ′ )v j v k − 2 f12 hi′ δk′ (Γ;ji ′ )v j v k′ 2 G G G  ′ 2 2  ∂ f12 h2 ′ i j′ k l = − 2 f1 hi′ − Γ;k (Γ + KF2 (v2 ) ′ )v v ;j G G2 ∂v l′ f 2 h2 = 1 2 KF2 (v2 ), G where in the last equality we used ′ ′ ′ j′ l′ ∂ (Γ i )v j = 2f −1 ∂f1 v l′ ∂ (hs′ i h Γ;k ;j ′ 1 s′ ;j ′ )v ′ ′ l k l ∂v ∂z ∂v

−

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= 2f1−1 Similarly, we have

∂f1 ∂z k

1757

  ′ ′ ′ ′ ′ ′ ′ v l′ − ht i hs r hr′ t′ l′ hs′ ;j ′ + hs i hs′ l′ ;j ′ v j = 0.

2 f22 g 2 i j ν G δ (Γ )v v = KF1 (v1 ), i ν ;j G2 G2 ′ ∂ 2 ln f2 j ′ k′ 4 2 v v . − 2 Gi δν (Γ;ji ′ )v j v ν = − 2 f22 g G G ∂z j ′ z k′ −

Thus we get (4.2).



Corollary 4.2 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of (M1 , F1 ) and (M2 , F2 ). If KF1 (π1 (v)) = KF2 (π2 (v)) ≡ c, ln f1 and ln f2 are pluriharmonic functions on M1 and M2 , respectively. Then the holomorphic sectional curvature of (f2 M1 ×f1 M2 , F ) along ′ v = (v i , v i ) is KF (v) ≡ c. (4.3) The Ricci scalar curvature of F along a nonzero vector v ∈ Tz1,0 (M1 × M2 ) associated to 0 the Chern-Finsler connection is given by [21]: RicF (v) = −

n X

µ µ α χ(Γ;µ ) = −δα (Γ;µ )v .

(4.4)

µ=1

Theorem 4.3 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of (M1 , F1 ) and (M2 , F2 ). Then the Ricci scalar curvature of (f2 M1 ×f1 M2 , F ) associated to the Chern-Finsler connection is given by RicF = RicF1 + RicF2 , where RicF1 and RicF2 are Ricci scalar curvatures of (M1 , F1 ) and (M2 , F2 ), respectively. Using (4.4), we have  ∂  ∂  ∂ ′ ∂  k j ∂ j′ ∂ k′ i RicF (v) = −v i v Γ Γ − Γ;ij − Γ;ij (Γ − − (Γ;k ) − ′) ;k ;i ;i ′ ′ i j i j j j ∂z ∂v ∂z ∂v ∂v ∂v  ∂  ∂ ∂  k ∂  k′ ∂ ′ ′ − Γ;ij ′ − Γ;ij ′ −v i′ (Γ;k ) − v i′ (Γ;k′ ). − Γ;ij ′ ′ ′ ′ ∂v j ∂z i ∂v j ∂z i ∂v j ′

Proof

∂f2 j Using Lemma 3.1, and notice that Γ;ij ′ = 2f2−1 ∂z i′ v , it follows that   ∂  k ∂f2 j ∂  k v i′ − Γ;ij ′ v (Γ;k ) = v i′ − 2f2−1 (Γ;k ). ∂v j ∂z i′ ∂v j

k k k ) = 0. Thus Since Γ;k (z, λv) = λΓ;k (z, v) for any nonzero λ ∈ C, we have v j ∂ j (Γ;k ∂v   ∂ k v i′ − Γ;ij ′ (Γ;k ) = 0. ∂v j Similar calculations gives  ′ ∂  k′ v i − Γ;ij (Γ;k′ ) = 0. ∂v j ′ Thus  ∂  ∂  ∂  k j′ ∂ k′ − Γ;ij (Γ;k ) − v i′ − Γ (Γ;k RicF (v) = −v i ′) ′ ;i ′ ′ i j i j ∂z ∂v ∂z ∂v = RicF1 (v1 ) + RicF2 (v2 ).



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Doubly Warped Product of Special Complex Finsler Manifolds

Let (M1 , F1 ) and (M2 , F2 ) be two K¨ ahler Finsler (or weakly K¨ahler Finsler, complex Berwald, weakly complex Berwald, complex Landsberg) manifold, one may want to know whether the DWP-complex Finsler manifold (f2 M1 ×f1 M2 , F ) is also a K¨ahler Finsler (or weakly K¨ahler Finsler, complex Berwald, weakly complex Berwald, complex Landsberg) manifold. In this section, we shall give a complete answer to these questions. Definition 5.1 (see [1]) Let F be a strongly pseudoconvex complex Finsler metric on a α α complex manifold M . F is called a strongly K¨ ahler Finsler metric iff Γµ;β − Γβ;µ = 0; called a α α µ K¨ahler Finsler metric iff (Γµ;β − Γβ;µ )v = 0; called a weakly K¨ahler Finsler metric iff α α Gα (Γµ;β − Γβ;µ )v β = 0.

