Directory UMM :Journals:Journal_of_mathematics:BJGA:

Chunping Zhong

Abstract. Let M be a smooth manifold with a Finsler metric F and ˜

g be the naturally induced Riemann metric on the slit tangent bundle ˜

M. The Weitzenb¨ock formulas of the horizontal Laplacian △h and the vertical Laplacian△v are obtained in terms of the Cartan connection of (M, F). The relationship between the Hodge-Laplace operator △ of ˜g and the operators △h,△v, △mix are investigated. As application, the relationship between the eigenvalues of △ and △h,△v are established, and a Bochner-type vanishing theorem of horizontal differential form on

˜

M is obtained.

M.S.C. 2010: 53C40, 53C60.

Key words: Finsler manifold; horizontal Laplacian; vertical Laplacian.

1

Introduction

LetM be a real manifold of dimensionm, we denote byT M its tangent bundle, and π : T M → M the canonical projection, the cotangent bundle of M is denoted by T∗_{M. We assume that M is endowed with a Finsler metricF in the sense of [1], see}

also [3] and [7]. It is known that there is not a canonical way to define Laplacian on (M, F), we refer to [2] for more details. Usually Laplacians on Finsler manifolds are constructed either on the base manifoldM or the slit tangent bundle ˜M =T M− {o}, whereodenotes the zero section ofT M. In [8], Munteanu obtained the Weitzenb¨ock formulas of horizontal and vertical Laplacians for 1-forms on a Riemann vector bundle p: E →M with compact fiber spaces F =p−1_{(x) for x∈ M. Very recently, this}

idea was developed in [10] to investigate vanishing theorem of holomorphic forms on K¨ahler Finsler manifolds. Note that since most curvature components of Finsler metrics depend not only on base pointx∈M but also on directiony∈TxM [9], and a Finsler metricF on M naturally induces a Riemann metric ˜g on ˜M, it is natural to define a Laplacian on the tangent bundleT M of a Finsler manifold (M, F). The purpose of this paper is to introduce the horizontal and vertical Laplacians of a Finsler metric on the slit tangent bundle ˜M and relate it to the Hodge-Laplace operator△ of the Riemann metric ˜gon ˜M. It is known that△is not a type preserving operator when acting on differential forms of type (p, q) on ˜M, i.e., those forms which are

∗

Balkan Journal of Geometry and Its Applications, Vol.16, No.1, 2011, pp. 170-181. c

horizontal type of p and vertical type of q on ˜M. Nevertheless, there are three components of △, i.e., the horizonal Laplacian △h, the vertical Laplacian △v and the mixed part △mix, which are type preserving [8]. Our results shows that both the local expressions of △h and△v depend on the choice of the Finsler connection associated to (M, F). Moreover, the horizontal Laplacian △h depends only on the horizontal curvature components of F, the vertical Laplacian △v depends only on the vertical curvature components ofF, the mixed Laplacian△mix depends only on the Riemann curvature ofF and is independent of the choice of Finsler connections associated to (M, F). We obtain the Weitzenb¨ock formula of△h and △v in terms of the Cartan connection of (M, F), and obtain a Bochner-type vanishing theorem of horizontal differential forms which are compactly supported in ˜M.

2

Preliminaries

In this section, we shall recall some basic facts of Finsler manifold. We refer to [1] for more details. LetF be a Finsler metric on a real manifoldM of dimensionm. Denote byV the vertical vector bundle of M, then there is a Riemann metricg onV which is induced by F, and a unique good vertical connection ∇ : X(V) → X(T∗_{M˜ ⊗ V)}

called the Cartan connection associated to (M, F). Note that associated to∇, there are horizontal bundle H and the horizontal map Θ : V → H. Using Θ one can transfers the natural Riemann metric g on V to H, and get a Riemann metric ˜g on the whole bundleTM˜, and consequently a linear connection, still denoted by ∇, onX(TM˜) andX(T∗_{M˜). Note that one can also consider the Berwald connection}

associated to (M, F), and it was shown in [5] that the Berwald connection and the Cartan connection play a different role when one considers the properties of adapted coordinates system related to them.

