Sasakian Finsler manifolds

(2013) 37: 319 – 339 c  T ¨UB˙ITAK doi:10.3906/mat-1006-371 h t t p : / / j o u r n a l s . t u b i t a k . g o v . t r / m a t h / Research Article

Sasakian Finsler manifolds

Ay¸se Funda YALINIZ1_{, Nesrin C¸ ALIS¸KAN}2,∗ 1

Department of Mathematics, Faculty of Arts and Sciences, Dumlupınar University, TR-43100, K¨utahya, Turkey 2_{Department of Mathematics, Graduate School of Natural and Applied Science, Dumlupınar University,}

TR-43100, K¨utahya, Turkey

Received: 20.06.2010 • Accepted: 04.10.2011 • Published Online: 19.03.2013 • Printed: 22.04.2013

Abstract: In this study, almost contact Finsler structures on vector bundle are defined and the condition of normality in terms of the Nijenhuis torsion Nφ of almost contact Finsler structure is obtained. It is shown that for a K -contact

structure on Finsler manifold ∇Xξ =−1₂φX and the flag curvature for plane sections containing ξ are equal to 1₄. By

using the Sasakian Finsler structure, the curvatures of a Finsler connection∇ on V are obtained. We prove that a locally symmetric Finsler manifold with K -contact Finsler structure has a constant curvature 1

4. Also, the Ricci curvature on Finsler manifold with K -contact Finsler structure is given. As a result, Sasakian structures in Riemann geometry and Finsler condition are generalized.

As a conclusion we can state that Riemannian Sasakian structures are compared to Sasakian Finsler structures and it is proven that they are adaptable.

Key words: Finsler connection, vector bundle, almost contact manifold, Sasakian manifold, nonlinear connection, Ricci tensor

1. Introduction

Let V (M ) ={ V, π, M} be a vector bundle of total space V with a (n+m)-dimensional C∞ manifold and with a base space M that is an n-dimensional C∞-manifold. The projection map π : V → M, u ∈ V → π (u) = x∈ M , where u = (x, y), and y ∈ Rm_{= π}−1_{(x) the fibre of V (M ) over x.}

A non-linear connection N on the total space V of V (M ) is a differentiable distribution N : V → Tu(V ) , u∈ V → Nu∈ Tu(V ) such that

Tu(V ) = Nu⊕ Vuv where Vuv={X ∈ Tu(V ) : π∗(X) = 0} . (1.1) Nu the horizontal distribution and Vv is the vertical distribution. Thus for all X ∈ Tu(V ) can be

separated by its components

X = XH+ XV where XH ∈ Nu, XV ∈ Vuv. (1.2)

Let xi_{, i=1,2,. . . ,n and y}a_{, a=1,2,. . . ,m be the coordinates of x and y such that} _xi_{, y}a _{are the}

coordinates of u∈V. The local base of Nu is δ δxi = ∂ ∂xi − N a i (x, y) ∂ ∂ya (1.3) ∗_{Correspondence: caliskan.nesrin@hotmail.com}

and that of Vv

u is where Nia(x, y) are the coefficients of N. Their dual bases are

 dxi_{, δy}a where δya _{= dy}a_{+ N}a i (x, y) dxi. (1.4) Let X = Xi_{(x, y)} δ δxi + ˜Xa(x, y)_∂y∂a,∀X ∈ Tu(V ). Then XH = Xi(x, y) δ δxi, X V _{= ˜X}a_{(x, y)} ∂ ∂ya, X˜ a _{= X}a_{+ N}a iXi. (1.5)

Let ω be a 1 -form ω = ˜ωi(x, y)dxi+ ωa(x, y)δya. Then ωH_{= ˜ω}

idxi, ˜ωi= ωi− Nia(x, y)ωa; ωV = ωaδya (1.6)

which gives

ωH(XV) = 0, ωV(XH) = 0 where ω = ωH+ ωV. (1.7) The Finsler tensor field of type

 p r q s



on V has the following local form [4]:

T = Ti1,...,ip,a1,...,ar j1,...,jq,b1,...,bs (x, y) δ δxi1 ⊗ ... ⊗ δ δxip ⊗ dx a1⊗ ... ⊗ dxar⊗ ∂ ∂yj1 ⊗ ... ⊗ ∂ ∂yjq ⊗ δy b1⊗ ... ⊗ δybs_{. (1.8)}

Definition 1.1 A Finsler connection on V is a linear connection ∇ = F Γ on V with the property that the

horizontal linear space Nu, u∈ V of the distribution N is parallel with respect to ∇ and the vertical spaces Vv

u , u∈ V are also parallel relative to ∇ [3].

