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On strongly convex weakly Kähler–Finsler metrics of constant flag curvature
Accepted Manuscript On strongly convex weakly Kähler-Finsler metrics of constant flag curvature
Hongchuan Xia, Chunping Zhong
PII: DOI: Reference:
S0022-247X(16)30176-7 http://dx.doi.org/10.1016/j.jmaa.2016.05.034 YJMAA 20452
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
20 April 2016
Please cite this article in press as: H. Xia, C. Zhong, On strongly convex weakly Kähler-Finsler metrics of constant flag curvature, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2016.05.034
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On strongly convex weakly K¨ahler-Finsler metrics of constant flag curvature Hongchuan Xiaa,b , Chunping Zhongb,∗ a College
of Mathematics and Information Science, Xinyang Normal University, Xinyang 464000, P.R. China b School of Mathematical Sciences, Xiamen University, Xiamen 361005, P.R. China
Abstract In this paper, we first give a necessary and sufficient condition for a strongly convex weakly K¨ahler-Finsler metric to be of constant flag curvature, and then prove that (i) a strongly convex weakly K¨ahler-Finsler metric of constant flag curvature is necessary of constant holomorphic curvature; (ii) a strongly convex K¨ahler-Berwald metric on a complex manifold of complex dimension n ≥ 2 has constant flag curvature if and only if it comes from a strongly convex locally complex Minkowski metric. We also give two examples of nontrivial strongly convex K¨ahler-Finsler metrics. Keywords: flag curvature, holomorphic curvature, strongly convex, complex Finsler metric 2010 MSC: 53C60, 53C40
1. Introduction and main results In Riemannian geometry, it is a simple fact that a K¨ahler metric of constant sectional curvature is necessary of constant holomorphic sectional curvature. In real Finsler geometry, flag curvature is a natural generalization of the sectional curvature in Riemannian geometry [1]. In complex Finsler geometry, the notion of holomorphic curvature is well defined for strongly pseudoconvex complex Finsler metrics [2], and even for upper semicontinuous complex Finsler metrics [2, 3]. In general, however, real and complex Finsler metrics are not tightly related as that of in Riemannian and Hermitian geometry, we refer to [2] for more details. For a real Finsler metric it is usually asked to be strongly convex along tangent directions, while for a complex Finsler metric it is asked to be strongly pseudoconvex along tangent directions. A complex Finsler metric is not necessarily a real Finsler metric, and vice verse. A complex Finsler metric on a complex manifold is called strongly convex if it is also a real Finsler metric [2]. Recently we systematically investigate the unitary invariant complex Finsler metrics and general complex (α, β) metrics [4–7], as a byproduct we find that there are lots of strongly convex complex Finsler metrics. Thus a natural question arises: for strongly convex complex Finsler manifolds whether there are relationships among the geometric objects in real and complex Finsler geometry? As we know, flag curvature and holomorphic curvature play fundamental roles respectively in real and complex Finsler geometry, one may wonder whether there exists relationship between the flag curvature and holomorphic curvature for a strongly convex complex Finsler metric? As our first main result we prove: ∗ Corresponding
author Email addresses: [email protected] (Hongchuan Xia), [email protected] (Chunping Zhong)
Preprint submitted to Journal of Mathematical Analysis and Applications
May 19, 2016
Theorem 1.1. Let F be a strongly convex weakly K¨ahler-Finsler metric on a complex manifold M. If F is of constant flag curvature Kˆ F = k, then F is of constant holomorphic curvature KF = k. Note that there are lots of real Finsler metrics which are of constant flag curvature in literature [8–10]. Theorem 1.1 indeed provides a possible approach to construct complex Finsler metrics of constant holomorphic curvature via real Finsler metrics of constant flag curvature. In particular, under condition that F is a K¨ahler-Berwald metric we get the following rigid result as our second main result. Theorem 1.2. Let F be a strongly convex K¨ahler-Berwald metric on a complex manifold M of complex dimension n ≥ 2. Then F is of constant flag curvature if and only if F is a locally complex Minkowski metric. Our last main result establish a relationship between real Einstein-Finsler metrics and the Ricci scalar curvature (see Definition 4.1). Theorem 1.3. Let F be a strongly convex K¨ahler-Berwald metric on a complex n-dimensional manifold M. Then F is a real Einstein-Finsler metric if and only if the Ricci scalar curvature Ric of F satisfying 2 Re Ric = (2n − 1)σ(z)G for some real scalar function σ(z) on M. The arrangements of this paper are as follows. In section 2, we introduce some necessary definitions and notions. In section 3, we give some lemmas which will be used in this paper. In section 4, we shall first establish a necessary and sufficient condition for a strongly convex weakly K¨ahler-Finsler metric to be of constant flag curvature. This condition is formulated into a system of six equations, which are used to prove our main results (Theorems 1.1–1.3). In section 5, we shall give two examples of strongly convex K¨ahler-Finsler metrics, one comes from a strongly convex complex Minkowski metric in Cn , the other is a strongly convex K¨ahler-Finsler metric which does not come from a strongly convex complex Minkowski metric.
2. Preliminaries Let’s recall some necessary notations and definitions, which can be found in [2]. Let M be a complex ndimensional manifold with the canonical complex structure J. Denote by T R M the real tangent bundle, and T C M the complexified tangent bundle of M. Then J acts complex linearly on T C M so that T C M = T 1,0 M ⊕T 0,1 M, where T 1,0 M is called the holomorphic tangent bundle of M. T 1,0 M is a complex manifold of complex dimension 2n, and we also denote by J the complex structure on T 1,0 M if it causes no confusion. Let {z1 , · · · , zn } be a set of local √ complex coordinates on M, with zα = xα + −1xα+n , so that {x1 , · · · , xn , x1+n , · · · , x2n } are local real coordinates √ on M. Denote by {z1 , · · · , zn , v1 , · · · , vn } the induced complex coordinates on T 1,0 M, with vα = uα + −1uα+n , so that {x1 , · · · , x2n , u1 , · · · , u2n } are local real coordinates on T R M. In the following, we shall denote M˜ the complement of zero section in T R M or T 1,0 M, depending on the real or complex situation. Unless otherwise stated we always assume that lowercase Greek indices such as α, β, γ etc., run 2
from 1 to n, whereas lowercase roman indices such as a, b, c, etc., run from 1 to 2n, and the Einstein summation convention is assumed throughout the paper. Set √ ∂ 1 ∂ ∂ = − −1 , ∂zα 2 ∂xα ∂xα+n √ ∂ 1 ∂ ∂ − −1 , ∂˙ α := α = ∂v 2 ∂uα ∂uα+n
√ ∂ 1 ∂ ∂ = + −1 , ∂xα+n ∂zα 2 ∂xα √ ∂ 1 ∂ ∂ = + −1 ∂˙ α := . ∂uα+n ∂vα 2 ∂uα
∂α :=
∂α :=
The bundles T 1,0 M and T R M are isomorphic. We choose the explicit isomorphism o : T 1,0 M → T R M with its inverse o : T R M → T 1,0 M, which are respectively given by ∀ v = vα
T R M u = vo = v + v, and T 1,0 M v = uo =
√ 1 (u − −1Ju), 2
∂ ∈ T 1,0 M ∂zα
∀ u = ua
∂ ∈ T R M. ∂xa
Definition 2.1 ([2]). A real Finsler metric on a manifold M is a function F : T R M → R+ satisfying the following properties: ˜ (a) G = F 2 is smooth on M; ˜ (b) F(p, u) > 0 for (p, u) ∈ M; (c) F(p, λu) = |λ|F(p, u) for all (p, u) ∈ T R M and λ ∈ R; (d) for any p ∈ M the indicatrix IF (p) = {u ∈ T p M| F(p, u) < 1} is strongly convex. Note that condition (d) is equivalent to the following matrix 2 ∂G (Gab ) = ∂ua ∂ub
(2.1)
˜ being positive definite on M. Definition 2.2 ([2]). A complex Finsler metric F on a complex manifold M is a continuous function F : T 1,0 M → R+ satisfying ˜ (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. Definition 2.3 ([2]). A complex Finsler metric F is called strongly pseudoconvex if the Levi matrix 2 ∂G (Gαβ ) = ∂vα ∂vβ ˜ is positive definite on M. Let F : T 1,0 M → R+ be a complex Finsler metric on a complex manifold M. Using the complex structure J on M and the bundle map o : T R M → T 1,0 M we can define a real function F o : T R M → R+ , F o (u) := F(uo ), ∀ u ∈ T R M. 3
(2.2)
Definition 2.4 ([2]). A complex Finsler metric F is called strongly convex if the associated function F o is a real Finsler metric. For a strongly convex complex Finsler metric F, we use the same symbol F to denote the associated real Finsler metric F o with the understanding that F(u) is defined by F(uo ) for u ∈ T R M. ˜ we shall use a semi-colon to distinguish between derivatives with In the following, for functions defined on M, respect to z and v in the complex setting (resp. x and u in the real setting). For example, Gα =
∂G , ∂vα
G;α =
∂G , ∂zα
Gα;β =
Ga =
∂G , ∂ua
G;a =
∂G , ∂xa
Ga;b
∂2G
, ∂vα ∂zβ ∂2G = a b. ∂u ∂x
ˆ a and Gα the real and complex spray coefficients associated to a strongly convex complex Finsler Denote G metric F, respectively, where ˆ a = 1 Gab (Gb;c uc − G;b ), G 2
Gα =
1 α β Γ v , 2 ;β
α Γ;β = GταGτ;β ,
α are called the Chern-Finsler nonlinear connection coefficients. Gαβ = ∂˙ β Gα are called the complex Berwald and Γ;β
nonlinear connection coefficients. The corresponding horizontal frames {δμ } and {δˇ μ } associated to the two nonlinear connections are respectively given by α ˙ ∂α , δμ = ∂μ − Γ;μ
δˇ μ = ∂μ − Gαμ ∂˙ α .
ˆ a by The real nonlinear connection coefficients Γˆba are related to G ˆa ∂G Γˆ ba = , ∂ub
ˆ a = 1 Γˆ a ub . G 2 b
α and complex Berwald connection coefficients Gαβμ The horizontal Chern-Finsler connection coefficients Γμ;β α are respectively related to Γ;β and Gαβ by α α = ∂˙ μ Γ;β , Γμ;β
Gαβμ = ∂˙ μ Gαβ ,
α μ α Γμ;β v = Γ;β ,
Gαβμ vμ = Gαβ .
α α α Γβ;μ , Γβ;μ Gαβμ . But for K¨ahler-Finsler metrics, the two It is clear that Gαβμ = Gαμβ , in general, however, Γμ;β α = Gαβμ (see Lemma 3.5). connection coefficients coincide, i.e., Γβ;μ
Definition 2.5 ([2]). Let F is a strongly pseudoconvex complex Finsler metric on a complex manifold M. F is α α called a strongly K¨ahler-Finsler metric if in local coordinates, Γβ;μ − Γμ;β = 0; called a K¨ahler-Finsler metric if α α α α − Γμ;β )vμ = 0; called a weakly K¨ahler-Finsler metric if Gα (Γβ;μ − Γμ;β )vμ = 0. (Γβ;μ
In [11], Chen and Shen proved that a K¨ahler-Finsler metric is actually a strongly K¨ahler-Finsler metric, so there are two notions of K¨ahler-Finsler metric in complex Finsler geometry. Definition 2.6 ([12, 13]). Let F is a strongly pseudoconvex complex Finsler metric on a complex manifold M. F α is called a complex Berwald metric if locally Γβ;μ depend only on the base manifold coordinates z = (z1 , · · · , zn );
F is called a weakly complex Berwald metric if locally Gαβμ depend only on z. 4
A complex Berwald metric is a weakly complex Berwald metric, the converse, however, is not true [13]. It was shown in [4, 5] that there are lots of non-Hermitian smooth weakly complex Berwald metrics. Theorem 2.1 ([10]). A real Finsler manifold (M, F) is of constant flag curvature k if and only if 1 a a b a R c = k Gδc − Gcb u u , 2
(2.3)
where Rac is the Riemann curvature tensor and given by Rac = 2
2ˆa ˆa ˆa ˆb ˆa ∂G ∂2 G ˆ b ∂ G − ∂G ∂G . − ub b c + 2G c b c ∂x ∂x ∂u ∂u ∂u ∂ub ∂uc
(2.4)
Definition 2.7 ([10]). A real Finsler metric F on a real n-dimensional manifold M is called an Einstein-Finsler metric if 2 , Ric = (n − 1)σ(x)F ˆ
where σ(x) ˆ is a real scalar function on M, Ric = Raa is the Ricci curvature of F. It is clear that a real Finsler metric of constant flag curvature is necessarily an Einstein-Finsler metric.
3. Some lemmas In the following, we set ⎧ ⎪ ⎪ ⎪ if 1 ≤ a ≤ n, ⎨ a, α=⎪ ⎪ ⎪ ⎩ a − n, if n ≤ a ≤ 2n,
⎧ ⎪ ⎪ ⎪ ⎨ b, β=⎪ ⎪ ⎪ ⎩ b − n,
if 1 ≤ b ≤ n, if n ≤ b ≤ 2n,
⎧ ⎪ ⎪ ⎪ if 1 ≤ c ≤ n, ⎨ c, γ=⎪ ⎪ ⎪ ⎩ c − n, if n ≤ c ≤ 2n.
Lemma 3.1. Let F be a strongly convex complex Finsler metric on a complex manifold M. Then F is a weakly K¨ahler-Finsler metric if and only if ⎧ 1 ⎪ ⎪ ⎪ Re Gβ = (Gβ + Gβ ), ⎪ ⎪ ⎪ 2 ⎨ ˆb = ⎪ G ⎪ 1 ⎪ β β β ⎪ ⎪ ⎪ ⎩ Im G = √ (G − G ), 2 −1
if 1 ≤ b ≤ n, if n + 1 ≤ b ≤ 2n,
where Gβ denote the complex conjugation of Gβ , i.e., Gβ = Gβ . Proof. Since a strongly convex complex Finsler metric F is a weakly K¨ahler-Finsler metric if and only if ( page 114, (2.6.10) in [2]) β ˙ o ∂β ) , ua Γˆ ab ∂˙ ob = (vα Γ;α
or equivalently, ˆ b ∂˙ o = (Gβ ∂˙ β )o . G b Hence our assertion follows.
5
Lemma 3.2. Let F be a strongly pseudoconvex complex Finsler metric on a complex manifold M. Then the holomorphic curvature KF of F along a nonzero vector v ∈ T z1,0 M is given by 2 α μ ν Gα ∂ν (Γ;μ )v v G2 4 = − 2 Gα ∂ν (Gα )vν G 2 = − 2 G;μν vμ vν − 4Gαβ Gα Gβ . G
KF (z, v) = −
(3.1) (3.2) (3.3)
Proof. The holomorphic curvature formula of F is given by KF (z, v) = −
2 α μ ν Gα δν (Γ;μ )v v G2
(3.4)
α for every nonzero vector v ∈ T z1,0 M (see page 109 in [2]). Here in (3.4), δν = ∂ν − Γ;ντ ∂˙ τ and Γ;μ = GναGν;μ . It is
easy to check that α ) = 0, Gα ∂˙ τ (Γ;μ
α Gα Γ;μ = G;μ ,
from which we have α α α α α ) = Gα ∂ν (Γ;μ ) = ∂ν (Gα Γ;μ ) − Gα;ν Γ;μ = G;μν − Gαβ Γ;νβ Γ;μ . Gα δν (Γ;μ
(3.5)
α μ v = 2Gα , Γ;νβ vν = 2Gβ , we obtain (3.1)-(3.3). Substituting (3.5) into (3.4) and notice that Γ;μ
Corollary 3.1. Let F be a strongly pseudoconvex complex Finsler metric on a complex manifold M. Then the holomorphic curvature KF of F along a nonzero vector v ∈ T z1,0 M satisfies KF (z, v) ≥ −
2 G;μν vμ vν , G2
(3.6)
and if the above equality holds, then KF (z, v) ≡ 0. Proof. The inequality (3.6) follows from (3.3), since (Gαβ ) is a positive definite Hermitian matrix. The equality KF (z, v) = −
2 G;μν vμ vν G2
holds if and only if Gαβ Gα Gβ ≡ 0, if and only if Gα ≡ 0,
∀ α = 1, · · · , n.