(5.1)

In [13], Chen and Shen proved that a K¨ ahler Finsler metric is actually a strongly K¨ahler Finsler metric. Thus there are only two K¨ ahlerian notions in Finsler setting, i.e., K¨ahler Finsler metric and weakly K¨ ahler Finsler metric. Now, let us consider the case of K¨ ahler Finsler manifolds and weakly K¨ahler Finsler manifolds. We have the following theorem. Theorem 5.2 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of (M1 , F1 ) and (M2 , F2 ). Then (f2 M1 ×f1 M2 , F ) is a K¨ ahler Finsler manifold if and only if (M1 , F1 ) and (M2 , F2 ) are K¨ahler Finsler manifolds and the functions f1 and f2 are positive constants. α α Proof Assume that (f2 M1 ×f1 M2 , F ) is a K¨ahler Finsler manifold, then Γβ;µ = Γµ;β . By Proposition 3.2, this is equivalent to 1

1

∂f2 i δ = 0, ∂z k′ j 2 2 ′ ′ ′ ∂f1 ′ Γji′ ;k′ = Γki′ ;j ′ , Γji′ ;k = 2f1−1 k δji ′ = 0. ∂z Thus (M1 , F1 ) and (M2 , F2 ) are K¨ ahler Finsler manifolds and the functions f1 and f2 are positive constant.  i i Γj;k = Γk;j ,

−1 i Γj;k ′ = 2f2

Theorem 5.3 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of two weakly K¨ahler Finsler manifolds (M1 , F1 ) and (M2 , F2 ). Then (f2 M1 ×f1 M2 , F ) is a weakly K¨ahler Finsler manifold if and only the functions f1 and f2 are positive constants. Proof By putting µ = k in (5.1) and using Proposition 3.2, after a long but trivial computation, we obtain α α Gα (Γβ;k − Γk;β )v β ′

′

′

′

′

i i i j i i i j = Gi (Γj;k − Γk;j )v j + Gi (Γji′ ;k − Γk;j + Gi′ (Γj;k − Γk;j )v j + Gi′ (Γji′ ;k − Γk;j ′ )v ′ )v ∂f2 ′ ∂f1 i i − Γk;j )v j − 2gk f2 j ′ v j + 2hf1 k . = f22 gi (Γj;k ∂z ∂z Similary, By putting µ = k ′ in (5.1), we obtain

′

∂f2 ∂f1 j v + 2gf2 k′ . ∂z j ∂z M2 , F ) is a weakly K¨ahler Finsler manifold if and only if ′

′

′

α β Gα (Γkα′ ;β − Γβ;k = f12 hi′ (Γji′ ;k′ − Γki′ ;j ′ )v j − 2hk′ f1 ′ )v

By Definition 5.1, (f2 M1 ×f1

i i f22 gi (Γj;k − Γk;j )v j − 2gk f2

∂f2 j ′ ∂f1 + 2hf1 k = 0, ′ v j ∂z ∂z

(5.2)

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′

′

f12 hi′ (Γji′ ;k′ − Γki′ ;j ′ )v j − 2hk′ f1

∂f2 ∂f1 j v + 2gf2 k′ = 0. ∂z j ∂z

1759 (5.3)

′

i i Note that gi (Γj;k − Γk;j )v j and f1 are independent of v k , thus differentiating (5.2) with respect ′ to v k , we get ∂f2 ∂f1 2hk′ f1 k = 2gk f2 k′ . (5.4) ∂z ∂z

Interchanging indices j and k in (5.2), and then contracting the obtained equality with v j , we get ∂f2 ′ ∂f1 i i (5.5) f22 gi (Γk;j − Γj;k )v k v j − 2gj f2 k′ v k v j + 2hf1 j v j = 0. ∂z ∂z ′

Contracting (5.3) with v k , we get ∂f2 ′ ∂f1 j k′ v v + 2gf2 k′ v k = 0. j ∂z ∂z Now subtracting (5.6) from (5.5) and using (5.4), we get ′

′

′

′

f12 hi′ (Γji′ ;k′ − Γki′ ;j ′ )v j v k − 2hk′ f1

′ ′ ′ ′ ∂f2 ′ ∂f1 j v = f12 hi′ (Γji′ ;k′ − Γki′ ;j ′ )v j v k + 2gf2 k′ v k . j ∂z ∂z Since (M1 , F1 ) and (M2 , F2 ) are weakly K¨ ahler Finsler manifolds, we have

i i f22 gi (Γk;j − Γj;k )v k v j + 2hf1

i i gi (Γk;j − Γj;k )v k = 0,

′

′

(5.6)

(5.7)

′

hi′ (Γji′ ;k′ − Γki′ ;j ′ )v j = 0.