Locally let (xi_{) = (x}1_{,· · ·, x}m_{) be the local coordinates on M and (x}i_{, y}i_{) =} (x1_{,· · · , x}m_{, y}1_{,· · ·, y}m_{) be the naturally induced coordinates on T M. Denote by} {δi,∂i}˙ the local frame forTM˜ and{dxi_{, δy}i_{}the dual frame forT}∗_{M˜, here}

δi= ∂ ∂xi −Γ

j

;i ∂

∂yj, ∂˙i = ∂ ∂yi, δy

i_=dyi_+Γi

;jdxj,

andΓ;ji is the nonlinear connection coefficients associated to the Cartan connection ∇. It is known that the Cartan connection∇satisfy

∇δk∂˙j = Γ

i

j;k∂˙i, ∇_∂˙_k∂˙j=Cjki ∂˙i, ∇δkδj =Γ

i

j;kδi, ∇_∂˙_kδj =Cjki δi, (2.1)

∇δkdx

i _{= −Γ}i

j;kdxj, ∇δkδy

i_=−Γi j;kδyj, (2.2)

∇_∂˙_kdxi = −Cjki dxj, ∇_∂˙_kδyi=−Cjki δyj, (2.3)

where in the above equations,Γi

j;k andCjki are the horizontal and vertical connection coefficients of the Cartan connection. That is,

Γ_ji_;k =1 2g

hi_[δj(g

hk) +δk(gjh)−δh(gjk)], Cjki = 1 2g

hi_∂˙ j(ghk), (2.4)

withghk= 1 2∂h˙ ∂k(F˙

2_{). It is clear that}

Note that{dxi_{, δy}i_{}is a local frame forT}∗_{M˜, it follows that the differential forms}

on ˜M can be represented in terms of {dxi_{, δy}i_{}. In [8], a differential formϕof type} (p, q) on ˜M is expressed in terms of {dxi_{, δy}i_{}as follows:}

(2.6) ϕ= 1

p!q!ϕIpJqdx

Ip_∧δyJq

whereIp=i1· · ·ip, Jq=j1· · ·jq,dxIp=dxi1∧ · · · ∧dxip, δyJq=δyj1∧ · · · ∧δyjq, and

ϕIpJq are anti-symmetric in their indexes inIpandJq, respectively. In the following,

we denote by∧p,q

c ( ˜M) the space of differential forms on ˜M which are of type (p, q) and the coefficientsϕIpJq are compactly supported in ˜M. Note that there is a naturally

point-wise inner producth·,·idefined in∧p,q

c ( ˜M) for every (x, y)∈M˜. More precisely, forϕ, ψ∈ ∧p,q

c ( ˜M),

hϕ, ψi= 1

p!q!ϕi1···ipj1···jqψr1···rps1···sqg

i1r1_{· · ·g}iprp_gj1s1_{· · ·g}jqsq_,

where the above equation is evaluated at (x, y)∈M˜. The point-wise inner product h·,·igives rise to a global inner product (·,·) defined in∧p,q

c ( ˜M). That is,

(2.7) (ϕ, ψ) =

Z

˜

M

hϕ, ψidV

whenever the integral of (2.7) exists. HeredV = det(gij)dx1_{∧· · ·∧dx}n_∧dy1_{∧· · ·∧dy}n is the natural volume form of the Riemann metric ˜g on ˜M. Note that

(2.8) df =δi(f)dxi+ ˙∂i(f)δyi

forf ∈C∞( ˜M) and

(2.9) d(δyi_{) =−}1 2R

i

jkdxj∧dxk−Gijkdxj∧δyk,

where Ri_jk = δk(Γi

;j)−δj(Γ;ik) and Gijk = ˙∂j(Γ;ik). Thus for ϕ ∈ ∧p,qc ( ˜M) we have dϕ = dhϕ+dvϕ+dmixϕ, where dhϕ ∈ ∧pc+1,q( ˜M), dvϕ ∈ ∧p,qc +1( ˜M) and dmixϕ ∈ ∧p+2,q−1

c ( ˜M). Let d∗h, d∗v and d∗mix be the formal adjoints of dh, dv and dmix, respectively with respect to the global inner product (2.7), i.e.,

(dhϕ, ψ) = (ϕ, d∗hψ),(dvϕ, ψ) = (ϕ, d∗vψ),(dmixϕ, ψ) = (ϕ, d∗mixψ). (2.10)

Let i be the substitution operator and e be the wedge product operator. The operatorsiand esatisfy