In this case, it follows from (3.2) that KF (z, v) ≡ 0. Definition 3.1. A function f (v) defined on Cn − {0} is called homogeneous of type (p, q) with respect to v = (v1 , · · · , vn ) if it satisfies f (λv) = λ p λq f (v),
∀ λ ∈ C − {0}, ∀ v ∈ Cn − {0}.
Lemma 3.3. Suppose that Ai (v) (i = 1, · · · , N) are homogeneous functions of type (pi , qi ) with respect to v =
N Ai (v) = 0 if and only if (v1 , · · · , vn ), where pi , qi are integers and (p s , q s ) (pt , qt ) for s t. Then i=1 Ai (v) = 0,
∀ i = 1, · · · , N. 6
Proof. If
N i=1
Ai (v) = 0, then N
Ai (λv) =
i=1
N
λ pi λqi Ai (v) = 0,
∀ λ ∈ C − {0}.
(3.7)
i=1
Without loss of generality, we can assume pi > 0, qi > 0 for any i, otherwise we can multiply |λ|2m in (3.7) for positive integer m big enough such that pi + m > 0, qi + m > 0. Then for every fixed s ∈ {1, · · · , N}, applying the operator
1 ∂ p s +q s p s !q s ! ∂λ p s ∂λq s
to (3.7), we get 1 ∂ ps +qs p s !q s ! ∂λ ps ∂λqs
⎡ N ⎤ ⎢⎢⎢ p qi ⎥⎥ i ⎢⎢⎣ λ λ Ai (v)⎥⎥⎥⎦ = A s (v) = 0, i=1
from which our assertion follows. Lemma 3.4. [2] Let F be a complex Finsler metric on a complex manifold M. Then F is a strongly convex complex Finsler metric if and only if Re Gαβ W α W β + Gαβ W α W β > 0,
∀ W = (W 1 , · · · , W n ) ∈ Cn , W 0.
Lemma 3.5. Let F be a strongly pseudoconvex complex Finsler metric on a complex manifold M. Then F is a K¨ahler-Finsler metric if and only if α , Gαμ = Γ;μ
or equivalently,
α Gαμν = Γν;μ .
Proof. Since 2Gα = Γ;να vν , differentiating it with respect to vμ gives α ν α v + Γ;μ , 2Gαμ = Γμ;ν
thus α α ν α α α ) = Γμ;ν v − Γ;μ = (Γμ;ν − Γν;μ )vν . 2(Gαμ − Γ;μ
Hence F is a K¨ahler-Finsler metric if and only if α , Gαμ = Γ;μ
if and only if α . Gαμν = Γν;μ
Corollary 3.2. Let F be a K¨ahler-Finsler metric on a complex manifold M. Then F is a complex Berwald metric if and only if F is a weakly complex Berwald metric. Corollary 3.3. Suppose that F is a weakly K¨ahler-Finsler metric as well as a weakly complex Berwald metric on a complex manifold M. Then it is necessarily a K¨ahler-Berwald metric.
7
Proof. By Proposition 4.3.9 in [14], F is a weakly K¨ahler-Finsler metric if and only if δˇ μG = 0, or equivalently, G;μ = GαμGα .
(3.8)
Note that for a weakly complex Berwald metric F, we have 2Gα = Gαβμ (z)vβ vμ , which is holomorphic with respect to v, thus differentiating (3.8) with respect to vβ yields Gβ;μ = GαμβGα + GαμGαβ = GαμGαβ , α , that is, F is a K¨ahler-Finsler metric. By Corollary 3.2, F is also a complex Berwald which implies that Gαμ = Γ;μ
metric. Lemma 3.6. Let F be a K¨ahler-Berwald metric on a complex manifold M. Then 2Gα;γ − Gαγ;β vβ + 2Gαγβ Gβ − Gαβ Gβγ = 0. Proof. By (2.3.16) in [2], we have α ) − δμ (Γ;να ) = 0. δν (Γ;μ
(3.9)
σ ˙ Since [δμ , ∂˙ α ] = Γα;μ ∂σ (see Lemma 2.3.3 in [2]), differentiating (3.9) with respect to vβ gives α ) − ∂˙ β δμ (Γ;να ) 0 = ∂˙ β δν (Γ;μ α α − (δμ ∂˙ β − [δμ , ∂˙ β ])Γ;μ = (δν ∂˙ β − [δν , ∂˙ β ])Γ;μ γ γ α α α α = δν (Γβ;μ ) − δμ (Γβ;ν ) + Γγ;ν Γβ;μ − Γγ;μ Γβ;ν .
(3.10)
α and Gαγμ coincides and are independent Since F is a K¨ahler-Berwald metric, it follows from Lemma 3.5 that Γγ;μ
of fiber coordinate v, so (3.10) reduces to ∂ν (Gαβμ ) − ∂μ (Gαβν ) + Gαγν Gγβμ − Gαγμ Gγβν = 0.
(3.11)
Contracting (3.11) with vβ andvμ successively yields 2Gα;ν − Gαν;μ vμ + 2Gαγν Gν − Gαγ Gγν = 0.
Remark 3.1. The term emerged in the right hand side of (3.10) is called the second curvature tensor in [15].
8
4. Main results Proposition 4.1. Let F be a strongly convex weakly K¨ahler-Finsler metric on a complex manifold M. Then F is of constant flag curvature Kˆ F = k if and only if the following system of equations hold: Gαγ;β vβ − 2Gαγβ Gβ + Gαβ Gβγ = 0,
(4.1)
2Gα;γ − Gαγ;β vβ + 2Gαγβ Gβ −
Gαβ Gβγ
= 0,
(4.2)
Gαγ;β vβ − 2Gαγβ Gβ −
Gαβ Gβγ
= 0,
(4.3)
1 Gα;γ − Gαβ Gβγ = − kvαGγ , 4 1 α β Gβ Gγ = − k(Gδαγ − vαGγ ), 4 1 Gαγ;β vβ − 2Gαγβ Gβ = − k(3Gδαγ − vαGγ ). 4 Proof. By Theorem 2.1, F is of constant flag curvature Kˆ F = k if and only if 1 Rac = k Gδac − Gcb ub ua . 2
(4.4) (4.5) (4.6)
(4.7)
In the following, for clarity we shall divide our proof into two steps. Step 1. Let’s first convert the formula (4.7) into complex setting, that is, rewrite the real tensors in (4.7) into complex tensors. Since F is a weakly K¨ahler-Finsler metric, by Lemma 3.1 we have ⎧ ⎪ ⎪ ⎪ 12 (Gβ + Gβ ), if 1 ≤ b ≤ n, ˆb =⎨ G ⎪ ⎪ ⎪ 1 β β ⎩ √ (G − G ), if n + 1 ≤ b ≤ 2n.
(4.8)
2 −1
Note that ⎧ ⎪ ⎪ ⎪ 2 Re ∂γ , if 1 ≤ c ≤ n, ∂ ⎨ = ⎪ ⎪ ∂xc ⎪ ⎩ −2 Im ∂γ , if n + 1 ≤ c ≤ 2n, ⎧ ⎪ ⎪ ⎪ if 1 ≤ c ≤ n, ∂ ⎨ 2 Re ∂˙ γ , = ⎪ ⎪ c ⎪ ∂u ⎩ −2 Im ∂˙ γ , if n + 1 ≤ c ≤ 2n, thus by (4.8) and (4.9), we have ⎧ ⎪ ⎪ ⎪ if 1 ≤ a, c ≤ n, Re[Gα;γ + Gα;γ ] = Re[Gα;γ + Gα;γ ], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ α α α α ⎪ a ˆ ⎪ − Im[G;γ + G;γ ] = − Im[G;γ − G;γ ], if 1 ≤ a ≤ n, n + 1 ≤ c ≤ 2n, ∂G ⎨ =⎪ ⎪ ⎪ ∂xc ⎪ Im[Gα;γ − Gα;γ ] = Im[Gα;γ + Gα;γ ], if n + 1 ≤ a ≤ 2n, 1 ≤ c ≤ n, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ α α α α ⎩ Re[G;γ − G;γ ] = Re[G;γ − G ], if n + 1 ≤ a, c ≤ 2n. ;γ Similarly, by (4.8) and (4.10), we have ⎧ ⎪ ⎪ ⎪ if 1 ≤ a, c ≤ n, Gαγ + Gαγ + Gαγ + Gαγ , ⎪ ⎪ ⎪ ⎪ √ ⎪ ⎪ α α α α ˆa ⎪ ⎪ if 1 ≤ a ≤ n, n + 1 ≤ c ≤ 2n, ∂G ⎨ −1(Gγ − Gγ + Gγ − Gγ ), 2 c =⎪ √ ⎪ ⎪ α α α α ∂u ⎪ − −1(Gγ + Gγ − Gγ − Gγ ), if n + 1 ≤ a ≤ 2n, 1 ≤ c ≤ n, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Gαγ − Gα − Gαγ + Gα , if n + 1 ≤ a, c ≤ 2n. γ γ 9
(4.9)
(4.10)
(4.11)
(4.12)
Since ub ∂x∂b = (vβ ∂β )o = vβ ∂β + vβ ∂β , we obtain ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2ˆa ⎪ G ∂ ⎨ b u = ⎪ ⎪ ∂xb ∂uc ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
α β if 1 ≤ a, c ≤ n, v , Re Gαγ;β vβ + Gαγ;β vβ + Gαγ;β vβ + Gγ;β α β α β α β α β − Im Gγ;β v − Gγ;β v + Gγ;β v − Gγ;β v , if 1 ≤ a ≤ n, n + 1 ≤ c ≤ 2n, α β Im Gαγ;β vβ + Gαγ;β vβ − Gαγ;β vβ − Gγ;β if n + 1 ≤ a ≤ 2n, 1 ≤ c ≤ n, v , α β α β α β α β if n + 1 ≤ a, c ≤ 2n. Re Gγ;β v − Gγ;β v − Gγ;β v + Gγ;β v ,
(4.13)
ˆ b ∂b = (Gβ ∂˙ β )o = Gβ ∂˙ β + Gβ ∂˙ , this together with (4.12) yields Note that G β ∂u ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2ˆa ⎪ ∂ G ⎨ b ˆ 2G = ⎪ ⎪ b c ⎪ ∂u ∂u ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
if 1 ≤ a, c ≤ n, 2 Re Gαγβ Gβ + Gαγβ Gβ + Gαγβ Gβ + Gαγβ Gβ , α β α β α β α β −2 Im Gγβ G − Gγβ G + Gγβ G − Gγβ G , if 1 ≤ a ≤ n, n + 1 ≤ c ≤ 2n, 2 Im Gαγβ Gβ + Gαγβ Gβ − Gαγβ Gβ − Gαγβ Gβ , if n + 1 ≤ a ≤ 2n, 1 ≤ c ≤ n, 2 Re Gαγβ Gβ − Gαγβ Gβ − Gαγβ Gβ + Gαγβ Gβ , if n + 1 ≤ a, c ≤ 2n.
(4.14)
From (4.12), if 1 ≤ a, c ≤ n, we have ⎤ ⎡ n 2n ˆ a ∂G ˆ b 1 ⎢⎢⎢ ˆ a ∂G ˆb ˆ a ∂G ˆ b ⎥⎥⎥ ∂G ∂ G ∂ G ⎥⎥⎥ = ⎢⎢⎢⎣ + 4 b=1 ∂ub ∂uc b=n+1 ∂ub ∂uc ⎦ ∂ub ∂uc 1 α = (Gβ + Gαβ + Gαβ + Gαβ )(Gβγ + Gβγ + Gβγ + Gβγ ) 4
+(Gαβ − Gαβ + Gαβ − Gαβ )(Gβγ + Gβγ − Gβγ − Gγβ ) = Re (Gαβ + Gαβ )(Gβγ + Gβγ ) . By similar reasons, we have
ˆb ˆ a ∂G ∂G = − Im (Gαβ + Gαβ )(Gβγ − Gβγ ) , if 1 ≤ a ≤ n, n + 1 ≤ c ≤ 2n, ∂ub ∂uc ˆ a ∂G ˆb ∂G = Im (Gαβ − Gαβ )(Gβγ + Gβγ ) , if n + 1 ≤ a ≤ 2n, 1 ≤ c ≤ n, c b ∂u ∂u ˆ a ∂G ˆb ∂G = Re (Gαβ − Gαβ )(Gβγ − Gβγ ) , if n + 1 ≤ a, c ≤ 2n. c b ∂u ∂u Therefore we have ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a ˆb ˆ ⎪ ∂G ∂G ⎨ = ⎪ ⎪ ⎪ ∂ub ∂uc ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
β β β β if 1 ≤ a, c ≤ n, Re Gαβ Gγ + Gαβ Gγ + Gαβ Gγ + Gαβ Gγ , α β α β α β α β − Im Gβ Gγ − Gβ Gγ + Gβ Gγ − Gβ Gγ , if 1 ≤ a ≤ n, n + 1 ≤ c ≤ 2n, Im Gαβ Gβγ + Gαβ Gβγ − Gαβ Gβγ − Gαβ Gβγ , if n + 1 ≤ a ≤ 2n, 1 ≤ c ≤ n, α β α β α β α β if n + 1 ≤ a, c ≤ 2n. Re Gβ Gγ − Gβ Gγ − Gβ Gγ + Gβ Gγ ,
By (2.4), (4.11), (4.13), (4.14) and (4.15), we obtain ⎧ ⎪ ⎪ ⎪ I = Re[A1 − (B1 + B2 ) + (C1 + C2 ) − (D1 + D2 )], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ II = − Im[A2 − (B1 − B2 ) + (C1 − C2 ) − (D1 − D2 )], ⎨ Rac = ⎪ ⎪ ⎪ ⎪ III = Im[A1 − (B3 + B4 ) + (C3 + C4 ) − (D3 + D4 )], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ IV = Re[A2 − (B3 − B4 ) + (C3 − C4 ) − (D3 − D4 )],
10
(4.15)
if 1 ≤ a, c ≤ n, if 1 ≤ a ≤ n, n + 1 ≤ c ≤ 2n, if n + 1 ≤ a ≤ 2n, 1 ≤ c ≤ n, if n + 1 ≤ a, c ≤ 2n,
(4.16)
where A1 = 2 Gα;γ + Gα;γ ,
A2 = 2 Gα;γ − Gα;γ ,
B1 = Gαγ;β vβ + Gαγ;β vβ ,
B2 = Gαγ;β vβ + Gαγ;β vβ ,
B3 = Gαγ;β vβ − Gαγ;β vβ , B4 = Gαγ;β vβ − Gαγ;β vβ , α Gβ , C1 = 2 Gαγβ Gβ + Gαγβ Gβ , C2 = 2 Gαγβ Gβ + Gγβ α C3 = 2 Gαγβ Gβ − Gαγβ Gβ , C4 = 2 Gαγβ Gβ − Gγβ Gβ , D1 = Gαβ Gβγ + Gαβ Gβγ , D2 = Gαβ Gβγ + Gαβ Gβγ , D4 = Gαβ Gβγ − Gαβ Gβγ . D3 = Gαβ Gβγ − Gαβ Gβγ , On the other hand, since F is a strongly convex complex Finsler metric,it follows that G(x, λu) = λ2G(x, u),
∀ λ ∈ R − {0}.