A linear connection ∇ on V is a Finsler connection on V if and only if  ∇XYH V = 0,∇XYV H = 0,∀X, Y ∈ Tu(V ) . (1.9)

A linear connection ∇ on V is a Finsler connection on V if and only if [4] ∇XY =  ∇XYH H +∇XYV V ,∀X, Y ∈ Tu(V ) , (1.10a) ∇Xω =  ∇XωH H +∇XωV V ,∀ω ∈ T_u∗(V ) and X ∈ Tu(V ) . (1.10b)

Remark 1.1 Let ∇ on V is a Finsler connection on V. We get immediately that [6]

Y ∈ V_uv⇒ ∀X ∈ Tu(V ) ;∇XY ∈ Vuv, Y ∈ Nu⇒ ∀X ∈ Tu(V ) ;∇XY ∈ Nu. (1.11)

For a Finsler connection ∇ on V, there is an associated pair of operators; h- and v-covariant derivation in the algebra of Finsler tensor fields. For eachX∈ Tu(V ) , set

∇H XY =∇XHY,∇_XHf = XH(f) ,∀Y ∈ T_u(V ) ,∀f ∈ F (V ) . (1.12) If ω∈ T_u∗(V ), we define (∇H Xω) (Y ) = XH(ω (Y ))− ω  ∇H XY  ,∀Y ∈ Tu(V ) . (1.13)

So, we may extend the action of the operator ∇H

X to any Finsler tensor field by asking these questions:

does ∇H

X preserve the type of Finsler tensor fields, is it R-linear, does it satisfy the Leibniz rule with respect

to tensor product and does it commute with all contractions? We keep the notation ∇H

X for this operator on

the algebra of Finsler tensor fields. We call it the operator of h-covariant derivation. In a similar way, for every vector field X ∈ Tu(V ) set

∇V

XY =∇XVY,∇_XVf = XV (f) ,∀Y ∈ T_u(V ) ,∀f ∈ F (V ) . (1.14)

If ω∈ T_u∗(V ), we define

(∇V_Xω) (Y ) = XV (ω (Y ))− ω∇V_XY,∀Y ∈ Tu(V ) . (1.15)

We extend the action of ∇V

X to any Finsler tensor field in a similar way, as for ∇HX. We obtain an

operator on the algebra of Finsler tensor fields on V ; this will be denoted also by ∇V

X and will be called the operator of v-covariant derivation [1].

Definition 1.2 Let ω ∈ T_u∗(V ) be a differential q-form on V, ∇ is a linear connection on V and T is the torsion tensor of ∇. Then its exterior differential dω is also defined as [4]:

(dω) (X1, ..., Xq+1) = q+1 Σ i=1(−1) i+1_(∇ Xω)  X1, ..., ˜Xi, ..., Xq+1  ,∀Xi∈ Tu(V ) − Σ 1≤i≤j≤q+1(−1) i+j_ω_{T (X} i, Xj) , X1, ..., ˜Xi, ..., ˜Xj, ..., Xq+1  . (1.16)

Proposition 1.1 If ω∈ T_u∗(V ) is a 1-form and ∇ is a Finsler connection on V, then its exterior differential is given by [3] (dω)XH_{, Y}H₌_∇H Xω   YH₋_∇H Yω   XH_{+ ω}_T_XH_{, Y}H_, (dω)XV_{, Y}H₌_∇V Xω   YH₋_∇H Yω   XV_{+ ω}_T_XV_{, Y}H_, (dω)XV_{, Y}V₌_∇V Xω   YV₋_∇V Yω   XV_{+ ω}_T_XV_{, Y}V_{,∀X, Y ∈ T} u(V ) . (1.17)