Thus we obtain the following differential equation 1 ∂f2 ′ 1 ∂f1 j f1 v = f2 k ′ v k . g ∂z j h ∂z

(5.8)

Note that the left hand side of (5.8) depends only on (z1 , v1 ), while the right hand side of (5.8) depends only on (z2 , v2 ), thus we get 1 ∂f1 j 1 ∂f2 ′ f1 v = f2 k′ v k = c. g ∂z j h ∂z

(5.9)

Suppose c 6= 0, then f1

∂f1 j v = cg, ∂z j

f2

∂f2 k′ v = ch. ∂z k′

(5.10) ′

Differentiating the above two equations with respect to v i , v j and v i , v j ′ , respectively, we get gij = 0,

hi′ j ′ = 0.

This is a contradiction since (gij ) and (hi′ j ′ ) are positive definite matrices. Thus it necessary that c = 0, which implies that f1 and f2 are positive constants.  As an immediate consequence of relations (2.4), we have Theorem 5.4 The DWP-complex Finsler manifold (f2 M1 ×f1 M2 , F ) is a Hermitian manifold if and only if (M1 , F1 ) and (M2 , F2 ) are Hermitian manifolds. For the case of complex Berwald manifold, weakly complex Berwald manifold and complex Landsberg manifolds, we need the following definitions. Definition 5.5 (see [4]) A complex Finsler manifold (M, F ) is called a complex Berwald α manifold if the horizontal connection coefficients Γβ;µ (z, v) of the Chern-Finsler connection are independent of fibre coordinates v and its associated Hermitian metric hF is a K¨ahler metric on M .

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Remark 5.6 In [2], Aikou gave the definition of complex Berwald manifold in which there is no requirement of the K¨ ahler Finsler condition. But different from real Finsler geometry, in complex Finsler geometry, there exists two different covariant derivative for Cartan tensor, Cijk|h and Cijk|h . The requirement of the K¨ ahler Finsler condition in [4] implies that Cijk|h = 0. Later the above definition of complex Berwald manifold was given by Aikou himself [4], and was widely used in various topics [3, 5]. In this paper, complex Berwald manifold is in the sense of Aikou [4]. Definition 5.7 (see [34]) Let F be a strongly pseudoconvex complex Finsler metric on M . If locally the connection coefficients Gα βµ (z, v) of the associated complex Berwald connection are independent of fibre coordinates v, then F is called a weakly complex Berwald metric. Definition 5.8 (see [5]) Let F be a complex Finsler metric on complex manifold M . F is said to be a complex Landsberg metric if it satisfies Gγνµ = Lγνµ .

(5.11)

Theorem 5.9 The DWP-complex Finsler manifold (f2 M1 ×f1 M2 , F ) is a complex Berwald manifold if and only if (M1 , F1 ) and (M2 , F2 ) are complex Berwald manifolds and the functions f1 and f2 are positive constants. Proof According to Theorem 5.2 and Definition 5.5, (f2 M1 ×f1 M2 , F ) is a complex Berwald manifold if and only if (M1 , F1 ) and (M2 , F2 ) are K¨ahler manifolds, the functions f1 α α and f2 are positive constants and Γβ;µ (z, v) = Γβ;µ (z). Thus by the relations of the coefficient α Γβ;µ of the Chern-Finsler connection in Proposition 3.2, we have 1

1

i i Γj;k (π1 (z), dπ1 (v)) =Γj;k (dπ1 (v)), 2

2

i i Γj;k (π2 (z), dπ2 (v)) =Γj;k (dπ2 (v)).