(4.17)
Differentiating (4.17) with respect to λ and then setting λ = 1, we get Gb (x, u)ub = 2G(x, u).
(4.18)
Differentiating (4.18) with respect to uc , we get Gcb ub = Gc . Since
⎧ ⎪ ⎪ ⎪ ⎨ Gγ + Gγ , Gc = ⎪ √ ⎪ ⎪ ⎩ −1(Gγ − Gγ ),
we have
if 1 ≤ c ≤ n, if n + 1 ≤ c ≤ 2n,
⎧ ⎪ ⎪ ⎪ ⎨ Gγ + Gγ , Gcb ub = Gc = ⎪ √ ⎪ ⎪ ⎩ −1(Gγ − Gγ ),
(4.19)
if 1 ≤ c ≤ n, if n + 1 ≤ c ≤ 2n.
Thus 1 1 k(Gδac − Gcb ub ua ) = k(2Gδac − Gc ua ). 2 2 Substituting
⎧ ⎪ ⎪ ⎪ ⎨ u =⎪ ⎪ ⎪ ⎩
1 α 2 (v
a
+ vα ),
√1 (vα 2 −1
(4.20)
if 1 ≤ a ≤ n,
− vα ),
if n + 1 ≤ a ≤ 2n,
into (4.20), we obtain ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎨ a b a k Gδc − Gcb u u = ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
if 1 ≤ a, c ≤ n, 2Gδαγ − Gγ vα − Gγ vα , II = Im 12 k vαGγ + vαGγ , if 1 ≤ a ≤ n, n + 1 ≤ c ≤ 2n, = − Im 1 k vαGγ − vαGγ , if n + 1 ≤ a ≤ 2n, 1 ≤ c ≤ n, III 2 1 α α α IV = Re 2 k 2Gδγ − v Gγ + v Gγ , if n + 1 ≤ a, c ≤ 2n.
I = Re
1 2k
(4.21)
Using Theorem 2.1 and comparing (4.16) with (4.21), we conclude that F is of constant flag curvature k if and only if I = I,
II = II,
III = III, 11
IV = IV.
(4.22)
Step 2. Note that terms on the left and right hand sides of the equalities in (4.22) are homogeneous functions with respect to v. Terms on the left hand sides of the equalities in (4.22) are of type (0, 2), (1, 1), (2, 0), (3, −1) and its complex conjugations, while terms on the right hand sides of the equalities in (4.22) are of type (1, 1), (0, 2) and its complex conjugations. It is easy to check that, with respect to the fiber variables v, there holds (1) Gαγ;β vβ , Gαγβ Gβ , Gαβ Gβγ , vαGγ are homogeneous functions of type (0, 2); α β (2) Gγ;β v , Gαγβ Gβ , Gαβ Gβγ , Gδαγ , vαGγ are homogeneous functions of type (1, 1);
(3) Gα;γ + Gα;γ , Gαγ;β vβ , Gαγβ Gβ , Gαβ Gβγ are homogeneous functions of type(2, 0); (4) Gαγ;β vβ , Gαγβ Gβ , Gαβ Gβγ are homogeneous functions of type (3, −1). In order to use Lemma 3.3, we only need to consider those terms in equalities in (4.22) which are of type (2, 0), (1, 1), (3, −1), since the complex conjugation of a homogeneous function of type (p, q) with respect to v is a homogeneous function of type (q, p) with respect to v. It is easy to check that the components in I which are of homogeneous functions of type (2, 0) with respect to v are Λ1 − Λ2 + Λ3 − Λ4 , where Λ1 = 2 Gα;γ + Gα;γ , Λ3 = 2 Gαγβ Gβ + Gαγβ Gβ ,
Λ2 = Gαγ;β vβ + Gαγ;β vβ , Λ4 = Gαβ Gβγ + Gαβ Gβγ .
The components in I which are of homogeneous functions of type (1, 1) with respect to v are −Λ5 + Λ6 − Λ7 , where
Λ6 = 2 Gαγβ Gβ + Gαγβ Gβ ,
Λ5 = Gαγ;β vβ + Gαγ;β vβ ,
Λ7 = Gαβ Gβγ + Gαβ Gβγ .
The components in I which are homogeneous functions of type (3, −1) with respect to v are Σ, where Σ = −Gαγ;β vβ + 2Gαγβ Gβ − Gαβ Gβγ .
(4.23)
The components in I which are homogeneous functions of type (2, 0), (1, 1) and (3, −1) with respect to v are respectively 1 − kvαGγ , 2
1 k 4Gδαγ − vαGγ − vαGγ 2
and
0.
Therefore I = I if and only if 1 Λ1 − Λ2 + Λ3 − Λ4 = − kvαGγ , 2 1 −Λ5 + Λ6 − Λ7 = k 4Gδαγ − vαGγ − vαGγ , 2 Σ = 0.
12
(4.24) (4.25) (4.26)
The components in II which are of homogeneous functions of type (2, 0) with respect to v are −Ξ1 + Ξ2 − Ξ3 + Ξ4 , where Ξ1 = 2 Gα;γ − Gα;γ , Ξ3 = 2 Gαγβ Gβ − Gαγβ Gβ ,
Ξ2 = Gαγ;β vβ − Gαγ;β vβ , Ξ4 = Gαβ Gβγ − Gαβ Gβγ .
The components in II which are of homogeneous functions of type (1, 1) with respect to v are −Ξ5 + Ξ6 + Ξ7 , where Ξ5 = Gαγ;β vβ − Gαγ;β vβ ,
Ξ6 = 2 Gαγβ Gβ − Gαγβ Gβ ,
Ξ7 = Gαβ Gβγ − Gαβ Gβγ .
The components in II which are homogeneous functions of type (3, −1) with respect to v are Σ, where Σ is given by (4.23). The components in II which are homogeneous functions of type (2, 0), (1, 1) and (3, −1) with respect to v are respectively 1 − kvαGγ , 2
1 α k v G γ − vα G γ 2
and
0.
Thus II = II if and only if 1 −Ξ1 + Ξ2 − Ξ3 + Ξ4 = − kvαGγ , 2 1 −Ξ5 + Ξ6 + Ξ7 = k vαGγ − vαGγ , 2 Σ = 0.
(4.27) (4.28) (4.29)
if and only if By similar arguments, it is easy to check that III = III 1 Λ1 − Λ2 + Λ3 − Λ4 = − kvαGγ , 2 1 Ξ5 − Ξ6 + Ξ7 = − k(vαGγ − vαGγ ), 2 Σ = 0.
(4.30) (4.31) (4.32)
if and only if IV = IV 1 α kv Gγ , 2 1 −Λ5 + Λ6 − Λ7 = k(4Gδαγ − vαGγ + vαGγ ), 2
Ξ1 − Ξ2 + Ξ3 − Ξ4 =
−Σ = 0.
(4.33) (4.34) (4.35)
Note that (4.24) and (4.30) are the same equations, (4.25) and (4.34) are the same equations, (4.27) and (4.33) are the same equations, (4.28) and (4.31) are the same equations, (4.26), (4.29), (4.32) and (4.35) are the same
13
equations. Therefore a strongly convex weakly K¨ahler-Finsler metric F is of constant flag curvature k if and only if the following system of equations holds: 1 Λ1 − Λ2 + Λ3 − Λ4 = − kvαGγ , 2 1 Ξ1 − Ξ2 + Ξ3 − Ξ4 = kvαGγ , 2 1 −Λ5 + Λ6 − Λ7 = k 4Gδαγ − vαGγ − vαGγ , 2 1 Ξ5 − Ξ6 + Ξ7 = − k(vαGγ − vαGγ ), 2 −Σ = 0, which is equivalent to the following system of equations: 2Gα;γ − Gαγ;β vβ + 2Gαγβ Gβ − Gαβ Gβγ = 0, 1 2Gα;γ − Gαγ;β vβ + 2Gαγβ Gβ − Gαβ Gγβ = − kvαGγ , 2 1 Gαγ;β vβ − 2Gαγβ Gβ + Gαβ Gβγ = − k 2Gδαγ − vαGγ , 2 Gαγ;β vβ − 2Gαγβ Gβ + Gαβ Gβγ = 0.
(4.36) (4.37) (4.38) (4.39)
In the following we shall simplify the equations (4.37) and (4.38). Since Gα are homogeneous functions of type (2, 0) with respect to v, it follows that Gβγ vγ = 2Gβ ,
Gαγ vγ = 0, from which we get Gαγβ vγ = −Gαβ ,
Gαβ Gγβ vγ = 2Gαβ Gβ .
Now contracting (4.37) with vγ or contracting (4.38) with vγ , we get 1 Gα;β vβ − 2Gαβ Gβ = − kvαG. 4
(4.40)
Differentiating (4.40) with respect to vγ yields 1 Gαγ;β vβ + Gα;γ − 2Gαγβ Gβ − 2Gαβ Gβγ = − kvαGγ . 4
(4.41)
Gαγ;β vβ − 2Gαγβ Gβ − Gαβ Gβγ = 0.
(4.42)
1 Gα;γ − Gαβ Gγβ = − kvαGγ . 4
(4.43)
(4.41) × 2 − (4.37) implies
(4.42) + (4.37) yields
Hence (4.37) is equivalent to (4.42) and (4.43). Similarly, it is easy to check that (4.38) is equivalent to the following system of equations: 1 Gαβ Gβγ = − k(Gδαγ − vαGγ ), 4 1 Gαγ;β vβ − 2Gαγβ Gβ = − k(3Gδαγ − vαGγ ). 4 14
Therefore we conclude that F is of constant flag curvature Kˆ F = k if and only if (4.1)-(4.6) hold. This completes the proof. Now we are ready to prove our first main result in this paper. Theorem 4.1. Let F be a strongly convex weakly K¨ahler-Finsler metric on a complex manifold M. If F is of constant flag curvature Kˆ F = k, then F is of constant holomorphic curvature KF = k. Proof. Contracting (4.4) with vγ , we have 1 Gα;γ vγ − 2Gαβ Gβ = − kvαG, 4
(4.44)
where we use the following identities: Gβγ vγ = 2Gβ ,
Gγ vγ = G.
Note that Gα G α =
1 α μ 1 1 1 Γ v Gα = GταGτ;μ vμGα = vτGτ;μ vμ = G;μ vμ , 2 ;μ 2 2 2
so that Gα;γ Gα = ∂γ (GαGα ) − GαGα;γ , GαβGα = ∂˙ β (GαGα ) − GαGαβ . Thus contracting (4.44) with Gα , we get 1 1 1 μ ν α γ μ α G;μν v v − Gα;γ G v − 2 Gβ;μ v − G Gαβ Gβ = − kG2 . 2 2 4 Using Lemma 3.2, we obtain k = KF −
4 α β α γ β μ G G − G G v − G G v 4G . α;γ αβ β;μ G2
(4.45)
Now since β α γ G v = 2Gαβ Gα Gβ , Gα;γ Gα vγ = Gαβ Γ;γ α β μ G v = 2Gαβ Gα Gβ . Gβ;μ Gβ vμ = Gαβ Γ;μ
It follows that k = KF . This completes the proof. Proposition 4.2. Let F be a strongly convex weakly K¨ahler-Finsler metric on a complex n-dimensional manifold M. Then F is a real Einstein-Finsler metric if and only if 2Gα;α − Gαα;β vβ + 2Gααβ Gβ − Gαβ Gβα = 0, α β v ) − 2Gαβ Gβα = (2n − 1)σ(z)G 2(Gααβ Gβ + Gααβ Gβ ) − (Gαα;β vβ + Gα;β
for some real scalar function σ(z) on M. 15
Proof. By Proposition 4.1, the Riemann curvature tensor Rac of F is given by (4.16), so its Ricci curvature is Ric = Raa
= Re 2 Gα;α + Gα;α − Gαα;β vβ + Gαα;β vβ + Gαα;β vβ + Gαα;β vβ α Gβ − Gαβ Gβα + Gαβ Gβα + Gαβ Gβα + Gαβ Gβα +2 Gααβ Gβ + Gααβ Gβ + Gααβ Gβ + Gαβ + Re 2 Gα;α − Gα;α − Gαα;β vβ − Gαα;β vβ − Gαα;β vβ + Gαα;β vβ α Gβ − Gαβ Gβα − Gαβ Gβα − Gαβ Gβα + Gαβ Gβα +2 Gααβ Gβ − Gααβ Gβ − Gααβ Gβ + Gαβ α Gβ − Gαβ Gβα + Gαβ Gβα . = 2 Re 2Gα;α − Gαα;β vβ + Gαα;β vβ + 2 Gααβ Gβ + Gαβ
Hence F is a real Einstein-Finsler metric if and only if 2 Re[2Gα;α − Gαα;β vβ + Gαα;β vβ + 2 Gααβ Gβ + Gααβ Gβ − Gαβ Gβα + Gαβ Gβα ] = (2n − 1)σG
(4.46)
for a real scalar function σ(z) on M. Note that Gα;α , Gαα;β vβ , Gααβ Gβ , Gαβ Gβα are homogeneous functions of type (2, 0) with respect to v, and the rest terms in equality (4.46) are homogeneous functions of type (1, 1) with respect to v. Using Lemma 3.3 and by comparing the types of homogeneity in (4.46), our assertion follows. Note that for a weakly complex Berwald metric F, its complex spray coefficients Gα are holomorphic with respect to the fiber coordinate v, thus Gαμ = 0. In this case, equations (4.1)-(4.6) can be greatly simplified. So it is natural to consider the case that F is a weakly complex Berwald metric in Proposition 4.1. According to Corollary 3.3, we actually should focus on K¨ahler-Berwald metrics. We have the following theorem. Theorem 4.2. Let F be a strongly convex K¨ahler-Berwald metric on a complex manifold M of complex dimension n ≥ 2. Then F is of constant flag curvature if and only if F is a locally complex Minkowski metric. Proof. Theorem 4.9 in [13] stated that a strongly convex weakly K¨ahler-Finsler metric is a weakly complex Berwald metric if and only if it is a real Berwald metric. Hence F is a real Berwald metric. If F is of constant flag curvature Kˆ F = k, under our condition for F, (4.5) reduces to k(Gδαγ − vαGγ ) = 0, which implies that k = 0 whenever n ≥ 2, namely, F has vanishing flag curvature. It is a well-known fact that a real Berwald metric with zero flag curvature is actually a locally real Minkowski metric (see Proposition 10.5.1 in [8]), this means that F, namely, F o is a locally real Minkowski metric, hence F is a locally complex Minkowski metric. Conversely, if F is a locally complex Minkowski metric, it results that F is also a locally real Minkowski metric since F is strongly convex. So F has vanishing flag curvature. This completes the proof. There are two notions of Ricci scalar curvature associated to a strongly pseudoconvex complex Finsler metric in literature, we refer to [13, 14, 16] for more details.