In the canonical coordinatesxi_{, y}a_{, there exists a well determined set of differentiable functions on V.} Fi jk(x, y) , Fabk(x, y) ; Cjai (x, y) ; Cbca (x, y) such that ∇H δ δxk δ δxj = F_jki (x, y)_δxδi, ∇Hδ δxk ∂ ∂yb = F_bka (x, y)_∂y∂a, ∇V ∂ ∂ya δ δxj = Cjai (x, y)δxδi,∇V∂ ∂yc ∂ ∂yb = C_bca (x, y)_∂y∂a where Fi

jk(x, y) , Fabk(x, y) are called coefficients of h-connections ∇H and Cbca (x, y), Cjai (x, y) are called coefficients of v-connections∇V_.

The torsion tensor field T of a Finsler-connection is characterised by five Finsler tensor fields: 

TXH, YH H, TXH, YH V, TXH, YV H, TXH, YV V , TXV, YV V .

Proposition 1.2 If the Finsler connection on V is without torsion then we have [3]

2. Almost contact Finsler structure on vector bundle

Let φ be an almost contact structure on V given by the tensor field of type 

1 1 1 1



with the properties 1. φ.φ =−In+ ηH⊗ ξH+ ηV ⊗ ξV

2. φ ξH_{= 0, φ ξ}V _{= 0}

3. ηH_ξH_{+ η}V_ξV_{= 1}

4. ηH_{φ X}H_{= 0, η}V _{φ X}H_{= 0, η}H_{φ X}V_{= 0, η}V _{φ X}V_{= 0,}

(2.1)

where η is 1-form and ξ is vector field [2].

Proposition 2.1 If φ is an almost contact Finsler structure on V, there exists a unique decomposition of φ

in the Finsler tensor fields,

φ = φ1+ φ2+ φ3+ φ4= φ1 _φ2 φ3 _φ4  (2.2) where φ1_{(ω, X) = φ} _ωH_{, X}H_{, φ}2_{(ω, X) = φ}_ωH_{, X}V_, φ3_{(ω, X) = φ} _ωV_{, X}H_{, φ}4_{(ω, X) = φ}_ωV_{, X}V_{∀X ∈ T} u(V ) ,∀ω ∈ Tu∗(V ) . (2.3) We can write φXH= φ1XH= φ3XH, φXV= φ2XV= φ4XV. (2.4) Let G be the Finsler metric structure on V which is symmetric, positive definite and non-degenerate on V. The metric-structure G on V is decomposed as:

G = GH_{+ G}V _(2.5) where GH _{is of type} 0 0 2 0 

, symmetric, positive definite and non-degenerate on Nu and GV is of type

0 0 0 2

, symmetric, positive definite and non-degenerate on Vv

u i.e. for X, Y ∈ Tu(V )

G (X, Y ) = GH(X, Y ) + GV (X, Y ) (2.6) where GH_{(X, Y ) = G}_XH_{, Y}H_{, G}V _{(X, Y ) = G}_XV_{, Y}V_{. .}

Now, if the Finsler metric structure G on V satisfies

G (φX, φY ) = G (X, Y )− η (X) η (Y ) , GH_{(φX, φY ) = G}H_{(X, Y )− η}H_XH_ηH_YH_, GV _{(φX, φY ) = G}V_{(X, Y )− η}V_XV_ηV _YV_, (2.7) which is equivalent to GH_{(X, ξ) = η}H_{(X) , G}V _{(X, ξ) = η}V _{(X) ,} GH_{(φX, φY ) =−G}H_φ2_{X, Y}_{, G}V_{(φX, φY ) =−G}V _φ2_{X, Y}_, (2.8)

then (φ, η, ξ, G) is called almost contact metrical Finsler structure on V [5]. Now, we define

Ω (X, Y ) = G (X, φY ) , ΩXH, YH= GH(X, φY ) , ΩXV, YV= GV (X, φY ) (2.9) and call it the fundamental 2-form.

Proposition 2.2 The fundamental 2-form, defined above, satisfies [5]

ΩφXH_{, φY}H_{= Ω}_XH_{, Y}H_{, Ω}_φXV_{, φY}V_{= Ω}_XV_{, Y}V_,

ΩXH_{, Y}H_=−Ω_YH_{, X}H_{, Ω}_XV_{, Y}V_=−Ω_YV_{, X}V_{∀X, Y ∈ T} u(V ) .