(2.11) i(δj)dxk =δ_jk, i(δj)δyk = 0, i( ˙∂j)dxk= 0, i( ˙∂j)δyk =δ_jk,

(2.12) e(dxi_)ϕ=dxi_{∧ϕ, e(δy}i_)ϕ=δyi_∧ϕ

forϕ∈ ∧p,q

c ( ˜M), and are anti-commutative, i.e.,

i(X)i(Y)ϕ=−i(Y)i(X)ϕ, e(φ)e(ω)ϕ=−e(ω)e(φ)ϕ (2.13)

forX, Y ∈ X(TM˜) and φ, ω∈ X(T∗_{M˜). In the calculation ofd}

h, dv, dmix and their formal adjointd∗

3

Levi-Civita connection of

g

˜

Lemma 3.1. Let D be the Levi-Civita connection of the Riemann metric ˜g on M˜. Then in terms of the local frame{δi,∂i}˙ and its dual frame {dxi_{, δy}i_{}, we have}

Proof. We only need to prove (3.1)-(3.4) since one can obtain (3.5)-(3.8) by the fol-lowing formula

DXω(Y) =X(ω(Y))−ω(DXY), ∀X, Y ∈ X(TM˜), ω∈ X(T∗M˜).

As well known, the Levi-Civita connection D of ˜g is characterized by the following Koszul formula it is easy to check that

consequently we have

This completes (3.2) and (3.4). By the torsion-freeness ofD, we get (3.3). ¤

4

Laplacians in terms of Cartan connection

Letd be the exterior differential operator on ˜M and d∗ _{be the formal adjoint of d}

with respect to the global inner product (·,·) defined by (2.7). In this section we shall first give a representation of d and d∗ in terms of the horizontal and vertical covariant derivatives of the Cartan connection∇. Then we shall derive the horizontal Laplacian△h and the vertical Laplacian△v in terms of ∇, respectively.

Lemma 4.1. Let (M, F) be a real Finsler manifold with the Cartan connection∇. Then

Proof. In deed, it follows from (3.13) and (3.14) that

ghiLk_ji=−1

Lemma 4.3. Let (M, F) be a real Finsler manifold with the Cartan connection∇. Let∇also denote the induced linear connection onTM˜ andT∗_{M˜, respectively. Then}

for everyϕ∈ ∧p,q

Lemma 4.4. Let (M, F) be a real Finsler manifold with the Cartan connection∇, andd∗ be the formal adjoint of the exterior differential operatordonM˜ with respect to the inner product(·,·)defined by (2.7). Then

(4.10)

d∗=−gkii(δk)∇δi+g

ki_Lh

jie(δyj)i(δk)i( ˙∂h) +gkiLhiki(δh)

−gkii( ˙∂k)∇_∂˙_i−gki

³

C_kih +1 2R

h ki

´

i( ˙∂h) +gkiC_jihe(dxj_{)i(δk)i( ˙∂h)}

−1 2g

ki_glh_Rs

ligsje(δyj)i(δk)i(δh).

Proof. In terms of the local frame{dxi_{, δy}i_{} forT}∗_{M˜, the dual operatord}∗ _ofdcan

be expressed in terms of the Levi-Civita connectionDof ˜gin an invariant form:

(4.11) d∗ϕ=−gkii(δk)Dδiϕ−g

ki_{i( ˙∂k)D}

˙

∂iϕ, ϕ∈ ∧

p,q c ( ˜M).

Using this and (3.5)-(3.8) and Lemma 4.2, one gets (4.10). ¤

Theorem 4.5. Let (M, F)be a real Finsler manifold with the Cartan connection∇. Letϕbe a differential form of type (p, q)which is compactly supported in M˜. Then

△hϕ = −gki∇δi∇δkϕ−g

ki_e(dxs_{)i(δk)(∇δ}

s∇δi− ∇δi∇δs)ϕ+g

ki_Gh ik∇δhϕ

+gki³Lh_ji|s+L_jsh_|i+Lt_jiLh_st−Lt_jsLh_it´e(dxs)e(δyj)i(δk)i( ˙∂h)ϕ

+gkiLh_ik|se(dxs)i(δh)ϕ+gki³Lt_ks|i−Lh_isLt_hk−Lh_ikLt_hs´e(δys)i( ˙∂t)ϕ

−gki_Lh

jiLtkse(δyj)e(δys)i( ˙∂h)i( ˙∂t)ϕ.

Proof. By (4.1) and (4.10), we get

dh = e(dxs_)∇δ

s−L

t

rse(dxr)e(δys)i( ˙∂t), d∗_h = −gkii(δk)∇δi+g

ki_Lh

jie(δyj)i(δk)i( ˙∂h) +gkiLhiki(δh).