16
Definition 4.1 ([16]). Let F be a strongly pseudoconvex complex Finsler metric defined on a complex manifold M. Then the Ricci scalar curvature Ric of the Chern-Finsler connection associated to F is defined by Ric(z, v) = −δα (Γμ;μ )vα , ˘ of the complex Berwald connection associated to F is defined by and the Ricci scalar curvature Ric ˘ Ric(z, v) = −δˇ α (Gμμ )vα , where the index μ is summed from 1 to n. ˘ However, by Lemma In general, the two Ricci scalar curvatures are not necessarily real-valued and Ric Ric. ˘ if F is a K¨ahler-Finsler metric. 3.5 we have Ric = Ric Theorem 4.3. Let F be a strongly convex K¨ahler-Berwald metric on a complex n-dimensional manifold M. Then F is a real Einstein-Finsler metric if and only if the Ricci scalar curvature Ric of F satisfying 2 Re Ric = (2n − 1)σ(z)G for some real scalar function σ(z) on M. Proof. Since F is a K¨ahler-Berwald metric, it follows from Lemma 3.6 that Gα;α − Gαα;β vβ + 2Gααβ Gβ − Gαβ Gβα = 0,
Gαμ = 0,
which together with Proposition 4.2 implies that F is a real Einstein-Finsler metric if and only if α β v = (2n − 1)σ(z)G Gαα;β vβ + Gα;β
(4.47)
holds for some real scalar function σ(z) on M. Note that the Ricci scalar curvature of F is ˘ = −δˇ α (Gμμ )vα = −∂α (Gμμ )vα = Gα vβ . Ric = Ric α;β
(4.48)
By plunging (4.48) into (4.47), our assertion follows. Corollary 4.1. Let F be a strongly convex K¨ahler-Berwald metric on a complex n-dimensional manifold M. If F does not comes from a Hermitian metric, then F is a real Einstein-Finsler metric if and only if the Ricci scalar curvature Ric of F satisfying Re Ric = 0. Proof. By Theorem 4.3, F is a real Einstein-Finsler metric if and only if (2n − 1)σ(z)G = 2 Re Ric = Gαα;β vβ + Gαα;β vβ = Gααμ;β (z)vμ vβ + Gααμ;β (z)vβ vμ
(4.49)
holds for some real scalar function σ(z) on M. Differentiating (4.49) with respect to vγ and vν successively yields (2n − 1)σ(z)Gγν = 0.
(4.50)
Since F does not come from a Hermitian metric, we have Gγν 0, this together with (4.50) implies that σ(z) = 0. Thus (4.49) holds if and only if σ(z) = 0, namely, Re Ric = 0. 17
5. Examples of strongly convex K¨ahler-Finsler metric In the following, we shall give two nontrivial examples of strongly convex K¨ahler-Finsler metrics, where one comes from a strongly convex complex Minkowski metric in Cn and the other is a strongly convex K¨ahler-Finsler metric which does not come from a strongly convex complex Minkowski metric. Proposition 5.1. Let z = (z1 , · · · , zn ) ∈ Cn , v = (v1 , · · · , vn ) ∈ T z1,0 Cn and let ε ∈ (0, +∞) be a constant. Define Fε (z, v) = |v1 |2 + · · · + |vn |2 + ε |v1 |4 + · · · + |vn |4 . Then (i) Fε is a strongly pseudoconvex complex Finsler metric for every ε ∈ (0, +∞); (ii) Fε is a strongly convex complex Finsler metric for every ε ∈ (0, 1); (iii) Fε is a strongly convex K¨ahler-Finsler metric with vanishing flag curvature and holomorphic curvature for every ε ∈ (0, 1). 1
Proof. Denote G = Fε2 = |v1 |2 + · · · + |vn |2 + εA 2 , where A = |v1 |4 + · · · + |vn |4 . It is easy to check that Gαβ = δαβ + 2εA− 2 |vα |2 δαβ − εA− 2 |vα |2 vα |vβ |2 vβ , 1
3
− 12
− 32
Gαβ = εA (vα )2 δαβ − εA |vα |2 vα |vβ |2 vβ ,
(5.2)
where δαβ = 1 for α = β and δαβ = 0 for α β. Thus for V = (V 1 , · · · , V n ) ∈ Cn , 2 n n n α β α2 − 12 α2 α2 − 32 α 2 α α |V | + 2εA |v | |V | − εA |v | v V . Gαβ V V = α=1 α=1 α=1 Note that
(5.1)
(5.3)
2 ⎛ ⎞2 n n n n ⎜⎜ ⎟⎟ α 2 α |v | vα V ≤ ⎜⎜⎝⎜ |vα |3 |V α |⎟⎟⎟⎠ ≤ 2 |vα |6 |V α |2 ≤ 2A |vα |2 |V α |2 . α=1 α=1 α=1 α=1
This together with (5.3) yields n
Gαβ V α V β ≥
|V α |2 ≥ 0
α=1
with equality holds if and only if V = 0 ∈ Cn . This proves (i). Next by (5.2), we have ⎛ n ⎞2 ⎛ n ⎞2 ⎜⎜ ⎟⎟⎟ ⎜⎜ α 2 ⎟⎟ α β − 32 ⎜ α⎟ − 32 ⎜ β 2 β ⎜ α β ⎜ ⎟ 2 Re Gαβ V V = −εA ⎜⎝ |v | v V ⎟⎠ − εA ⎜⎜⎝ |v | v V ⎟⎟⎟⎠ α=1
+ εA
− 12
β=1
n
(vα V α )2 + εA
α=1
− 12
n (vα V α )2 . α=1
Thus 2 Re Gαβ V α V β + Gαβ V α V β =2
n
α2
|V | + 4εA
α=1
+ εA− 2 1
n α=1
− 12
n
α2
α2
|v | |V | − εA
α=1
(vα V α )2 + εA− 2 1
− 32
⎡ ⎞⎤2 ⎛ n ⎢⎢⎢ ⎟⎥ ⎜ ⎢⎢⎣2 Re ⎜⎜⎜⎜⎝ |vα |2 vα V α ⎟⎟⎟⎟⎠⎥⎥⎥⎥⎦
n (vα V α )2 . α=1
18
α=1
(5.4)
Since
⎡ ⎞⎤2 ⎡ n ⎤ ⎛ n n n ⎢⎢⎢ ⎟⎥ ⎢ α 3 α ⎥⎥⎥2 ⎜ ⎢⎢⎣2 Re ⎜⎜⎜⎜⎝ |vα |2 vα V α ⎟⎟⎟⎟⎠⎥⎥⎥⎥⎦ ≤ ⎢⎢⎢⎢⎣2 |v | |V |⎥⎥⎦ ≤ 4 |vα |6 |V α |2 ≤ 4A |vα |2 |V α |2 α=1
and 2
n
α=1
|vα |2 |V α |2 +
α=1
α=1
n n n (vα )2 (V α )2 + (vα )2 (V α )2 = (vα V α + vα V α )2 ≥ 0, α=1
α=1
(5.5)
α=1
(5.6)
α=1
equality (5.4) yields n n 1 |V α |2 − 2εA− 2 |vα |2 |V α |2 2 Re Gαβ V α V β + Gαβ V α V β ≥ 2 α=1
α=1
= 2A
n 1 (A 2 − ε|vα |2 )|V α |2
≥ 2A− 2
n (1 − ε)|vα |2 |V α |2 ≥ 0,
− 12
α=1
1
α=1
with equality holds if and only if vα V α = 0 for α = 1, · · · , n, if and only if V = 0. By Lemma 3.4, Fε is a strongly convex complex Finsler metric for every ε ∈ (0, 1). This completes the proof of (ii). The assertion (iii) is obvious since Fε is a strongly convex complex Minkowski metric. In the following we shall give an example of strongly convex K¨ahler-Finsler metric which does not come from a complex Minkowski metric. Let M be a complex m-dimensional manifold endowed with a Hermitian metric α 2 (ξ) = aαβ (z)ξα ξβ ,
∀ ξ = ξα
∂ ∈ T z1,0 M, ∂zα
and N be a complex n-dimensional manifold endowed with a Hermitian metric β 2 (η) = bμν (w)ημ ην ,
∀ η = ημ
∂ ∈ T 1,0 N. ∂wμ
Here {z1 , · · · , zm } and {w1 , · · · , wn } are local complex coordinates on M and N, respectively; {z1 , · · · , zm , ξ1 , · · · , ξm } and {w1 , · · · , wn , η1 , · · · , ηn } are the induced complex coordinates on T 1,0 M and T 1,0 N, respectively. For constants ε > 0 and k > 1, one can define the following Szab´o metric on the product complex manifold M × N: 1 α2k (ξ) + β 2k (η)) k . Fε = α 2 (ξ) + β 2 (η) + ε(α
(5.7)
In [17], it was proved that the Szab´o metric Fε is a strongly pseudoconvex complex Berwald metric on M × N, and Fε is a strongly pseudoconvex K¨ahler-Finsler metric if and only if α and β are K¨ahler metrics on M and N, respectively. In this paper, we shall prove the strong convexity of Fε . Proposition 5.2. Let Fε be the Szab´o metric defined by (5.7). Then (i) Fε is a strongly convex complex Berwald metric on M × N; (ii) Fε is a strongly convex K¨ahler-Finsler metric on M × N if and only if α and β are K¨ahler metrics on M and N, respectively.
19
Proof. Denote upper index T the transpose of a vector and “∗” the Hermitian transpose of a vector in a finite dimensional complex linear space. Set H1 := (aαβ (z)), H2 := (bμν (w)) and denote ξ := (ξ1 , · · · , ξm )T ,
ξ∗ := (ξ1 , · · · , ξm ),
η := (η1 , · · · , ηn )T ,
η∗ := (η1 , · · · , ηn ),
ξ˜ := (ξ1 , · · · , ξm )T ,
ξ˜∗ := (ξ1 , · · · , ξm ),
η˜ := (η1 , · · · , ηn )T ,
η˜ ∗ := (η1 , · · · , ηn ),
where ξα = aαβ (z)ξβ ,
ημ = bμν (w)ην ,
Set G = Fε2 , a simple calculation gives ⎛ 2 ∂2 G ⎜⎜⎜ ∂ G ⎜⎜⎜ ∂ξα ∂ξβ ∂ξα ∂ην H := ⎜⎜ 2 ⎝ ∂G ∂2 G ⎛ ⎜⎜⎜ := ⎜⎜⎜⎜ H ⎜⎝
∂ημ ∂ξβ
∂ημ ∂ην
∂2 G ∂ξα ∂ξβ
∂2 G ∂ξα ∂ην
∂2 G ∂ημ ∂ξβ
∂2 G ∂ημ ∂ην
⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎠ (m+n)×(m+n)
⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎠
(m+n)×(m+n)
α, β = 1, · · · , m;
⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟, 2 α ∗ ⎟ c2 H2 + c3 β 2 η˜ η˜ ⎠ ⎞ −ξ˜η˜ T ⎟⎟⎟⎟⎟ ⎟⎟ , ⎠ α2 T ⎟ η ˜ η ˜ 2 β
⎛ ⎜⎜⎜ c H + c β 2 ξ˜ξ˜∗ 3 α2 ⎜ 1 1 = ⎜⎜⎜⎜ ⎝ −c η˜ ξ˜∗ 3
⎛ 2 ⎜⎜⎜ β ξ˜ξ˜T ⎜ 2 = c3 ⎜⎜⎜⎜ α ⎝ −η˜ ξ˜T
μ, ν = 1, · · · , n.
−c3 ξ˜η˜ ∗
where α2k (ξ) + β 2k (η)] k −1α 2(k−1) (ξ), c1 = 1 + ε[α 1
α2k (ξ) + β 2k (η)] k −1β 2(k−1) (η), c2 = 1 + ε[α 1
α2k (ξ) + β 2k (η)] k −2α 2(k−1) (ξ)ββ2(k−1) (η). c3 = (k − 1)ε[α 1
For
⎛ ⎞ ⎜⎜⎜ V ⎟⎟⎟ ⎜⎜⎜ 1 ⎟⎟⎟ V = ⎜⎜ 0 ∈ Cm+n , ⎝ V ⎟⎟⎠ 2
where V1 = (V11 , · · · , V1m )T ∈ Cm , V2 = (V21 , · · · , V2n )T ∈ Cn ,
it is easy to check that ∗ )T 2 Re V ∗ HV + V ∗ H(V = 2c1α 2 (V1 ) + 2c2β 2 (V2 ) + c3
#
$2
α β
ξ, V1 α + ξ, V1 α −
η, V2 β + η, V2 β α β
> 0,
where α 2 (V1 ) = aαβ (z)V1α V1β ,
β 2 (V2 ) = bμν (w)V2μ V2ν ,
ξ, V1 α = aαβ (z)ξα V1β ,
η, V2 β = bμν (w)ημ V2ν .
Hence by Lemma 3.4, Fε is a strongly convex complex Finsler metric. Thus by the result in [17], we complete the proof of the assertions (i) and (ii).
Acknowledgement This work is supported by the Program for New Century Excellent Talents in University (No. NCET-13-0510); the National Natural Science Foundation of China (Grant No. 11271304, 11171277); the Fujian Province Natural Science Funds for Distinguished Young Scholar (Grant No. 2013J06001); the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. 20
References [1] Z. Shen, Projectively flat Finsler metrics of constant flag curvature, Trans. Amer. Math. Soc. 355 (4) (2003) 1713–1728. doi:10.1090/S0002-9947-02-03216-6. URL http://dx.doi.org/10.1090/S0002-9947-02-03216-6 [2] M. Abate, G. Patrizio, Finsler metrics—a global approach With applications to geometric function theory, Vol. 1591 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1994. [3] M. Suzuki, The holomorphic curvature of intrinsic metrics, Math. Rep. Toyama Univ. 4 (1981) 107–114. [4] C. Zhong, On unitary invariant strongly pseudoconvex complex Finsler metrics, Differential Geom. Appl. 40 (2015) 159–186. doi:10.1016/j.difgeo.2015.02.002. URL http://dx.doi.org/10.1016/j.difgeo.2015.02.002 [5] H. Xia, C. Zhong, A classification of unitary invariant weakly complex Berwald metrics of constant holomorphic curvature, Differential Geom. Appl. 43 (2015) 1–20. [6] H. Xia, C. Zhong, On unitary invariant weakly complex Berwald metrics with vanishing holomorphic curvature and closed Chin. Ann. Math. Ser. B 37 (2) (2016) 161–174. doi:10.1007/s11401-016-1007-z. URL http://dx.doi.org/10.1007/s11401-016-1007-z [7] H. Xia, C. Zhong, On a class of smooth complex finsler metrics, Results in Mathematics (2016) 1– 30doi:10.1007/s00025-016-0543-x. URL http://dx.doi.org/10.1007/s00025-016-0543-x [8] D. Bao, S.-S. Chern, Z. Shen, An introduction to Riemann-Finsler geometry, Vol. 200 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. doi:10.1007/978-1-4612-1268-3. URL http://dx.doi.org/10.1007/978-1-4612-1268-3 [9] Z.