This is equivalent to the conditions that (M1 , F1 ) and (M2 , F2 ) are complex Berwald manifolds and the functions f1 and f2 are positive constants.  In the sense of Aikou [3, 4], a complex Finsler manifold (M, F ) is said to be modeled α on a complex Minkowski space if the connection coefficients Γβ;µ (z, v) of the Chern-Finsler α α connection depend only on the coordinates of the base manifold M : Γβ;µ = Γβ;µ (z). So, according to Proposition 3.2, we have Corollary 5.10 The DWP-complex Finsler manifold (f2 M1 ×f1 M2 , F ) is modeled on a complex Minkowski space if and only if (M1 , F1 ) and (M2 , F2 ) are modeled on a complex Minkowski spaces. In the sense of Aikou [2], a complex Finsler manifold (M, F ) is said to be complex locally Minkowski, if there exits an open cover {U, XU } such that on each πT−1 (U ) the function F is a function of the fibre-coordinate only. A complex Finsler manifold (M, F ) is complex locally Minkowski if and only if it is modeled on a complex Minkowski space and the complex Rund α connection (i.e., the connection coefficients Γβ;µ (z, v) of the associated Chern-Finsler connection) on (M, F ) is holomorphic. In [4], Aikou gave an example of complex manifold which is modeled on a complex Minkowski space, but not complex locally Minkowski. According to Proposition 3.2, we have

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Corollary 5.11 If lnf1 and lnf2 are pluriharmonic functions on M1 and M2 , respectively, then the DWP-complex Finsler manifold (f2 M1 ×f1 M2 , F ) is a complex locally Minkowski if and only if (M1 , F1 ) and (M2 , F2 ) are complex locally Minkowski manifolds. Theorem 5.12 The DWP-complex Finsler manifold (f2 M1 ×f1 M2 , F ) is a weakly complex Berwald manifold if and only if (M1 , F1 ) and (M2 , F2 ) are weakly complex Berwald manifolds. Proof According to Definition 5.7, (f2 M1 ×f1 M2 , F ) is a weakly complex Berwald maniα fold if and only if Gα βµ (z, v) = Gβµ (z). Thus by the relations of the complex Berwald connection in Proposition 3.6, we have 1

1

Gij;k (π1 (z), dπ1 (v)) =Gij;k (dπ1 (v)),

2

2

Gij;k (π2 (z), dπ2 (v)) =Gij;k (dπ2 (v)),

which is equivalent to the condition that (M1 , F1 ) and (M2 , F2 ) are weakly complex Berwald manifolds.  Remark 5.13 It was shown in [34] that the complex Wrona metric is a weakly complex Berwald metric, but not a complex Berwald metric. This assertion was proved by showing α µ α v ≡ 0 while Γ;µ do not vanish identically. It was also shown in [35] that that Gα = 21 Γ;µ there are lots of unitary invariant strongly pseudoconvex complex Finsler metric which are weakly complex Berwald metrics. Theorem 5.12 provide us an effective way to construct weakly complex Berwald manifolds. Theorem 5.14 The DWP-complex Finsler manifold (f2 M1 ×f1 M2 , F ) is a complex Landsberg manifold if and only if (M1 , F1 ) and (M2 , F2 ) are complex Landsberg manifolds and the functions f1 and f2 are positive constants. Proof According to Definition 5.8, Propositions 3.6 and 3.8, (f2 M1 ×f1 M2 , F ) is a complex Landsberg manifold if and only if 1

1

∂f2 i ∂f2 δ = Lij ′ k = 0, Gijk′ = f2−1 k′ δji = Lijk′ = 0, ∂z j ′ k ∂z 2 2 ′ ′ ′ ′ ′ ′ ∂f1 ′ ∂f1 ′ Gij ′ k′ = Lji ′ k′ , Gij ′ k = f1−1 k δji ′ = Lij ′ k = 0, Gijk′ = f1−1 j δki ′ = Lijk′ = 0, ∂z ∂z which is equivalent to the condition that (M1 , F1 ) and (M2 , F2 ) are complex Landsberg manifolds and the functions f1 and f2 are positive constants.  Gijk = Lijk ,

Gij ′ k = f2−1

A complex n-dimensional complex Finsler space (M, F ) is called a G-Landsberg spaces if it is complex Landsberg and the spray coefficient Gα are holomorphic with respect to v, i.e., ∂˙µ (Gα ) = 0 [5]. Thus we have Corollary 5.15 The DWP-complex Finsler manifold (f2 M1 ×f1 M2 , F ) is a complex GLandsberg manifold if and only if (M1 , F1 ) and (M2 , F2 ) are complex G-Landsberg manifolds and the functions f1 and f2 are positive constants. Remark 5.16 In real Finsler Geometry, every Berwald space is a Landsberg space, the converse, however, is still an open problem [10]. This problem was studied by several authors [20, 25, 28, 29]. In complex Finsler geometry, there is also notions of complex Berwald metric and complex Landsberg metric. Every complex Berwald metric is complex Landsberg metric [5]. One may want to know whether every complex Landsberg metric is also a complex Berwald