Thus

dh◦d∗h = −e(dxs)∇δsg

ki_i(δ

k)∇δi+e(dx

s_)∇δ

sg

ki_Lh

jie(δyj)i(δk)i( ˙∂h) +e(dxs_)∇δ

sg

ki_Lh

iki(δh) +gkiLtrse(dxr)e(δys)i( ˙∂t)i(δk)∇δi

−gkiLh_jiLt_rse(dxr)e(δys)i( ˙∂t)e(δyj)i(δk)i( ˙∂h)

−gki_Lh

ikLtrse(dxr)e(δys)i( ˙∂t)i(δh).

Since the Cartan connection∇is horizontal metrical, it follows that

(4.12) gki_|s=δs(gki_{) +g}ti_Γk

t;s+gktΓti;s= 0,

we obtain

dh◦d∗h = gktΓti;se(dxs)i(δk)∇δi−g

ki_e(dxs_)i(δk)∇δ

s∇δi

+gkiP_jih_|se(dxs)e(δyj)i(δk)i( ˙∂h) +gkiLhjie(dxs)e(δyj)i(δk)i( ˙∂h)∇δs

+gkiLh_ik|se(dxs)i(δh) +gkiLh_ike(dxs)i(δh)∇δs

−gkiLt_rse(dxr)e(δys)i(δk)i( ˙∂t)∇δi−g

ki_Lh

jiLjrse(dxr)e(δys)i(δk)i( ˙∂h) −gki_Lh

jiLtrse(dxr)e(δys)e(δyj)i(δk)i( ˙∂t)i( ˙∂h) +gki_Lh

ikLtrse(dxr)e(δys)i(δh)i( ˙∂t).

On the other hand, it is easy to check that

d∗_h◦dh = gkiΓks;i∇δs−g

ki_Γs

l;ie(dxl)i(δk)∇δs−g

ki_∇δ

i∇δk+g

ki_e(dxs_)i(δ

k)∇δi∇δs

+gki_Lt

ks|ie(δys)i( ˙∂t) +gkiLtrs|ie(dxr)e(δys)i(δk)i( ˙∂t) +gkiLt_kse(δys)i( ˙∂t)∇δi+g

ki_Lt

rse(dxr)e(δys)i(δk)i( ˙∂t)∇δi

−gki_Lh

jie(δyj)i( ˙∂h)∇δk−g

ki_Lh

jie(dxs)e(δyj)i(δk)i( ˙∂h)∇δs

+gkiLh_jiLt_khe(δyj)i( ˙∂t)−gkiLh_jiLt_kse(δyj)e(δys)i( ˙∂h)i( ˙∂t)

+gkiLh_jiLt_rhe(dxr)e(δyj)i(δk)i( ˙∂t)

+gki_Lh

jiLtrse(dxr)e(δyj)e(δys)i(δk)i( ˙∂h)i( ˙∂t) +gki_Lh

ik∇δh−g

ki_Lh

ike(dxs)i(δh)∇δs

−gkiLh_ikLt_hse(δys)i( ˙∂t)−gkiLh_ikLt_rse(dxr)e(δys)i(δh)i( ˙∂t),

whereLt

rs|idenote the horizontal covariant derivative ofLtrs with respect to∇. Now sumdh◦d∗_h andd∗_h◦dh together and rearrange the resulted terms and indexes, we obtain the expression of△h in Theorem 4.5. ¤

Remark 4.6. The horizontal Laplacian for functionsf ∈C∞( ˜M)was first derived in [4], and the horizontal and vertical Laplacians for 1-forms on the total spaceE of Riemann vector bundlep:E→M was first derived in [8] under the assumption that the fiber spaceF =p−1_{(x), x∈M, is compact.}

Note that ifF is a Landsberg metric thenLi_jk= 0. Thus we have

Corollary 4.7. Let (M, F) be a Landsberg manifold with the Cartan connection∇, andϕbe a differential form of type(p, q)which is compactly supported inM˜. Then

△hϕ = −gki_∇δ

i∇δkϕ−g

ki_e(dxs_{)i(δk)(∇δ}

s∇δi− ∇δi∇δs)ϕ+g

ki_Gh ik∇δhϕ.