Shen,
Lectures on Finsler geometry,
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doi:10.1142/9789812811622. URL http://dx.doi.org/10.1142/9789812811622 [10] S.-S. Chern, Z. Shen, Riemann-Finsler geometry, Vol. 6 of Nankai Tracts in Mathematics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. doi:10.1142/5263. URL http://dx.doi.org/10.1142/5263 [11] B. Chen, Y. Shen, K¨ahler Finsler metrics are actually strongly K¨ahler, Chin. Ann. Math. Ser. B 30 (2) (2009) 173–178. doi:10.1007/s11401-008-0092-z. URL http://dx.doi.org/10.1007/s11401-008-0092-z [12] T. Aikou, On complex Finsler manifolds, Rep. Fac. Sci. Kagoshima Univ. Math. Phys. Chem. (24) (1991) 9–25. 21
[13] C. Zhong, On real and complex Berwald connections associated to strongly convex weakly K¨ahler-Finsler metric, Differential Geom. Appl. 29 (3) (2011) 388–408. doi:10.1016/j.difgeo.2011.03.006. URL http://dx.doi.org/10.1016/j.difgeo.2011.03.006 [14] G. Munteanu, Complex spaces in Finsler, Lagrange and Hamilton geometries, Vol. 141 of Fundamental Theories of Physics, Kluwer Academic Publishers, Dordrecht, 2004. doi:10.1007/978-1-4020-2206-7. URL http://dx.doi.org/10.1007/978-1-4020-2206-7 [15] H. Rund, The curvature theory of direction-dependent connections on complex manifolds, Tensor (N.S.) 24 (1972) 189–205. [16] L. Sun, C. Zhong, Characterizations of complex Finsler connections and weakly complex Berwald metrics, Differential Geom. Appl. 31 (5) (2013) 648–671. doi:10.1016/j.difgeo.2013.07.003. URL http://dx.doi.org/10.1016/j.difgeo.2013.07.003 [17] Y. Chen, R. Yan, The szabo´ metric on the product of complex manifolds, Acta Math. Sin.(Chinese Series) 50 (4) (2007) 801–804.
22
Hongchuan Xia, Chunping Zhong
PII: DOI: Reference:
S0022-247X(16)30176-7 http://dx.doi.org/10.1016/j.jmaa.2016.05.034 YJMAA 20452
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
20 April 2016
Please cite this article in press as: H. Xia, C. Zhong, On strongly convex weakly Kähler-Finsler metrics of constant flag curvature, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2016.05.034
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On strongly convex weakly K¨ahler-Finsler metrics of constant flag curvature Hongchuan Xiaa,b , Chunping Zhongb,∗ a College
of Mathematics and Information Science, Xinyang Normal University, Xinyang 464000, P.R. China b School of Mathematical Sciences, Xiamen University, Xiamen 361005, P.R. China
Abstract In this paper, we first give a necessary and sufficient condition for a strongly convex weakly K¨ahler-Finsler metric to be of constant flag curvature, and then prove that (i) a strongly convex weakly K¨ahler-Finsler metric of constant flag curvature is necessary of constant holomorphic curvature; (ii) a strongly convex K¨ahler-Berwald metric on a complex manifold of complex dimension n ≥ 2 has constant flag curvature if and only if it comes from a strongly convex locally complex Minkowski metric. We also give two examples of nontrivial strongly convex K¨ahler-Finsler metrics. Keywords: flag curvature, holomorphic curvature, strongly convex, complex Finsler metric 2010 MSC: 53C60, 53C40
1. Introduction and main results In Riemannian geometry, it is a simple fact that a K¨ahler metric of constant sectional curvature is necessary of constant holomorphic sectional curvature. In real Finsler geometry, flag curvature is a natural generalization of the sectional curvature in Riemannian geometry [1]. In complex Finsler geometry, the notion of holomorphic curvature is well defined for strongly pseudoconvex complex Finsler metrics [2], and even for upper semicontinuous complex Finsler metrics [2, 3]. In general, however, real and complex Finsler metrics are not tightly related as that of in Riemannian and Hermitian geometry, we refer to [2] for more details. For a real Finsler metric it is usually asked to be strongly convex along tangent directions, while for a complex Finsler metric it is asked to be strongly pseudoconvex along tangent directions. A complex Finsler metric is not necessarily a real Finsler metric, and vice verse. A complex Finsler metric on a complex manifold is called strongly convex if it is also a real Finsler metric [2]. Recently we systematically investigate the unitary invariant complex Finsler metrics and general complex (α, β) metrics [4–7], as a byproduct we find that there are lots of strongly convex complex Finsler metrics. Thus a natural question arises: for strongly convex complex Finsler manifolds whether there are relationships among the geometric objects in real and complex Finsler geometry? As we know, flag curvature and holomorphic curvature play fundamental roles respectively in real and complex Finsler geometry, one may wonder whether there exists relationship between the flag curvature and holomorphic curvature for a strongly convex complex Finsler metric? As our first main result we prove: ∗ Corresponding
author Email addresses: [email protected] (Hongchuan Xia), [email protected] (Chunping Zhong)
Preprint submitted to Journal of Mathematical Analysis and Applications
May 19, 2016
Theorem 1.1. Let F be a strongly convex weakly K¨ahler-Finsler metric on a complex manifold M. If F is of constant flag curvature Kˆ F = k, then F is of constant holomorphic curvature KF = k. Note that there are lots of real Finsler metrics which are of constant flag curvature in literature [8–10]. Theorem 1.1 indeed provides a possible approach to construct complex Finsler metrics of constant holomorphic curvature via real Finsler metrics of constant flag curvature. In particular, under condition that F is a K¨ahler-Berwald metric we get the following rigid result as our second main result. Theorem 1.2. Let F be a strongly convex K¨ahler-Berwald metric on a complex manifold M of complex dimension n ≥ 2. Then F is of constant flag curvature if and only if F is a locally complex Minkowski metric. Our last main result establish a relationship between real Einstein-Finsler metrics and the Ricci scalar curvature (see Definition 4.1). Theorem 1.3. Let F be a strongly convex K¨ahler-Berwald metric on a complex n-dimensional manifold M. Then F is a real Einstein-Finsler metric if and only if the Ricci scalar curvature Ric of F satisfying 2 Re Ric = (2n − 1)σ(z)G for some real scalar function σ(z) on M. The arrangements of this paper are as follows. In section 2, we introduce some necessary definitions and notions. In section 3, we give some lemmas which will be used in this paper. In section 4, we shall first establish a necessary and sufficient condition for a strongly convex weakly K¨ahler-Finsler metric to be of constant flag curvature. This condition is formulated into a system of six equations, which are used to prove our main results (Theorems 1.1–1.3). In section 5, we shall give two examples of strongly convex K¨ahler-Finsler metrics, one comes from a strongly convex complex Minkowski metric in Cn , the other is a strongly convex K¨ahler-Finsler metric which does not come from a strongly convex complex Minkowski metric.
2. Preliminaries Let’s recall some necessary notations and definitions, which can be found in [2]. Let M be a complex ndimensional manifold with the canonical complex structure J. Denote by T R M the real tangent bundle, and T C M the complexified tangent bundle of M. Then J acts complex linearly on T C M so that T C M = T 1,0 M ⊕T 0,1 M, where T 1,0 M is called the holomorphic tangent bundle of M. T 1,0 M is a complex manifold of complex dimension 2n, and we also denote by J the complex structure on T 1,0 M if it causes no confusion. Let {z1 , · · · , zn } be a set of local √ complex coordinates on M, with zα = xα + −1xα+n , so that {x1 , · · · , xn , x1+n , · · · , x2n } are local real coordinates √ on M. Denote by {z1 , · · · , zn , v1 , · · · , vn } the induced complex coordinates on T 1,0 M, with vα = uα + −1uα+n , so that {x1 , · · · , x2n , u1 , · · · , u2n } are local real coordinates on T R M. In the following, we shall denote M˜ the complement of zero section in T R M or T 1,0 M, depending on the real or complex situation. Unless otherwise stated we always assume that lowercase Greek indices such as α, β, γ etc., run 2
from 1 to n, whereas lowercase roman indices such as a, b, c, etc., run from 1 to 2n, and the Einstein summation convention is assumed throughout the paper. Set √ ∂ 1 ∂ ∂ = − −1 , ∂zα 2 ∂xα ∂xα+n √ ∂ 1 ∂ ∂ − −1 , ∂˙ α := α = ∂v 2 ∂uα ∂uα+n
√ ∂ 1 ∂ ∂ = + −1 , ∂xα+n ∂zα 2 ∂xα √ ∂ 1 ∂ ∂ = + −1 ∂˙ α := . ∂uα+n ∂vα 2 ∂uα
∂α :=
∂α :=
The bundles T 1,0 M and T R M are isomorphic. We choose the explicit isomorphism o : T 1,0 M → T R M with its inverse o : T R M → T 1,0 M, which are respectively given by ∀ v = vα
T R M u = vo = v + v, and T 1,0 M v = uo =
√ 1 (u − −1Ju), 2
∂ ∈ T 1,0 M ∂zα
∀ u = ua
∂ ∈ T R M. ∂xa
Definition 2.1 ([2]). A real Finsler metric on a manifold M is a function F : T R M → R+ satisfying the following properties: ˜ (a) G = F 2 is smooth on M; ˜ (b) F(p, u) > 0 for (p, u) ∈ M; (c) F(p, λu) = |λ|F(p, u) for all (p, u) ∈ T R M and λ ∈ R; (d) for any p ∈ M the indicatrix IF (p) = {u ∈ T p M| F(p, u) < 1} is strongly convex. Note that condition (d) is equivalent to the following matrix 2 ∂G (Gab ) = ∂ua ∂ub
(2.1)
˜ being positive definite on M. Definition 2.2 ([2]). A complex Finsler metric F on a complex manifold M is a continuous function F : T 1,0 M → R+ satisfying ˜ (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. Definition 2.3 ([2]). A complex Finsler metric F is called strongly pseudoconvex if the Levi matrix 2 ∂G (Gαβ ) = ∂vα ∂vβ ˜ is positive definite on M. Let F : T 1,0 M → R+ be a complex Finsler metric on a complex manifold M. Using the complex structure J on M and the bundle map o : T R M → T 1,0 M we can define a real function F o : T R M → R+ , F o (u) := F(uo ), ∀ u ∈ T R M. 3
(2.2)
Definition 2.4 ([2]). A complex Finsler metric F is called strongly convex if the associated function F o is a real Finsler metric. For a strongly convex complex Finsler metric F, we use the same symbol F to denote the associated real Finsler metric F o with the understanding that F(u) is defined by F(uo ) for u ∈ T R M. ˜ we shall use a semi-colon to distinguish between derivatives with In the following, for functions defined on M, respect to z and v in the complex setting (resp. x and u in the real setting). For example, Gα =
∂G , ∂vα
G;α =
∂G , ∂zα
Gα;β =
Ga =
∂G , ∂ua
G;a =
∂G , ∂xa
Ga;b
∂2G
, ∂vα ∂zβ ∂2G = a b. ∂u ∂x
ˆ a and Gα the real and complex spray coefficients associated to a strongly convex complex Finsler Denote G metric F, respectively, where ˆ a = 1 Gab (Gb;c uc − G;b ), G 2
Gα =
1 α β Γ v , 2 ;β
α Γ;β = GταGτ;β ,
α are called the Chern-Finsler nonlinear connection coefficients. Gαβ = ∂˙ β Gα are called the complex Berwald and Γ;β
nonlinear connection coefficients. The corresponding horizontal frames {δμ } and {δˇ μ } associated to the two nonlinear connections are respectively given by α ˙ ∂α , δμ = ∂μ − Γ;μ
δˇ μ = ∂μ − Gαμ ∂˙ α .
ˆ a by The real nonlinear connection coefficients Γˆba are related to G ˆa ∂G Γˆ ba = , ∂ub
ˆ a = 1 Γˆ a ub . G 2 b
α and complex Berwald connection coefficients Gαβμ The horizontal Chern-Finsler connection coefficients Γμ;β α are respectively related to Γ;β and Gαβ by α α = ∂˙ μ Γ;β , Γμ;β
Gαβμ = ∂˙ μ Gαβ ,
α μ α Γμ;β v = Γ;β ,
Gαβμ vμ = Gαβ .
α α α Γβ;μ , Γβ;μ Gαβμ . But for K¨ahler-Finsler metrics, the two It is clear that Gαβμ = Gαμβ , in general, however, Γμ;β α = Gαβμ (see Lemma 3.5). connection coefficients coincide, i.e., Γβ;μ
Definition 2.5 ([2]). Let F is a strongly pseudoconvex complex Finsler metric on a complex manifold M. F is α α called a strongly K¨ahler-Finsler metric if in local coordinates, Γβ;μ − Γμ;β = 0; called a K¨ahler-Finsler metric if α α α α − Γμ;β )vμ = 0; called a weakly K¨ahler-Finsler metric if Gα (Γβ;μ − Γμ;β )vμ = 0. (Γβ;μ
In [11], Chen and Shen proved that a K¨ahler-Finsler metric is actually a strongly K¨ahler-Finsler metric, so there are two notions of K¨ahler-Finsler metric in complex Finsler geometry. Definition 2.6 ([12, 13]). Let F is a strongly pseudoconvex complex Finsler metric on a complex manifold M. F α is called a complex Berwald metric if locally Γβ;μ depend only on the base manifold coordinates z = (z1 , · · · , zn );
F is called a weakly complex Berwald metric if locally Gαβμ depend only on z. 4
A complex Berwald metric is a weakly complex Berwald metric, the converse, however, is not true [13]. It was shown in [4, 5] that there are lots of non-Hermitian smooth weakly complex Berwald metrics. Theorem 2.1 ([10]). A real Finsler manifold (M, F) is of constant flag curvature k if and only if 1 a a b a R c = k Gδc − Gcb u u , 2
(2.3)
where Rac is the Riemann curvature tensor and given by Rac = 2
2ˆa ˆa ˆa ˆb ˆa ∂G ∂2 G ˆ b ∂ G − ∂G ∂G . − ub b c + 2G c b c ∂x ∂x ∂u ∂u ∂u ∂ub ∂uc
(2.4)
Definition 2.7 ([10]). A real Finsler metric F on a real n-dimensional manifold M is called an Einstein-Finsler metric if 2 , Ric = (n − 1)σ(x)F ˆ
where σ(x) ˆ is a real scalar function on M, Ric = Raa is the Ricci curvature of F. It is clear that a real Finsler metric of constant flag curvature is necessarily an Einstein-Finsler metric.
3. Some lemmas In the following, we set ⎧ ⎪ ⎪ ⎪ if 1 ≤ a ≤ n, ⎨ a, α=⎪ ⎪ ⎪ ⎩ a − n, if n ≤ a ≤ 2n,
⎧ ⎪ ⎪ ⎪ ⎨ b, β=⎪ ⎪ ⎪ ⎩ b − n,
if 1 ≤ b ≤ n, if n ≤ b ≤ 2n,
⎧ ⎪ ⎪ ⎪ if 1 ≤ c ≤ n, ⎨ c, γ=⎪ ⎪ ⎪ ⎩ c − n, if n ≤ c ≤ 2n.