(2.10)

Proposition 2.3 Let ∇ be a Finsler connection on V and Ω be the fundamental 2-form which satisfies

Ω (X, Y ) = dη (X, Y ) i.e. ΩXH_{, Y}H₌_∇H Xη   YH₋_∇H Yη   XH_{+ η}_T_XH_{, Y}H_, ΩXV_{, Y}H₌_∇V Xη   YH₋_∇H Yη   XV_{+ η}_T_XV_{, Y}H_, ΩXV_{, Y}V₌_∇V Xη   YV₋_∇V Yη   XV_{+ η}_T_XV_{, Y}V_. (2.11)

Then, the almost contact metrical Finsler structure is called almost Sasakian Finsler structure and the Finsler connection ∇satisfying (2.11) is called almost Sasakian Finsler connection on V [5].

Theorem 2.1 Let Ω be the fundamental 2-form and almost Sasakian Finsler connection ∇ on V is torsion

free. Then [5] ΩXH_{, Y}H₌_∇H Xη   YH₋_∇H Yη   XH_, ΩXV_{, Y}H₌_∇V Xη   YH₋_∇H Yη   XV_, ΩXV_{, Y}V₌_∇V Xη   YV₋_∇V Yη   XV_{,∀X, Y ∈ T} u(V ) . (2.12)

Proof From Proposition 1.2 and equations in (2.11), we have (2.12). 2

Definition 2.1 An almost Sasakian Finsler structure on V is said to be a Sasakian Finsler structure if the

1-form η is a killing vector field, i.e.  ∇H Xη   YH₊_∇H Yη   XH_{= 0,}_∇V Xη   YH₊_∇H Yη   XV_{= 0,}  ∇V Xη   YV₊_∇V Yη   XV_{= 0∀X, Y ∈ T} u(V ) . (2.13)

The Finsler connection ∇ on V is torsion free, which is called Sasakian Finsler connection [5].

Theorem 2.2 Let ∇ be the torsion free Finsler connection together with a Sasakian Finsler structure on V

and Ω is to be the fundamental 2-form; then ΩXH_{, Y}H_{= 2}_∇H Xη   YH₌₋₂_∇H Yη   XH_, ΩXH_{, Y}V_{= 2}_∇H Xη   YV₌₋₂_∇V Yη   XH_, ΩXV_{, Y}V_{= 2}_∇V Xη   YV₌₋₂_∇V Yη   XV_{,∀X, Y ∈ T} u(V ) . (2.14)

Proof From (2.12) and (2.13) we have (2.14) [5]. 2

Example 2.1 Let V (M ) = { V, π, M } be a vector bundle with the total space V = R10 _{is a 10-dimensional} C∞-manifold and the base space M = R5 _{is a 5-dimensional C}∞_{-manifold. Let x}i _{, 1 ≤ i ≤ 5 and} ya_{, 1 ≤ a ≤ 5 be the coordinates of u = (x, y) ∈ V , that is u =} _x1_{, x}2_{, x}3_{, x}4_{, x}5_{, y}1_{, y}2_{, y}3_{, y}4_{, y}5 _{∈ V .} The local base of Nu is

 _δ

δx1,_δxδ2,_δxδ3,_δxδ4,_δxδ5



and the local base of Vv u is



∂

∂y1,_∂y∂2,_∂y∂3,_∂y∂4,_∂y∂5

 such that Tu(V ) = Nu⊕ Vuv. Then XH _{= X}1 δ δx1+ X2 δ_δx2 + X3 δ_δx3 + X4 δ_δx4 + X5 δ_δx5, XV _{= ˜X}1 ∂

∂y1+ ˜X2 ∂_∂y2 + ˜X3 ∂_∂y3 + ˜X4 ∂_∂y4 + ˜X5 ∂_∂y5  XH ∈ Nu, XV ∈ Vuv.