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metric. By Theorems 5.9 and 5.14, it follows that these two class of manifolds are the same type complex Finsler manifolds under warped product operations, thus showing that it is impossible to construct such metrics by this method. But, if we have two complex Landsberg manifolds M1 and M2 which are not complex Berwald manifolds, we can use the above Theorems to produce such manifolds of dimension dimC M1 + dimC M2 , as follows. Let M1 and M2 be two complex Landsberg manifolds, but they are not complex Berwald manifolds. Now, for any positive constant functions f1 : M1 → R+ and f2 : M2 → R+ , Theorem 5.14 ensures that the DWP-complex Finsler manifold (f2 M1 ×f1 M2 , F ) is a complex Landsberg manifold. Theorem 5.9 guarantees that it is not a complex Berwald manifold, otherwise, M1 and M2 must be complex Berwald manifolds, which is a contradiction to the choice of M1 and M2 . Practical application of this method depends on the existence of a non-Berwald complex Landsberg manifold to start with. If there is, we would prefer to select one with the lowest dimension. Let us have complex Landsberg manifolds which are not Berwald, ordered according to their dimension. Then, by well-ordering principle, there exists a natural number that is the lowest dimension of such manifolds, and there exists a manifold of this type, which has this dimension. We have the following corollary of the theorems. Corollary 5.17 Let C be the collection of complex Landsberg manifolds which are not complex Berwald manifold. Assume that C is non-empty. Let m0 be the lowest dimension. Let M be a complex Finsler manifold in C with dimension m0 , then M cannot be of warped product type. Proof Let, on the contrary, M be of warped product type, that is, M = (f2 M1 ×f1 M2 , F ) for some functions f1 : M1 → R+ and f2 : M2 → R+ , then, by Theorem 5.14, M1 and M2 are complex Landsberg manifold, and according to Theorem 5.9, M1 and M2 are not complex Berwald manifolds, otherwise M itself should be a complex Berwald manifold. But, as dimC M1 ≥ 1, M1 itself is a member of C, whose dimension should less than m0 , contradicting the choice of m0 . 

6

Geodesics of Doubly Warped Product Complex Finsler Manifold In this section, we shall investigate geodesics of DWP-complex Finsler manifold.

A real geodesic σ α (t) with an affine parameter t of a complex Finsler manifold (M, F ) satisfies the following equation [1]: γ γ α µ σ˙ = Gνα Gβγ (Γµ;ν σ ¨ α (t) + Γ;µ − Γν;µ )σ˙ β σ˙ µ .

(6.1)

Proposition 6.1 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of (M1 , F1 ) ′ and (M2 , F2 ). If σ(t) = (σ α (t)) = (σ k (t), σ k (t)) is a geodesic of (f2 M1 ×f1 M2 , F ), then it is necessary that h ∂f1 i ∂f2 r′ k k µ i r l − Γ l )σ σ˙ = f2−2 g sk f22 gil (Γr;s σ ¨ k (t) + Γ;µ ˙ + 2f1 h s − 2f2−1 σ˙ v , (6.2) s;r ˙ σ ∂z ∂z r′ h i ′ ′ ′ ∂f2 ∂f1 ′ ′ ′ ′ k′ µ σ ¨ k (t) + Γ;µ σ˙ = f1−2 hs k f12 hi′ l′ (Γrl′ ;s′ − Γsl′ ;r′ )σ˙ i σ˙ r′ + 2f2 g . − 2f1−1 r σ˙ r v k . ′ ∂z ∂z s

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Proof By putting α = k in (6.1), using (2.4), (2.5) and Proposition 3.2, after a long computation, we obtain γ γ k µ σ ¨ k (t) + Γ;µ σ˙ = Gνk Gβγ (Γµ;ν − Γν;µ )σ˙ β σ˙ µ h i r i r′ l )σ l − Γ l )σ ˙ + Gil (Γrl′ ;s − Γs;r ˙ = f2−2 g sk Gil (Γr;s ′ ˙ σ s;r ˙ σ ′

′

i r i r′ l )σ l − Γ l )σ +Gi′ l (Γr;s ˙ + Gi′ l (Γrl′ ;s − Γs;r σ˙ ′ ˙ s;r ˙ σ i r′ i r l )σ l′ − Γ l′ )σ +Gil′ (Γr;s ˙ + Gil′ (Γrl′ ;s − Γs;r ˙ ′ ˙ σ s;r ˙ σ ′

′

i ′ i′ r ′ i′ r l′ )σ l′ − Γ l′ )σ ˙ + Gi′ l′ (Γrl′ ;s − Γs;r σ˙ +Gi′ l′ (Γr;s ′ ˙ s;r ˙ σ h ∂f2 r′ k ∂f1 i i r l − Γ l )σ = f2−2 g sk f22 gil (Γr;s ˙ + 2f1 h s − 2f2−1 σ˙ v . s;r ˙ σ ∂z ∂z r′

Similary, by putting α = k ′ in (6.1), using (2.4), (2.5) and Proposition 3.2, we obtain

h ′ ′ ′ ∂f2 i ∂f1 ′ ′ ′ ′ k′ µ σ ¨ k (t) + Γ;µ σ˙ = f1−2 hs k f12 hi′ l′ (Γrl′ ;s′ − Γsl′ ;r′ )σ˙ i σ˙ r′ + 2f2 g − 2f1−1 r σ˙ r v k , ′ ∂z ∂z s which completes the proof.