Corollary 4.8. Let(M, F)be a real Finsler manifold with the Cartan connection∇, andϕbe a horizontal differential form of typepwhich is compactly supported in M˜. Then

△hϕ = −gki_∇δ

i∇δkϕ−g

ki_e(dxs_{)i(δk)(∇δ}

s∇δi− ∇δi∇δs)ϕ

+gkiGh_ik∇δhϕ+g

ki_Lh ik|se(dx

Theorem 4.9. Let (M, F)be a real Finsler manifold with the Cartan connection∇, andϕbe a differential form of type(p, q)which is compactly supported inM˜. Then

△vϕ = −gki_∇

Proof. By Lemma 4.1 and Lemma 4.4, we have

(4.13) dv=e(δys_)∇

Since∇ is v-metrical, it follows that the vertical covariant derivatives ofgki _vanish identically, i.e.,

gki_ks= ˙∂s(gki) +gtiC_tsk +gktC_tsi ≡0.

Using this fact and the properties of the operatorsiande, we obtain

dv◦d∗_v = gtiC_stke(δys_{)i( ˙∂k)∇}

By similar calculations, we have

Note thatgki_Ch

ji=ghiCjik andgkiks= 0, thus

gkiC_jih_ks+gkiC_jst C_tih+ghiC_jsk_ki−gtiC_jihC_tsk

= ghi³C_jik_ks+C_jsk_ki´−ghi³C_jitC_tsk −C_jst C_tik´.

Now it is easy to check thatCk

jiks=Cjskki. Thus sumdv◦d∗v andd∗v◦dv together and rearrange the resulted terms, we obtain△v in Theorem 4.9. ¤

Corollary 4.10. Let∇be the Cartan connection associated to a real Finsler manifold

(M, F), andϕbe a horizontal differential form of typepwhich is compactly supported inM˜. Then

△vϕ = −gki∇_∂˙_i∇_∂˙_kϕ−

1 2g

ki_Rh ki∇_∂˙_hϕ

−gki³C_ska _ki−C_sihC_kha +C_kihC_sha +1 2R

h kiCsha

´

e(dxs)i(δa)ϕ

−gki_Ct

jiCsthe(dxj)e(dxs)i(δk)i(δh)ϕ.

5

The mixed-type operator

In this section, we shall derive the local expression of△mix=dmix◦d∗

mix+d∗mix◦dmix.

Theorem 5.1. Letϕbe a differential form of type(p, q)which is compactly supported inM˜. Then

(5.1)

△mixϕ=1 4g

ki_glh_Rj

acRsligsje(dxa)e(dxc)i(δk)i(δh)ϕ

+1 2g

ki_glh_Rt

hkRsligsje(δyj)i( ˙∂t)ϕ

+gkiglhRt_hcRs_ligsje(dxc_)e(δyj_{)i(δk)i( ˙∂t)ϕ.}

Proof. On the one hand,

dmix◦d∗_mix = 1 4g

ki_glh_Rj

acRsligsje(dxa)e(dxc)i(δk)i(δh)

−1 4g

ki_glh_Rt

acRsligsje(dxa)e(dxc)e(δyj)i(δk)i(δh)i( ˙∂t).

On the other hand,

d∗_mix◦dmix = 1 4g

ki_glh_Rt

acRsligsje(δyj)i(δk)i(δh)e(dxa)e(dxc)i( ˙∂t).

Using the relationshipi(δh)e(dxa_{) =δ}a

h−e(dxa)i(δh) repeatedly, we get

i(δk)i(δh)e(dxa_)e(dxc_{) = δ}a

Thus

d∗_mix◦dmix = 1 4g

ki_glh_(Rt

hk−Rtkh)Rsligsje(δyj)i( ˙∂t)

+1 4g

ki_glh_Rt

hcRsligsje(dxc)e(δyj)i(δk)i( ˙∂t)

−1 4g

ki_glh_Rt

kcRsligsje(dxc)e(δyj)i(δh)i( ˙∂t)

−1 4g

ki_glh_Rt

ahRlisgsje(dxa)e(δyj)i(δk)i( ˙∂t)

+1 4g

ki_glh_Rt

akRsligsje(dxa)e(δyj)i(δh)i( ˙∂t)

+1 4g

ki_glh_Rt

acRsligsje(dxa)e(dxc)e(δyj)i(δk)i(δh)i( ˙∂t).

Sumdmix◦d∗_mixandd∗_mix◦dmixtogether, and use the factRi

jk=−Rikj, we get (5.1).