Lemma 3.1. Let F be a strongly convex complex Finsler metric on a complex manifold M. Then F is a weakly K¨ahler-Finsler metric if and only if ⎧ 1 ⎪ ⎪ ⎪ Re Gβ = (Gβ + Gβ ), ⎪ ⎪ ⎪ 2 ⎨ ˆb = ⎪ G ⎪ 1 ⎪ β β β ⎪ ⎪ ⎪ ⎩ Im G = √ (G − G ), 2 −1
if 1 ≤ b ≤ n, if n + 1 ≤ b ≤ 2n,
where Gβ denote the complex conjugation of Gβ , i.e., Gβ = Gβ . Proof. Since a strongly convex complex Finsler metric F is a weakly K¨ahler-Finsler metric if and only if ( page 114, (2.6.10) in [2]) β ˙ o ∂β ) , ua Γˆ ab ∂˙ ob = (vα Γ;α
or equivalently, ˆ b ∂˙ o = (Gβ ∂˙ β )o . G b Hence our assertion follows.
5
Lemma 3.2. Let F be a strongly pseudoconvex complex Finsler metric on a complex manifold M. Then the holomorphic curvature KF of F along a nonzero vector v ∈ T z1,0 M is given by 2 α μ ν Gα ∂ν (Γ;μ )v v G2 4 = − 2 Gα ∂ν (Gα )vν G 2 = − 2 G;μν vμ vν − 4Gαβ Gα Gβ . G
KF (z, v) = −
(3.1) (3.2) (3.3)
Proof. The holomorphic curvature formula of F is given by KF (z, v) = −
2 α μ ν Gα δν (Γ;μ )v v G2
(3.4)
α for every nonzero vector v ∈ T z1,0 M (see page 109 in [2]). Here in (3.4), δν = ∂ν − Γ;ντ ∂˙ τ and Γ;μ = GναGν;μ . It is
easy to check that α ) = 0, Gα ∂˙ τ (Γ;μ
α Gα Γ;μ = G;μ ,
from which we have α α α α α ) = Gα ∂ν (Γ;μ ) = ∂ν (Gα Γ;μ ) − Gα;ν Γ;μ = G;μν − Gαβ Γ;νβ Γ;μ . Gα δν (Γ;μ
(3.5)
α μ v = 2Gα , Γ;νβ vν = 2Gβ , we obtain (3.1)-(3.3). Substituting (3.5) into (3.4) and notice that Γ;μ
Corollary 3.1. Let F be a strongly pseudoconvex complex Finsler metric on a complex manifold M. Then the holomorphic curvature KF of F along a nonzero vector v ∈ T z1,0 M satisfies KF (z, v) ≥ −
2 G;μν vμ vν , G2
(3.6)
and if the above equality holds, then KF (z, v) ≡ 0. Proof. The inequality (3.6) follows from (3.3), since (Gαβ ) is a positive definite Hermitian matrix. The equality KF (z, v) = −
2 G;μν vμ vν G2
holds if and only if Gαβ Gα Gβ ≡ 0, if and only if Gα ≡ 0,
∀ α = 1, · · · , n.
In this case, it follows from (3.2) that KF (z, v) ≡ 0. Definition 3.1. A function f (v) defined on Cn − {0} is called homogeneous of type (p, q) with respect to v = (v1 , · · · , vn ) if it satisfies f (λv) = λ p λq f (v),
∀ λ ∈ C − {0}, ∀ v ∈ Cn − {0}.
Lemma 3.3. Suppose that Ai (v) (i = 1, · · · , N) are homogeneous functions of type (pi , qi ) with respect to v =
N Ai (v) = 0 if and only if (v1 , · · · , vn ), where pi , qi are integers and (p s , q s ) (pt , qt ) for s t. Then i=1 Ai (v) = 0,
∀ i = 1, · · · , N. 6
Proof. If
N i=1
Ai (v) = 0, then N
Ai (λv) =
i=1
N
λ pi λqi Ai (v) = 0,
∀ λ ∈ C − {0}.
(3.7)
i=1
Without loss of generality, we can assume pi > 0, qi > 0 for any i, otherwise we can multiply |λ|2m in (3.7) for positive integer m big enough such that pi + m > 0, qi + m > 0. Then for every fixed s ∈ {1, · · · , N}, applying the operator
1 ∂ p s +q s p s !q s ! ∂λ p s ∂λq s
to (3.7), we get 1 ∂ ps +qs p s !q s ! ∂λ ps ∂λqs
⎡ N ⎤ ⎢⎢⎢ p qi ⎥⎥ i ⎢⎢⎣ λ λ Ai (v)⎥⎥⎥⎦ = A s (v) = 0, i=1
from which our assertion follows. Lemma 3.4. [2] Let F be a complex Finsler metric on a complex manifold M. Then F is a strongly convex complex Finsler metric if and only if Re Gαβ W α W β + Gαβ W α W β > 0,
∀ W = (W 1 , · · · , W n ) ∈ Cn , W 0.
Lemma 3.5. Let F be a strongly pseudoconvex complex Finsler metric on a complex manifold M. Then F is a K¨ahler-Finsler metric if and only if α , Gαμ = Γ;μ
or equivalently,
α Gαμν = Γν;μ .
Proof. Since 2Gα = Γ;να vν , differentiating it with respect to vμ gives α ν α v + Γ;μ , 2Gαμ = Γμ;ν
thus α α ν α α α ) = Γμ;ν v − Γ;μ = (Γμ;ν − Γν;μ )vν . 2(Gαμ − Γ;μ
Hence F is a K¨ahler-Finsler metric if and only if α , Gαμ = Γ;μ
if and only if α . Gαμν = Γν;μ
Corollary 3.2. Let F be a K¨ahler-Finsler metric on a complex manifold M. Then F is a complex Berwald metric if and only if F is a weakly complex Berwald metric. Corollary 3.3. Suppose that F is a weakly K¨ahler-Finsler metric as well as a weakly complex Berwald metric on a complex manifold M. Then it is necessarily a K¨ahler-Berwald metric.
7
Proof. By Proposition 4.3.9 in [14], F is a weakly K¨ahler-Finsler metric if and only if δˇ μG = 0, or equivalently, G;μ = GαμGα .
(3.8)
Note that for a weakly complex Berwald metric F, we have 2Gα = Gαβμ (z)vβ vμ , which is holomorphic with respect to v, thus differentiating (3.8) with respect to vβ yields Gβ;μ = GαμβGα + GαμGαβ = GαμGαβ , α , that is, F is a K¨ahler-Finsler metric. By Corollary 3.2, F is also a complex Berwald which implies that Gαμ = Γ;μ
metric. Lemma 3.6. Let F be a K¨ahler-Berwald metric on a complex manifold M. Then 2Gα;γ − Gαγ;β vβ + 2Gαγβ Gβ − Gαβ Gβγ = 0. Proof. By (2.3.16) in [2], we have α ) − δμ (Γ;να ) = 0. δν (Γ;μ
(3.9)
σ ˙ Since [δμ , ∂˙ α ] = Γα;μ ∂σ (see Lemma 2.3.3 in [2]), differentiating (3.9) with respect to vβ gives α ) − ∂˙ β δμ (Γ;να ) 0 = ∂˙ β δν (Γ;μ α α − (δμ ∂˙ β − [δμ , ∂˙ β ])Γ;μ = (δν ∂˙ β − [δν , ∂˙ β ])Γ;μ γ γ α α α α = δν (Γβ;μ ) − δμ (Γβ;ν ) + Γγ;ν Γβ;μ − Γγ;μ Γβ;ν .
(3.10)
α and Gαγμ coincides and are independent Since F is a K¨ahler-Berwald metric, it follows from Lemma 3.5 that Γγ;μ
of fiber coordinate v, so (3.10) reduces to ∂ν (Gαβμ ) − ∂μ (Gαβν ) + Gαγν Gγβμ − Gαγμ Gγβν = 0.
(3.11)
Contracting (3.11) with vβ andvμ successively yields 2Gα;ν − Gαν;μ vμ + 2Gαγν Gν − Gαγ Gγν = 0.
Remark 3.1. The term emerged in the right hand side of (3.10) is called the second curvature tensor in [15].
8
4. Main results Proposition 4.1. Let F be a strongly convex weakly K¨ahler-Finsler metric on a complex manifold M. Then F is of constant flag curvature Kˆ F = k if and only if the following system of equations hold: Gαγ;β vβ − 2Gαγβ Gβ + Gαβ Gβγ = 0,
(4.1)
2Gα;γ − Gαγ;β vβ + 2Gαγβ Gβ −
Gαβ Gβγ
= 0,
(4.2)
Gαγ;β vβ − 2Gαγβ Gβ −
Gαβ Gβγ
= 0,
(4.3)
1 Gα;γ − Gαβ Gβγ = − kvαGγ , 4 1 α β Gβ Gγ = − k(Gδαγ − vαGγ ), 4 1 Gαγ;β vβ − 2Gαγβ Gβ = − k(3Gδαγ − vαGγ ). 4 Proof. By Theorem 2.1, F is of constant flag curvature Kˆ F = k if and only if 1 Rac = k Gδac − Gcb ub ua . 2
(4.4) (4.5) (4.6)
(4.7)
In the following, for clarity we shall divide our proof into two steps. Step 1. Let’s first convert the formula (4.7) into complex setting, that is, rewrite the real tensors in (4.7) into complex tensors. Since F is a weakly K¨ahler-Finsler metric, by Lemma 3.1 we have ⎧ ⎪ ⎪ ⎪ 12 (Gβ + Gβ ), if 1 ≤ b ≤ n, ˆb =⎨ G ⎪ ⎪ ⎪ 1 β β ⎩ √ (G − G ), if n + 1 ≤ b ≤ 2n.
(4.8)
2 −1
Note that ⎧ ⎪ ⎪ ⎪ 2 Re ∂γ , if 1 ≤ c ≤ n, ∂ ⎨ = ⎪ ⎪ ∂xc ⎪ ⎩ −2 Im ∂γ , if n + 1 ≤ c ≤ 2n, ⎧ ⎪ ⎪ ⎪ if 1 ≤ c ≤ n, ∂ ⎨ 2 Re ∂˙ γ , = ⎪ ⎪ c ⎪ ∂u ⎩ −2 Im ∂˙ γ , if n + 1 ≤ c ≤ 2n, thus by (4.8) and (4.9), we have ⎧ ⎪ ⎪ ⎪ if 1 ≤ a, c ≤ n, Re[Gα;γ + Gα;γ ] = Re[Gα;γ + Gα;γ ], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ α α α α ⎪ a ˆ ⎪ − Im[G;γ + G;γ ] = − Im[G;γ − G;γ ], if 1 ≤ a ≤ n, n + 1 ≤ c ≤ 2n, ∂G ⎨ =⎪ ⎪ ⎪ ∂xc ⎪ Im[Gα;γ − Gα;γ ] = Im[Gα;γ + Gα;γ ], if n + 1 ≤ a ≤ 2n, 1 ≤ c ≤ n, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ α α α α ⎩ Re[G;γ − G;γ ] = Re[G;γ − G ], if n + 1 ≤ a, c ≤ 2n. ;γ Similarly, by (4.8) and (4.10), we have ⎧ ⎪ ⎪ ⎪ if 1 ≤ a, c ≤ n, Gαγ + Gαγ + Gαγ + Gαγ , ⎪ ⎪ ⎪ ⎪ √ ⎪ ⎪ α α α α ˆa ⎪ ⎪ if 1 ≤ a ≤ n, n + 1 ≤ c ≤ 2n, ∂G ⎨ −1(Gγ − Gγ + Gγ − Gγ ), 2 c =⎪ √ ⎪ ⎪ α α α α ∂u ⎪ − −1(Gγ + Gγ − Gγ − Gγ ), if n + 1 ≤ a ≤ 2n, 1 ≤ c ≤ n, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Gαγ − Gα − Gαγ + Gα , if n + 1 ≤ a, c ≤ 2n. γ γ 9
(4.9)
(4.10)
(4.11)
(4.12)
Since ub ∂x∂b = (vβ ∂β )o = vβ ∂β + vβ ∂β , we obtain ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2ˆa ⎪ G ∂ ⎨ b u = ⎪ ⎪ ∂xb ∂uc ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
α β if 1 ≤ a, c ≤ n, v , Re Gαγ;β vβ + Gαγ;β vβ + Gαγ;β vβ + Gγ;β α β α β α β α β − Im Gγ;β v − Gγ;β v + Gγ;β v − Gγ;β v , if 1 ≤ a ≤ n, n + 1 ≤ c ≤ 2n, α β Im Gαγ;β vβ + Gαγ;β vβ − Gαγ;β vβ − Gγ;β if n + 1 ≤ a ≤ 2n, 1 ≤ c ≤ n, v , α β α β α β α β if n + 1 ≤ a, c ≤ 2n. Re Gγ;β v − Gγ;β v − Gγ;β v + Gγ;β v ,
(4.13)
ˆ b ∂b = (Gβ ∂˙ β )o = Gβ ∂˙ β + Gβ ∂˙ , this together with (4.12) yields Note that G β ∂u ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2ˆa ⎪ ∂ G ⎨ b ˆ 2G = ⎪ ⎪ b c ⎪ ∂u ∂u ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
if 1 ≤ a, c ≤ n, 2 Re Gαγβ Gβ + Gαγβ Gβ + Gαγβ Gβ + Gαγβ Gβ , α β α β α β α β −2 Im Gγβ G − Gγβ G + Gγβ G − Gγβ G , if 1 ≤ a ≤ n, n + 1 ≤ c ≤ 2n, 2 Im Gαγβ Gβ + Gαγβ Gβ − Gαγβ Gβ − Gαγβ Gβ , if n + 1 ≤ a ≤ 2n, 1 ≤ c ≤ n, 2 Re Gαγβ Gβ − Gαγβ Gβ − Gαγβ Gβ + Gαγβ Gβ , if n + 1 ≤ a, c ≤ 2n.