Corollary 6.2 Let (M1 ×f1 M2 , F ) be a product complex Finsler manifold of (M1 , F1 ) ′ and (M2 , F2 ). If {σ k (t)} and {σ k (t)} are real geodesics of F1 and F2 , respectively. Then ′ {σ α (t)} = {σ k (t), σ k (t)} is a geodesic of F if and only if f1 is a positive constant function on M1 . Theorem 6.3 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of (M1 , F1 ) and (M2 , F2 ). (i) If the warping function f1 (resp.f2 ) is a positive constant on M1 (resp. M2 ), then any geodesic of (M1 , F1 ) (resp. (M2 , F2 )) is a geodesic of (f2 M1 ×f1 M2 , F ), that is to say, (M1 , F1 ) (resp. (M2 , F2 )) is a totally geodesic subspace of the doubly warped product complex Finsler space (f2 M1 ×f1 M2 , F ). (ii) If the warping function f1 (resp. f2 ) is a positive constant on M1 (resp. M2 ), then the projection of any geodesic of the DWP-complex Finsler manifold (f2 M1 ×f1 M2 , F ) onto M1 (resp.M2 ) is a geodesic of (M1 , F1 ) (resp. (M2 , F2 )). Proof Since the proof of (ii) is similar to (i), we only prove (i). If (σ k (t)) is a geodesic of (M1 , F1 ), then i r l − Γ l )σ σ ¨ k (t) + Γ;ik σ˙ i = g sk gil (Γr;s ˙ . s;r ˙ σ ′

′

(6.3) ′

Consider the curve (σ k (t), σ k ), where σ k is a constant. We have σ˙ i = 0, thus the differential equation (6.2) reduces to h ∂f1 i i r l − Γ l )σ ˙ σ ˙ + 2f h , σ ¨ k (t) + Γ;ik σ˙ i = f2−2 g sk f22 gil (Γr;s 1 s;r ∂z s ′

(6.4)

k µ where we used Γ;µ σ˙ = Γ;ik σ˙ i + Γ;ik′ σ˙ i = Γ;ik σ˙ i . Notice that the warping function f1 is a positive constant on M1 , the differential equation (6.4) coincides with (6.3). So that the curve ′  (z k (t), z k ) is a geodesic of (f2 M1 ×f1 M2 , F ).

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Projective Flatness of Doubly Warped Product Complex Finsler Manifold

Let F (x, u) be a real Finsler metric defined on a domain D ⊂ Rn . F is called projectively flat if its geodesic are straight lines [14]. This is equivalent to the following system of PDEs [15] n X ∂2F ∂F ub = b a ∂x ∂u ∂xa

for a = 1, · · · , n,

b=1

where x = (x1 , · · · , xn ) ∈ D and u = (u1 , · · · , un ) ∈ Tx D, F is called dually flat if and only if F satisfies [26] n X ∂2F 2 b ∂F 2 u = 2 a for a = 1, · · · , n. b a ∂x ∂u ∂x b=1

For a strongly convex complex Finsler metric, it makes sense to talk about its projectiveness. Let (M1 , F1 ) and (M2 , F2 ) be two locally projectively flat or dually flat Finsler manifold, one may want to know whether the DWP-complex Finsler manifold (f2 M1 ×f1 M2 , F ) is also a locally projectively flat or dually flat complex Finsler manifold. For this problem we need the following proposition. Proposition 7.1 (see [35]) Let F be a strongly convex complex Finsler metric defined on a domain D ⊂ Cn . Then (i) F is locally projectively flat if and only if F;α = Fα;β v β + Fα;β v β ,

∀α = 1, · · · , n.

(7.1)

(ii) F is locally dually flat if and only if 2G;α = Gα;β v β + Gα;β v β ,

∀α = 1, · · · , n.