¤

Remark 5.2. It follows from Theorem 4.5 and Theorem 4.9 that the local expressions of△h and△v depend on the choice of Finsler connection associated to(M, F), while the expression of△mixis independent of the choice of Finsler connections associated to(M, F).

Corollary 5.3. Letϕbe a horizontal form of typepwhich is compactly supported in

˜ M. Then

△mixϕ = 1 4g

ki_glh_Rj

acRsligsje(dxa)e(dxc)i(δk)i(δh)ϕ.

Especially, for every horizontal1-formϕ=ϕidxi which is compactly supported inM˜, we have△mixϕ≡0.

6

Some properties of the type preserving operators

Let△ be the Hodge-Laplace operator of ˜g on ˜M. It is clear that △,△h,△v,△mix satisfy

(△ϕ, ψ) = (△hϕ, ψ) + (△vϕ, ψ) + (△mixϕ, ψ), (6.1)

which implies that△ϕ= 0 if and only if

(6.2) △hϕ= 0,△vϕ= 0,△mixϕ= 0, ∀ϕ∈ ∧p,q c ( ˜M).

This is essentially arisen from the type-preserving property of △h,△v and △mix. Especially iff ∈C∞

c ( ˜M), then

0≤(df, df) = (△f, f) = (△hf, f) + (△vf, f), (6.3)

Proposition 6.1. Letλ, λh andλv be constants such that

Theorem 6.2. Let f be a scalar field on M˜ which is not constant and satisfies

△hf =λhf,△vf =λvf with λh, λv being a constant such that λ2

h+λ2v 6= 0. Then

the constantλ=:λh+λv must be positive.

Proof. It is easy to check that 1

Sincef is compactly supported in ˜M, thus

(6.5) 0 =

Substituting the identity△f =△hf+△vf in (6.5), we get

Z

is,f is a constant. This is a contradiction to the assumption. ¤

Theorem 6.3. There exists no horizontal 1-formϕ=ϕidxi _{with compact support in} ˜

Thus if condition (6.6) and (6.7) are satisfied, we have

−1 2△kϕk

2_=Ttlϕt_ϕl_+gki_glt_(ϕ

l|iϕt|k+ϕlkkϕtki)≥0,

Remark 6.4. Note that a functionf ∈C∞( ˜M)which is both horizontal and vertical parallel with respect to the Cartan connection ∇ is necessary a constant. It follows from Theorem 6.3 that the only exceptions are vector fields which are both horizontal and vertical parallel, and there are no such vector fields other than zero if the quadratic formTtlϕtϕl is positive definite.

Acknowledgment. This work was supported by Program for New Century Excel-lent TaExcel-lents in Fujian Province University, and the Natural Science Foundation of China(Grant No. 10971170, 10601040).

References

[1] M. Abate and G. Patrizio, Finsler metrics—A global approach with applica-tions to geometric function theory, Lecture Notes in Mathematics, Volume 1591, Springer-Verlag, Berlin Heidelberg, 1994.

[2] P. Antonelli, B. Lackey(eds.),The Theory of Finslerian Laplacians and Applica-tions, Kluwer Academic Publisher, 1998.

[3] D. Bao, S.-S. Chern and Z. Shen,An introduction to Riemann-Finsler geometry, GTM 200, Springer-Verlag New York, Inc. 2000.

[4] D. Bao and B. Lackey, A Geometric Inequality and a Weitzenb¨ock Formula, 245-275: P. Antonelli, B. Lackey(eds.)–The Theory of Finslerian Laplacians and Applications, Kluwer Academic Publisher, 1998.

[5] B. Bidabad, Complete Finsler manifolds and adapted coordinates, Balkan J. Geom. Appl., 14, 1(2009), 21-29.

[6] S. S. Chern and Z. Shen,Riemann-Finsler Geometry, WorldScientific, 2005. [7] M. Matsumoto,Foundations of Finsler geometry and special Finsler spaces,

Kai-seisha, Japan 1986.

[8] O. Munteanu,Weitzenb¨ock formulas for horizontal and vertical Laplacians, Hous-ton J. Math., 29, 4 (2003), 889-900.

[9] B. Najafi and A. Tayebi, Finsler metrics of constant scalar curvature and pro-jective invariants, Balkan J. Geom. Appl., 15, 2(2010), 82-91.

[10] C. Zhong,A vanishing theorem on Kaehler Finsler manifolds, Diff. Geom. Appl., 27 (2009), 551-565.

Author’s address:

Chunping Zhong

School of Mathematical Sciences, Xiamen University,