(4.14)
From (4.12), if 1 ≤ a, c ≤ n, we have ⎤ ⎡ n 2n ˆ a ∂G ˆ b 1 ⎢⎢⎢ ˆ a ∂G ˆb ˆ a ∂G ˆ b ⎥⎥⎥ ∂G ∂ G ∂ G ⎥⎥⎥ = ⎢⎢⎢⎣ + 4 b=1 ∂ub ∂uc b=n+1 ∂ub ∂uc ⎦ ∂ub ∂uc 1 α = (Gβ + Gαβ + Gαβ + Gαβ )(Gβγ + Gβγ + Gβγ + Gβγ ) 4
+(Gαβ − Gαβ + Gαβ − Gαβ )(Gβγ + Gβγ − Gβγ − Gγβ ) = Re (Gαβ + Gαβ )(Gβγ + Gβγ ) . By similar reasons, we have
ˆb ˆ a ∂G ∂G = − Im (Gαβ + Gαβ )(Gβγ − Gβγ ) , if 1 ≤ a ≤ n, n + 1 ≤ c ≤ 2n, ∂ub ∂uc ˆ a ∂G ˆb ∂G = Im (Gαβ − Gαβ )(Gβγ + Gβγ ) , if n + 1 ≤ a ≤ 2n, 1 ≤ c ≤ n, c b ∂u ∂u ˆ a ∂G ˆb ∂G = Re (Gαβ − Gαβ )(Gβγ − Gβγ ) , if n + 1 ≤ a, c ≤ 2n. c b ∂u ∂u Therefore we have ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a ˆb ˆ ⎪ ∂G ∂G ⎨ = ⎪ ⎪ ⎪ ∂ub ∂uc ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
β β β β if 1 ≤ a, c ≤ n, Re Gαβ Gγ + Gαβ Gγ + Gαβ Gγ + Gαβ Gγ , α β α β α β α β − Im Gβ Gγ − Gβ Gγ + Gβ Gγ − Gβ Gγ , if 1 ≤ a ≤ n, n + 1 ≤ c ≤ 2n, Im Gαβ Gβγ + Gαβ Gβγ − Gαβ Gβγ − Gαβ Gβγ , if n + 1 ≤ a ≤ 2n, 1 ≤ c ≤ n, α β α β α β α β if n + 1 ≤ a, c ≤ 2n. Re Gβ Gγ − Gβ Gγ − Gβ Gγ + Gβ Gγ ,
By (2.4), (4.11), (4.13), (4.14) and (4.15), we obtain ⎧ ⎪ ⎪ ⎪ I = Re[A1 − (B1 + B2 ) + (C1 + C2 ) − (D1 + D2 )], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ II = − Im[A2 − (B1 − B2 ) + (C1 − C2 ) − (D1 − D2 )], ⎨ Rac = ⎪ ⎪ ⎪ ⎪ III = Im[A1 − (B3 + B4 ) + (C3 + C4 ) − (D3 + D4 )], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ IV = Re[A2 − (B3 − B4 ) + (C3 − C4 ) − (D3 − D4 )],
10
(4.15)
if 1 ≤ a, c ≤ n, if 1 ≤ a ≤ n, n + 1 ≤ c ≤ 2n, if n + 1 ≤ a ≤ 2n, 1 ≤ c ≤ n, if n + 1 ≤ a, c ≤ 2n,
(4.16)
where A1 = 2 Gα;γ + Gα;γ ,
A2 = 2 Gα;γ − Gα;γ ,
B1 = Gαγ;β vβ + Gαγ;β vβ ,
B2 = Gαγ;β vβ + Gαγ;β vβ ,
B3 = Gαγ;β vβ − Gαγ;β vβ , B4 = Gαγ;β vβ − Gαγ;β vβ , α Gβ , C1 = 2 Gαγβ Gβ + Gαγβ Gβ , C2 = 2 Gαγβ Gβ + Gγβ α C3 = 2 Gαγβ Gβ − Gαγβ Gβ , C4 = 2 Gαγβ Gβ − Gγβ Gβ , D1 = Gαβ Gβγ + Gαβ Gβγ , D2 = Gαβ Gβγ + Gαβ Gβγ , D4 = Gαβ Gβγ − Gαβ Gβγ . D3 = Gαβ Gβγ − Gαβ Gβγ , On the other hand, since F is a strongly convex complex Finsler metric,it follows that G(x, λu) = λ2G(x, u),
∀ λ ∈ R − {0}.
(4.17)
Differentiating (4.17) with respect to λ and then setting λ = 1, we get Gb (x, u)ub = 2G(x, u).
(4.18)
Differentiating (4.18) with respect to uc , we get Gcb ub = Gc . Since
⎧ ⎪ ⎪ ⎪ ⎨ Gγ + Gγ , Gc = ⎪ √ ⎪ ⎪ ⎩ −1(Gγ − Gγ ),
we have
if 1 ≤ c ≤ n, if n + 1 ≤ c ≤ 2n,
⎧ ⎪ ⎪ ⎪ ⎨ Gγ + Gγ , Gcb ub = Gc = ⎪ √ ⎪ ⎪ ⎩ −1(Gγ − Gγ ),
(4.19)
if 1 ≤ c ≤ n, if n + 1 ≤ c ≤ 2n.
Thus 1 1 k(Gδac − Gcb ub ua ) = k(2Gδac − Gc ua ). 2 2 Substituting
⎧ ⎪ ⎪ ⎪ ⎨ u =⎪ ⎪ ⎪ ⎩
1 α 2 (v
a
+ vα ),
√1 (vα 2 −1
(4.20)
if 1 ≤ a ≤ n,
− vα ),
if n + 1 ≤ a ≤ 2n,
into (4.20), we obtain ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎨ a b a k Gδc − Gcb u u = ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
if 1 ≤ a, c ≤ n, 2Gδαγ − Gγ vα − Gγ vα , II = Im 12 k vαGγ + vαGγ , if 1 ≤ a ≤ n, n + 1 ≤ c ≤ 2n, = − Im 1 k vαGγ − vαGγ , if n + 1 ≤ a ≤ 2n, 1 ≤ c ≤ n, III 2 1 α α α IV = Re 2 k 2Gδγ − v Gγ + v Gγ , if n + 1 ≤ a, c ≤ 2n.
I = Re
1 2k
(4.21)
Using Theorem 2.1 and comparing (4.16) with (4.21), we conclude that F is of constant flag curvature k if and only if I = I,
II = II,
III = III, 11
IV = IV.
(4.22)
Step 2. Note that terms on the left and right hand sides of the equalities in (4.22) are homogeneous functions with respect to v. Terms on the left hand sides of the equalities in (4.22) are of type (0, 2), (1, 1), (2, 0), (3, −1) and its complex conjugations, while terms on the right hand sides of the equalities in (4.22) are of type (1, 1), (0, 2) and its complex conjugations. It is easy to check that, with respect to the fiber variables v, there holds (1) Gαγ;β vβ , Gαγβ Gβ , Gαβ Gβγ , vαGγ are homogeneous functions of type (0, 2); α β (2) Gγ;β v , Gαγβ Gβ , Gαβ Gβγ , Gδαγ , vαGγ are homogeneous functions of type (1, 1);
(3) Gα;γ + Gα;γ , Gαγ;β vβ , Gαγβ Gβ , Gαβ Gβγ are homogeneous functions of type(2, 0); (4) Gαγ;β vβ , Gαγβ Gβ , Gαβ Gβγ are homogeneous functions of type (3, −1). In order to use Lemma 3.3, we only need to consider those terms in equalities in (4.22) which are of type (2, 0), (1, 1), (3, −1), since the complex conjugation of a homogeneous function of type (p, q) with respect to v is a homogeneous function of type (q, p) with respect to v. It is easy to check that the components in I which are of homogeneous functions of type (2, 0) with respect to v are Λ1 − Λ2 + Λ3 − Λ4 , where Λ1 = 2 Gα;γ + Gα;γ , Λ3 = 2 Gαγβ Gβ + Gαγβ Gβ ,
Λ2 = Gαγ;β vβ + Gαγ;β vβ , Λ4 = Gαβ Gβγ + Gαβ Gβγ .
The components in I which are of homogeneous functions of type (1, 1) with respect to v are −Λ5 + Λ6 − Λ7 , where
Λ6 = 2 Gαγβ Gβ + Gαγβ Gβ ,
Λ5 = Gαγ;β vβ + Gαγ;β vβ ,
Λ7 = Gαβ Gβγ + Gαβ Gβγ .
The components in I which are homogeneous functions of type (3, −1) with respect to v are Σ, where Σ = −Gαγ;β vβ + 2Gαγβ Gβ − Gαβ Gβγ .
(4.23)
The components in I which are homogeneous functions of type (2, 0), (1, 1) and (3, −1) with respect to v are respectively 1 − kvαGγ , 2
1 k 4Gδαγ − vαGγ − vαGγ 2
and
0.
Therefore I = I if and only if 1 Λ1 − Λ2 + Λ3 − Λ4 = − kvαGγ , 2 1 −Λ5 + Λ6 − Λ7 = k 4Gδαγ − vαGγ − vαGγ , 2 Σ = 0.
12
(4.24) (4.25) (4.26)
The components in II which are of homogeneous functions of type (2, 0) with respect to v are −Ξ1 + Ξ2 − Ξ3 + Ξ4 , where Ξ1 = 2 Gα;γ − Gα;γ , Ξ3 = 2 Gαγβ Gβ − Gαγβ Gβ ,
Ξ2 = Gαγ;β vβ − Gαγ;β vβ , Ξ4 = Gαβ Gβγ − Gαβ Gβγ .
The components in II which are of homogeneous functions of type (1, 1) with respect to v are −Ξ5 + Ξ6 + Ξ7 , where Ξ5 = Gαγ;β vβ − Gαγ;β vβ ,
Ξ6 = 2 Gαγβ Gβ − Gαγβ Gβ ,
Ξ7 = Gαβ Gβγ − Gαβ Gβγ .
The components in II which are homogeneous functions of type (3, −1) with respect to v are Σ, where Σ is given by (4.23). The components in II which are homogeneous functions of type (2, 0), (1, 1) and (3, −1) with respect to v are respectively 1 − kvαGγ , 2
1 α k v G γ − vα G γ 2
and
0.
Thus II = II if and only if 1 −Ξ1 + Ξ2 − Ξ3 + Ξ4 = − kvαGγ , 2 1 −Ξ5 + Ξ6 + Ξ7 = k vαGγ − vαGγ , 2 Σ = 0.
(4.27) (4.28) (4.29)
if and only if By similar arguments, it is easy to check that III = III 1 Λ1 − Λ2 + Λ3 − Λ4 = − kvαGγ , 2 1 Ξ5 − Ξ6 + Ξ7 = − k(vαGγ − vαGγ ), 2 Σ = 0.
(4.30) (4.31) (4.32)
if and only if IV = IV 1 α kv Gγ , 2 1 −Λ5 + Λ6 − Λ7 = k(4Gδαγ − vαGγ + vαGγ ), 2
Ξ1 − Ξ2 + Ξ3 − Ξ4 =
−Σ = 0.
(4.33) (4.34) (4.35)
Note that (4.24) and (4.30) are the same equations, (4.25) and (4.34) are the same equations, (4.27) and (4.33) are the same equations, (4.28) and (4.31) are the same equations, (4.26), (4.29), (4.32) and (4.35) are the same
13
equations. Therefore a strongly convex weakly K¨ahler-Finsler metric F is of constant flag curvature k if and only if the following system of equations holds: 1 Λ1 − Λ2 + Λ3 − Λ4 = − kvαGγ , 2 1 Ξ1 − Ξ2 + Ξ3 − Ξ4 = kvαGγ , 2 1 −Λ5 + Λ6 − Λ7 = k 4Gδαγ − vαGγ − vαGγ , 2 1 Ξ5 − Ξ6 + Ξ7 = − k(vαGγ − vαGγ ), 2 −Σ = 0, which is equivalent to the following system of equations: 2Gα;γ − Gαγ;β vβ + 2Gαγβ Gβ − Gαβ Gβγ = 0, 1 2Gα;γ − Gαγ;β vβ + 2Gαγβ Gβ − Gαβ Gγβ = − kvαGγ , 2 1 Gαγ;β vβ − 2Gαγβ Gβ + Gαβ Gβγ = − k 2Gδαγ − vαGγ , 2 Gαγ;β vβ − 2Gαγβ Gβ + Gαβ Gβγ = 0.
(4.36) (4.37) (4.38) (4.39)
In the following we shall simplify the equations (4.37) and (4.38). Since Gα are homogeneous functions of type (2, 0) with respect to v, it follows that Gβγ vγ = 2Gβ ,
Gαγ vγ = 0, from which we get Gαγβ vγ = −Gαβ ,
Gαβ Gγβ vγ = 2Gαβ Gβ .
Now contracting (4.37) with vγ or contracting (4.38) with vγ , we get 1 Gα;β vβ − 2Gαβ Gβ = − kvαG. 4
(4.40)
Differentiating (4.40) with respect to vγ yields 1 Gαγ;β vβ + Gα;γ − 2Gαγβ Gβ − 2Gαβ Gβγ = − kvαGγ . 4
(4.41)
Gαγ;β vβ − 2Gαγβ Gβ − Gαβ Gβγ = 0.
(4.42)
1 Gα;γ − Gαβ Gγβ = − kvαGγ . 4
(4.43)
(4.41) × 2 − (4.37) implies
(4.42) + (4.37) yields
Hence (4.37) is equivalent to (4.42) and (4.43). Similarly, it is easy to check that (4.38) is equivalent to the following system of equations: 1 Gαβ Gβγ = − k(Gδαγ − vαGγ ), 4 1 Gαγ;β vβ − 2Gαγβ Gβ = − k(3Gδαγ − vαGγ ). 4 14
Therefore we conclude that F is of constant flag curvature Kˆ F = k if and only if (4.1)-(4.6) hold. This completes the proof. Now we are ready to prove our first main result in this paper. Theorem 4.1. Let F be a strongly convex weakly K¨ahler-Finsler metric on a complex manifold M. If F is of constant flag curvature Kˆ F = k, then F is of constant holomorphic curvature KF = k. Proof. Contracting (4.4) with vγ , we have 1 Gα;γ vγ − 2Gαβ Gβ = − kvαG, 4
(4.44)
where we use the following identities: Gβγ vγ = 2Gβ ,
Gγ vγ = G.
Note that Gα G α =
1 α μ 1 1 1 Γ v Gα = GταGτ;μ vμGα = vτGτ;μ vμ = G;μ vμ , 2 ;μ 2 2 2
so that Gα;γ Gα = ∂γ (GαGα ) − GαGα;γ , GαβGα = ∂˙ β (GαGα ) − GαGαβ . Thus contracting (4.44) with Gα , we get 1 1 1 μ ν α γ μ α G;μν v v − Gα;γ G v − 2 Gβ;μ v − G Gαβ Gβ = − kG2 . 2 2 4 Using Lemma 3.2, we obtain k = KF −
4 α β α γ β μ G G − G G v − G G v 4G . α;γ αβ β;μ G2
(4.45)
Now since β α γ G v = 2Gαβ Gα Gβ , Gα;γ Gα vγ = Gαβ Γ;γ α β μ G v = 2Gαβ Gα Gβ . Gβ;μ Gβ vμ = Gαβ Γ;μ
It follows that k = KF . This completes the proof. Proposition 4.2. Let F be a strongly convex weakly K¨ahler-Finsler metric on a complex n-dimensional manifold M. Then F is a real Einstein-Finsler metric if and only if 2Gα;α − Gαα;β vβ + 2Gααβ Gβ − Gαβ Gβα = 0, α β v ) − 2Gαβ Gβα = (2n − 1)σ(z)G 2(Gααβ Gβ + Gααβ Gβ ) − (Gαα;β vβ + Gα;β
for some real scalar function σ(z) on M. 15
Proof. By Proposition 4.1, the Riemann curvature tensor Rac of F is given by (4.16), so its Ricci curvature is Ric = Raa
= Re 2 Gα;α + Gα;α − Gαα;β vβ + Gαα;β vβ + Gαα;β vβ + Gαα;β vβ α Gβ − Gαβ Gβα + Gαβ Gβα + Gαβ Gβα + Gαβ Gβα +2 Gααβ Gβ + Gααβ Gβ + Gααβ Gβ + Gαβ + Re 2 Gα;α − Gα;α − Gαα;β vβ − Gαα;β vβ − Gαα;β vβ + Gαα;β vβ α Gβ − Gαβ Gβα − Gαβ Gβα − Gαβ Gβα + Gαβ Gβα +2 Gααβ Gβ − Gααβ Gβ − Gααβ Gβ + Gαβ α Gβ − Gαβ Gβα + Gαβ Gβα . = 2 Re 2Gα;α − Gαα;β vβ + Gαα;β vβ + 2 Gααβ Gβ + Gαβ
Hence F is a real Einstein-Finsler metric if and only if 2 Re[2Gα;α − Gαα;β vβ + Gαα;β vβ + 2 Gααβ Gβ + Gααβ Gβ − Gαβ Gβα + Gαβ Gβα ] = (2n − 1)σG
(4.46)
for a real scalar function σ(z) on M. Note that Gα;α , Gαα;β vβ , Gααβ Gβ , Gαβ Gβα are homogeneous functions of type (2, 0) with respect to v, and the rest terms in equality (4.46) are homogeneous functions of type (1, 1) with respect to v. Using Lemma 3.3 and by comparing the types of homogeneity in (4.46), our assertion follows. Note that for a weakly complex Berwald metric F, its complex spray coefficients Gα are holomorphic with respect to the fiber coordinate v, thus Gαμ = 0. In this case, equations (4.1)-(4.6) can be greatly simplified. So it is natural to consider the case that F is a weakly complex Berwald metric in Proposition 4.1. According to Corollary 3.3, we actually should focus on K¨ahler-Berwald metrics. We have the following theorem. Theorem 4.2. Let F be a strongly convex K¨ahler-Berwald metric on a complex manifold M of complex dimension n ≥ 2. Then F is of constant flag curvature if and only if F is a locally complex Minkowski metric. Proof. Theorem 4.9 in [13] stated that a strongly convex weakly K¨ahler-Finsler metric is a weakly complex Berwald metric if and only if it is a real Berwald metric. Hence F is a real Berwald metric. If F is of constant flag curvature Kˆ F = k, under our condition for F, (4.5) reduces to k(Gδαγ − vαGγ ) = 0, which implies that k = 0 whenever n ≥ 2, namely, F has vanishing flag curvature. It is a well-known fact that a real Berwald metric with zero flag curvature is actually a locally real Minkowski metric (see Proposition 10.5.1 in [8]), this means that F, namely, F o is a locally real Minkowski metric, hence F is a locally complex Minkowski metric. Conversely, if F is a locally complex Minkowski metric, it results that F is also a locally real Minkowski metric since F is strongly convex. So F has vanishing flag curvature. This completes the proof. There are two notions of Ricci scalar curvature associated to a strongly pseudoconvex complex Finsler metric in literature, we refer to [13, 14, 16] for more details.