(7.2)

It is obviously that F is a strongly convex complex Finsler metric whenever F1 and F2 are strongly convex complex Finsler metrics. Thus it makes sense to investigate the projective flatness or dual flatness of F . Theorem 7.2 Let (f2 M1 ×f1 M2 , F ) be a DWP-complex Finsler manifold of two strongly convex complex Finsler manifolds (M1 , F1 ) and (M2 , F2 ). (i) If (M1 , F1 ) and (M2 , F2 ) are locally projectively flat manifolds, then (f2 M1 ×f1 M2 , F ) is a locally projectively flat manifold if and only if the functions f1 and f2 are positive constants. (ii) If (M1 , F1 ) and (M2 , F2 ) are locally dually flat manifolds, then (f2 M1 ×f1 M2 , F ) is a ∂f2 j ′ 1 i locally dually flat manifold if and only if f1 h ∂f is a real value function. ∂z i v + f2 g ∂z j′ v Proof (i) According to Proposition 7.1, (f2 M1 ×f1 M2 , F ) is locally projectively flat if and only if ′

F;i = Fi,j v j + Fi,j v j + Fi,j ′ v j + Fi,j ′ v j ′ , j′

j

(7.3) j′

F;i′ = Fi′ ,j v + Fi′ ,j v j + Fi′ ,j ′ v + Fi′ ,j ′ v .

(7.4)

′

Contracting (7.3) and (7.4) with respect to v i and v i , respectively, and compare the obtained equations, we get ′

F;i v i = F;i′ v i .

(7.5)

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Using (2.2), and notice (M1 , F1 ) and (M2 , F2 ) are projectively flat manifolds, we obtain f22 F1

∂F1 ∂z j

v j + f1 F22

∂f1 i ∂F2 j ′ ∂f2 ′ v = f12 F2 v + f2 F12 i′ v i . ′ i ∂z ∂z ∂z j

(7.6)

′

Differentiating (7.6) with respect to v s and v t , we get 2f1 F2

∂f1 ∂F2 ∂f2 ∂F1 . = 2f2 F1 t′ ∂z s ∂v t′ ∂z ∂v s

(7.7)

′

Contracting (7.7) with respect to v s and v t , we get 1 ∂f1 s 1 ∂f2 ′ f1 s v = f2 t′ v t . g ∂z h ∂z

(7.8)

Since the left hand side of (7.8) depends only on (z1 , v1 ), while the right hand side of (7.8) depends only on (z2 , v2 ), it follows that 1 ∂f1 s 1 ∂f2 ′ f1 v = f2 t′ v t = c, g ∂z s h ∂z

(7.9)

which implies that the functions f1 and f2 are positive constants. (ii) According to Proposition 7.1, (f2 M1 ×f1 M2 , F ) is locally dually flat if and only if   2G = G v j + G v j + G ′ v j ′ + G ′ v j ′ ;i i,j i,j i,j i,j  2G;i′ = Gi′ ,j v j + G ′ v j + Gi′ ,j ′ v j ′ + G ′ ′ v j ′ i ,j

i ,j

Using (2.2), and notice (M1 , F1 ) and (M2 , F2 ) are locally dually flat complex Finsler manifolds, we find that this is equivalent to  ∂f ∂f2 j ′  ∂f1 2 j′ v + v , (7.10) 2f1 h i = f2 gi ∂z ∂z j ′ ∂z j ′  ∂f ∂f2 ∂f1 j  1 j 2f2 g i′ = f1 hi′ v + v . (7.11) ∂z ∂z j ∂z j ′

Contracting (7.10) and (7.11) with v i and v i , respectively, we obtain  ∂f ∂f1 ∂f2 j ′  2 j′ 2f1 h i v i = f2 g v + v , ′ ∂z ∂z j ∂z j ′  ∂f1 j ∂f1 j  ∂f2 ′ 2f2 g i′ v i = f1 h v + v . ∂z ∂z j ∂z j

(7.12) (7.13)

Interchanging indices i and j of (7.13), and adding (7.12), we get f1 h

∂f2 ′ ∂f2 j ′ ∂f1 i ∂f1 i v + f2 g j ′ v j = f1 h v + f2 g v . i i ∂z ∂z ∂z ∂z j ′

(7.14)

Since the left hand side of (7.14) is conjugate to the right hand side of (7.14), thus f1 h

∂f1 i ∂f2 ′ v + f2 g j ′ v j ∂z i ∂z

is a real value function.