16
Definition 4.1 ([16]). Let F be a strongly pseudoconvex complex Finsler metric defined on a complex manifold M. Then the Ricci scalar curvature Ric of the Chern-Finsler connection associated to F is defined by Ric(z, v) = −δα (Γμ;μ )vα , ˘ of the complex Berwald connection associated to F is defined by and the Ricci scalar curvature Ric ˘ Ric(z, v) = −δˇ α (Gμμ )vα , where the index μ is summed from 1 to n. ˘ However, by Lemma In general, the two Ricci scalar curvatures are not necessarily real-valued and Ric Ric. ˘ if F is a K¨ahler-Finsler metric. 3.5 we have Ric = Ric Theorem 4.3. Let F be a strongly convex K¨ahler-Berwald metric on a complex n-dimensional manifold M. Then F is a real Einstein-Finsler metric if and only if the Ricci scalar curvature Ric of F satisfying 2 Re Ric = (2n − 1)σ(z)G for some real scalar function σ(z) on M. Proof. Since F is a K¨ahler-Berwald metric, it follows from Lemma 3.6 that Gα;α − Gαα;β vβ + 2Gααβ Gβ − Gαβ Gβα = 0,
Gαμ = 0,
which together with Proposition 4.2 implies that F is a real Einstein-Finsler metric if and only if α β v = (2n − 1)σ(z)G Gαα;β vβ + Gα;β
(4.47)
holds for some real scalar function σ(z) on M. Note that the Ricci scalar curvature of F is ˘ = −δˇ α (Gμμ )vα = −∂α (Gμμ )vα = Gα vβ . Ric = Ric α;β
(4.48)
By plunging (4.48) into (4.47), our assertion follows. Corollary 4.1. Let F be a strongly convex K¨ahler-Berwald metric on a complex n-dimensional manifold M. If F does not comes from a Hermitian metric, then F is a real Einstein-Finsler metric if and only if the Ricci scalar curvature Ric of F satisfying Re Ric = 0. Proof. By Theorem 4.3, F is a real Einstein-Finsler metric if and only if (2n − 1)σ(z)G = 2 Re Ric = Gαα;β vβ + Gαα;β vβ = Gααμ;β (z)vμ vβ + Gααμ;β (z)vβ vμ
(4.49)
holds for some real scalar function σ(z) on M. Differentiating (4.49) with respect to vγ and vν successively yields (2n − 1)σ(z)Gγν = 0.
(4.50)
Since F does not come from a Hermitian metric, we have Gγν 0, this together with (4.50) implies that σ(z) = 0. Thus (4.49) holds if and only if σ(z) = 0, namely, Re Ric = 0. 17
5. Examples of strongly convex K¨ahler-Finsler metric In the following, we shall give two nontrivial examples of strongly convex K¨ahler-Finsler metrics, where one comes from a strongly convex complex Minkowski metric in Cn and the other is a strongly convex K¨ahler-Finsler metric which does not come from a strongly convex complex Minkowski metric. Proposition 5.1. Let z = (z1 , · · · , zn ) ∈ Cn , v = (v1 , · · · , vn ) ∈ T z1,0 Cn and let ε ∈ (0, +∞) be a constant. Define Fε (z, v) = |v1 |2 + · · · + |vn |2 + ε |v1 |4 + · · · + |vn |4 . Then (i) Fε is a strongly pseudoconvex complex Finsler metric for every ε ∈ (0, +∞); (ii) Fε is a strongly convex complex Finsler metric for every ε ∈ (0, 1); (iii) Fε is a strongly convex K¨ahler-Finsler metric with vanishing flag curvature and holomorphic curvature for every ε ∈ (0, 1). 1
Proof. Denote G = Fε2 = |v1 |2 + · · · + |vn |2 + εA 2 , where A = |v1 |4 + · · · + |vn |4 . It is easy to check that Gαβ = δαβ + 2εA− 2 |vα |2 δαβ − εA− 2 |vα |2 vα |vβ |2 vβ , 1
3
− 12
− 32
Gαβ = εA (vα )2 δαβ − εA |vα |2 vα |vβ |2 vβ ,
(5.2)
where δαβ = 1 for α = β and δαβ = 0 for α β. Thus for V = (V 1 , · · · , V n ) ∈ Cn , 2 n n n α β α2 − 12 α2 α2 − 32 α 2 α α |V | + 2εA |v | |V | − εA |v | v V . Gαβ V V = α=1 α=1 α=1 Note that
(5.1)
(5.3)
2 ⎛ ⎞2 n n n n ⎜⎜ ⎟⎟ α 2 α |v | vα V ≤ ⎜⎜⎝⎜ |vα |3 |V α |⎟⎟⎟⎠ ≤ 2 |vα |6 |V α |2 ≤ 2A |vα |2 |V α |2 . α=1 α=1 α=1 α=1
This together with (5.3) yields n
Gαβ V α V β ≥
|V α |2 ≥ 0
α=1
with equality holds if and only if V = 0 ∈ Cn . This proves (i). Next by (5.2), we have ⎛ n ⎞2 ⎛ n ⎞2 ⎜⎜ ⎟⎟⎟ ⎜⎜ α 2 ⎟⎟ α β − 32 ⎜ α⎟ − 32 ⎜ β 2 β ⎜ α β ⎜ ⎟ 2 Re Gαβ V V = −εA ⎜⎝ |v | v V ⎟⎠ − εA ⎜⎜⎝ |v | v V ⎟⎟⎟⎠ α=1
+ εA
− 12
β=1
n
(vα V α )2 + εA
α=1
− 12
n (vα V α )2 . α=1
Thus 2 Re Gαβ V α V β + Gαβ V α V β =2
n
α2
|V | + 4εA
α=1
+ εA− 2 1
n α=1
− 12
n
α2
α2
|v | |V | − εA
α=1
(vα V α )2 + εA− 2 1
− 32
⎡ ⎞⎤2 ⎛ n ⎢⎢⎢ ⎟⎥ ⎜ ⎢⎢⎣2 Re ⎜⎜⎜⎜⎝ |vα |2 vα V α ⎟⎟⎟⎟⎠⎥⎥⎥⎥⎦
n (vα V α )2 . α=1
18
α=1
(5.4)
Since
⎡ ⎞⎤2 ⎡ n ⎤ ⎛ n n n ⎢⎢⎢ ⎟⎥ ⎢ α 3 α ⎥⎥⎥2 ⎜ ⎢⎢⎣2 Re ⎜⎜⎜⎜⎝ |vα |2 vα V α ⎟⎟⎟⎟⎠⎥⎥⎥⎥⎦ ≤ ⎢⎢⎢⎢⎣2 |v | |V |⎥⎥⎦ ≤ 4 |vα |6 |V α |2 ≤ 4A |vα |2 |V α |2 α=1
and 2
n
α=1
|vα |2 |V α |2 +
α=1
α=1
n n n (vα )2 (V α )2 + (vα )2 (V α )2 = (vα V α + vα V α )2 ≥ 0, α=1
α=1
(5.5)
α=1
(5.6)
α=1
equality (5.4) yields n n 1 |V α |2 − 2εA− 2 |vα |2 |V α |2 2 Re Gαβ V α V β + Gαβ V α V β ≥ 2 α=1
α=1
= 2A
n 1 (A 2 − ε|vα |2 )|V α |2
≥ 2A− 2
n (1 − ε)|vα |2 |V α |2 ≥ 0,
− 12
α=1
1
α=1
with equality holds if and only if vα V α = 0 for α = 1, · · · , n, if and only if V = 0. By Lemma 3.4, Fε is a strongly convex complex Finsler metric for every ε ∈ (0, 1). This completes the proof of (ii). The assertion (iii) is obvious since Fε is a strongly convex complex Minkowski metric. In the following we shall give an example of strongly convex K¨ahler-Finsler metric which does not come from a complex Minkowski metric. Let M be a complex m-dimensional manifold endowed with a Hermitian metric α 2 (ξ) = aαβ (z)ξα ξβ ,
∀ ξ = ξα
∂ ∈ T z1,0 M, ∂zα
and N be a complex n-dimensional manifold endowed with a Hermitian metric β 2 (η) = bμν (w)ημ ην ,
∀ η = ημ
∂ ∈ T 1,0 N. ∂wμ
Here {z1 , · · · , zm } and {w1 , · · · , wn } are local complex coordinates on M and N, respectively; {z1 , · · · , zm , ξ1 , · · · , ξm } and {w1 , · · · , wn , η1 , · · · , ηn } are the induced complex coordinates on T 1,0 M and T 1,0 N, respectively. For constants ε > 0 and k > 1, one can define the following Szab´o metric on the product complex manifold M × N: 1 α2k (ξ) + β 2k (η)) k . Fε = α 2 (ξ) + β 2 (η) + ε(α
(5.7)
In [17], it was proved that the Szab´o metric Fε is a strongly pseudoconvex complex Berwald metric on M × N, and Fε is a strongly pseudoconvex K¨ahler-Finsler metric if and only if α and β are K¨ahler metrics on M and N, respectively. In this paper, we shall prove the strong convexity of Fε . Proposition 5.2. Let Fε be the Szab´o metric defined by (5.7). Then (i) Fε is a strongly convex complex Berwald metric on M × N; (ii) Fε is a strongly convex K¨ahler-Finsler metric on M × N if and only if α and β are K¨ahler metrics on M and N, respectively.
19
Proof. Denote upper index T the transpose of a vector and “∗” the Hermitian transpose of a vector in a finite dimensional complex linear space. Set H1 := (aαβ (z)), H2 := (bμν (w)) and denote ξ := (ξ1 , · · · , ξm )T ,
ξ∗ := (ξ1 , · · · , ξm ),
η := (η1 , · · · , ηn )T ,
η∗ := (η1 , · · · , ηn ),
ξ˜ := (ξ1 , · · · , ξm )T ,
ξ˜∗ := (ξ1 , · · · , ξm ),
η˜ := (η1 , · · · , ηn )T ,
η˜ ∗ := (η1 , · · · , ηn ),
where ξα = aαβ (z)ξβ ,
ημ = bμν (w)ην ,
Set G = Fε2 , a simple calculation gives ⎛ 2 ∂2 G ⎜⎜⎜ ∂ G ⎜⎜⎜ ∂ξα ∂ξβ ∂ξα ∂ην H := ⎜⎜ 2 ⎝ ∂G ∂2 G ⎛ ⎜⎜⎜ := ⎜⎜⎜⎜ H ⎜⎝
∂ημ ∂ξβ
∂ημ ∂ην
∂2 G ∂ξα ∂ξβ
∂2 G ∂ξα ∂ην
∂2 G ∂ημ ∂ξβ
∂2 G ∂ημ ∂ην
⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎠ (m+n)×(m+n)
⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎠
(m+n)×(m+n)
α, β = 1, · · · , m;
⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟, 2 α ∗ ⎟ c2 H2 + c3 β 2 η˜ η˜ ⎠ ⎞ −ξ˜η˜ T ⎟⎟⎟⎟⎟ ⎟⎟ , ⎠ α2 T ⎟ η ˜ η ˜ 2 β
⎛ ⎜⎜⎜ c H + c β 2 ξ˜ξ˜∗ 3 α2 ⎜ 1 1 = ⎜⎜⎜⎜ ⎝ −c η˜ ξ˜∗ 3
⎛ 2 ⎜⎜⎜ β ξ˜ξ˜T ⎜ 2 = c3 ⎜⎜⎜⎜ α ⎝ −η˜ ξ˜T
μ, ν = 1, · · · , n.
−c3 ξ˜η˜ ∗
where α2k (ξ) + β 2k (η)] k −1α 2(k−1) (ξ), c1 = 1 + ε[α 1
α2k (ξ) + β 2k (η)] k −1β 2(k−1) (η), c2 = 1 + ε[α 1
α2k (ξ) + β 2k (η)] k −2α 2(k−1) (ξ)ββ2(k−1) (η). c3 = (k − 1)ε[α 1
For
⎛ ⎞ ⎜⎜⎜ V ⎟⎟⎟ ⎜⎜⎜ 1 ⎟⎟⎟ V = ⎜⎜ 0 ∈ Cm+n , ⎝ V ⎟⎟⎠ 2
where V1 = (V11 , · · · , V1m )T ∈ Cm , V2 = (V21 , · · · , V2n )T ∈ Cn ,
it is easy to check that ∗ )T 2 Re V ∗ HV + V ∗ H(V = 2c1α 2 (V1 ) + 2c2β 2 (V2 ) + c3
#
$2
α β
ξ, V1 α + ξ, V1 α −
η, V2 β + η, V2 β α β
> 0,
where α 2 (V1 ) = aαβ (z)V1α V1β ,
β 2 (V2 ) = bμν (w)V2μ V2ν ,
ξ, V1 α = aαβ (z)ξα V1β ,
η, V2 β = bμν (w)ημ V2ν .
Hence by Lemma 3.4, Fε is a strongly convex complex Finsler metric. Thus by the result in [17], we complete the proof of the assertions (i) and (ii).
Acknowledgement This work is supported by the Program for New Century Excellent Talents in University (No. NCET-13-0510); the National Natural Science Foundation of China (Grant No. 11271304, 11171277); the Fujian Province Natural Science Funds for Distinguished Young Scholar (Grant No. 2013J06001); the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. 20
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