(7.15) 

References [1] Abate M, Patrizio G. Finsler Metrics—A Global Approach with Applications to Geometric Function Theory. Lecture Notes in Mathematics, Volume 1591. Berlin Aeidelberg: Springer-Verlag, 1994 [2] Aikou T. On complex Finsler manifolds. Rep Fac Sci Kagoshima Univ (Math Phys & Chem), 1991, 24: 9–25

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[3] Aikou T. Complex manifolds modeled on a complex Minkowski space. J Math Kyoto Univ (JMKYAZ), 1998, 35-1: 85–103 [4] Aikou T. Some remarks on locally conformal complex Berwald spaces. Contem Math, 1996, 196: 109–120 [5] Aldea N, Munteanu G. On complex Landsberg and Berwald spaces. J Geom Phys, 2012, 62(2): 368–380 [6] Asanov G S. Finslerian extensions of Schwarzschild metric. Fortschr Phys, 1992, 40: 667–693 [7] Asanov G S. Finslerian metric functions over the product R × M and their potential applications. Rep Math Phys, 1998, 41: 117–132 [8] Baagherzadeh Hushmandi A, Rezaii M M. On warped product Finsler spaces of Landsberg type. J Math Phys, 2011, 52, No. 9093506.17 [9] Baagherzadeh Hushmandi A, Rezaii M M, Morteza M. On the curvature of warped product Finsler spaces and the Laplacian of the Sasaki-Finsler metrics. J Geom Phys, 2012, 62(10): 2077–2098 [10] Bao D, Chern S S, Shen Z. An Introduction to Riemann-Finsler Geometry. GTM 200. Springer-Verlag, 2000 [11] Beem J K, Ehrlich P, Powell T G. Warped product manifolds in relativity in selected studies: a volume dedicated to the memory of Albert Einstein (T. M. Rassias and G. M. Rassias eds.). Armsterdarm: NorthHolland, 1982: 41–56 [12] Bishop R L, O’Neill B. Manifolds of negative curvature. Trans Amer Math Soc, 1969, 145: 1–49 [13] Chen B, Shen Y B. K¨ ahler Finsler metrics are actually strongly K¨ ahler. Chin Ann Math Ser B, 2009, 30(2): 173–178 [14] Chern S S, Shen Z. Riemann-Finsler geometry. World Scientific, 2005 ¨ [15] Hamel G. Uber die Geometrieen in denen die Geraden die K¨ urzesten sind. Math Ann, 1903, 57: 231–264 [16] Katanaev O M, Kl¨ osch T, Kummer W. Global properties of warped solutions in general relativity. Ann Physics, 1999, 276: 191–222 [17] Kobayashi S. Negative vector bundles and complex Finsler structures. Nagoya Math J, 1975, 57: 153–166 [18] Kozma L, Peter I R, Varga C. Warped product of Finsler-manifolds. Ann Univ Sci Budapest, 2001, 44: 157 [19] Matsumoto M. Foundations of Finsler Geometry and Special Finsler Spaces. Kaiseisha Press, 1986 [20] Matsumoto M. Remarks on Berwald and Landsberg spaces, in Finsler Geometry. Contem Math, 1996, 196: 79–82 [21] Munteanu G. Complex Spaces in Finsler, Lagrange and Hamilton Geometries. Kluwer Academic Publishers, 2004 [22] O’Neill B. Semi-Riemannian Geometry. New York: Academic Publisher, 1983 [23] Peyghan E, Tayebi A. On doubly warped product Finsler manifolds. Nonlinear Anal Real World Appl, 2012, 13(4): 1703–1720 [24] Rund H. The curvature theory of direction-dependent connections on complex manifolds. Tensor N S, 1972, 24: 189–205 [25] Shen Z. On a class of Landsberg metrics in Finsler geometry. Can J Math, 2009, 61(6): 1357–1374 [26] Shen Z. Riemann-Finsler geometry with applications to information geometry. Chin Ann Math Ser B, 2006, 27(1): 73–94 [27] Sun L, Zhong C. Characterizations of complex Finsler connections and weakly complex Berwald metrics. Differential Geom Appl, 2013, 31: 648–671 [28] Szabso Z I. All regular Landsberg metrics are Berwald. Ann Global Anal Geom, 2008, 34(4): 381–386 [29] Szabso Z I. Correction to all regular Landsberg metrics are Berwald. Ann Global Anal Geom, 2009, 35(3): 227–230 [30] Unal B. Doubly Warped Products. Differ Geom Appl, 2001, 15(3): 253–263 [31] Wong B. On the holomorphic curvature of some intrinsic metrics. Proc Amer Math Soc, 1977, 65: 57–61 [32] Wu H. A remark on holomorphic sectional curvature. Indiana Univ Math J, 1973, 22: 1103–1108 [33] Wu Z, Zhong C. Some results on product complex Finsler manifolds. Acta Math Sci, 2011, 31B(4): 1541– 1552 [34] Zhong C. On real and complex Berwald connections associated to strongly convex weakly K¨ ahler-Finsler metric. Differential Geom Appl, 2011, 29: 388–408 [35] Zhong C. On unitary invariant strongly pseudoconvex complex Finsler metrics. Differential Geom Appl, 2015, 40: 159–186