Let η be a 1-form, η = ηidxi + ˜ηaδya then ηH = η1dx1 + η2dx2 + η3dx3 + η4dx4 + η5dx5 and ηV _{= ˜η}

1δy1+ ˜η2δy2+ ˜η3δy3+ ˜η4δy4+ ˜η5δy5 where η = ηH+ ηV and ηH  XV_{= 0 , η}V _XH_{= 0 .} We put ηH₌ 1 3  dx5_{− x}3_dx1_{− x}4_dx2 _{and η}V ₌ 1 3 

δy5_{− y}3_δy1_{− y}4_δy2_. The structure vector field ξ is given by ξ = 3

 δ δx5 +_∂y∂5  and ξ is decomposed as ξH _{= 3} δ δx5 and ξV _{= 3} ∂ ∂y5.

The tensor field φH _{of type (1, 1) and φ}V _{of type (1, 1) by a matrix form is given by}

φH= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 0 0 0 −1 0 0 0 0 0 0 0 1 0 0 0 −1 0 0 −x4 _x3 _{0 0 0} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , φV = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 0 0 0 −1 0 0 0 0 0 0 0 1 0 0 0 −1 0 0 −y4 _y3 _{0 0 0} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .

We can see that ηH_ξH_{= 1 , φ}H_ξH_{= 0 , η}V _ξV_{= 1 , φ}V_ξV_{= 0 , η}H_ξV_{= 0 , η}V _ξH_{= 0 ,}



φH2_XH _=−XH_{+ η}H_XH_ξH_, _φV2_XV _=−XV _{+ η}V _XV_ξV _{and hence ( φ, ξ, η ) is almost contact} Finsler structure on R10_.

3. Integrability tensor field of the almost contact Finsler structure

The integrability tensor field of the almost contact Finsler structure on V is given by [4] ˜N (X, Y ) = [φX, φY ]− φ [φX, Y ]− φ [X, φY ] + φ2_{[X, Y ] + dη}H_{(X, Y ) ξ}H_{+ dη}V _{(X, Y ) ξ}V_{,∀X, Y ∈ T}

u(V ) .

We define four tensorsN(1)_{, N}(2)_{, N}(3) _{and N}(4)_{, respectively by∀X}H_{, Y}H_{, ξ}H_{∈ N}

u and∀XV, YV, ξV ∈ Vv u N(1)XH, YH= Nφ  XH, YH+ dηHXH, YHξH, (3.1a) N(2)XH, YH=LH_φXηH YH−LH_φYηH XH, (3.1b) N(3)_XH₌_LH ξ φ   XH_{, N}(4)_XH₌_LH ξ ηH   XH_{, (3.2a)}

N(1)XV, YV= Nφ  XV, YV+ dηV XV, YVξV, (3.2b) N(2)_XV_{, Y}V₌_LV φXηV   YV₋_LV φYηV   XV_{, (3.2c)} N(3)XV=LV_ξ φ XV, N(4)XV=L_ξVηV XV, (3.3a) N(1)XV, YH= Nφ  XV, YH+ dηV XV, YHξV + dηHXV, YHξH, (3.3b) N(2)XV, YH=LV_φXηH YH+LV_φXηV YH−LH_φYηH XV−LH_φYηV XV, (3.3c) N(3)XV=LH_ξ φ XV, N(4)XV=L_ξHηV XV, (3.3d) N(3)YH=LV_ξφ YH, N(4)YH=L_ξVηH YH. (3.3e) It is clear that the almost contact Finsler structure (φ, ξ, η) is normal if and only if these four tensors vanish.

Lemma 3.1 If N(1)_{= 0 , then N}(2)_{= N}(3)_{= N}(4) _{= 0.}

Proof If N(1)_{= 0 , then for X}H_{, Y}H_{, ξ}H_{∈ N}

u, from (3.1.a) we have



ξH, XH + φξH, φXH − ξHηHXHξH = 0. (3.4) Applying ηH _{to (3.4), we see that}

N(4)XH=LH_ξ ηH XH= ξHηHXH− ηHξH, XH = 0. From this equation, we also have

ηH_ξH_{, φX}H _{= 0. (3.5)}

On the other hand, applying φ to (3.4), we get

N(3)XH=LH_ξ φXH = φXH, ξH −φXH, ξH = 0. (3.6) Finally, from N(1)_{= 0 , by using (3.6), we derive}

0 =−φXH_{, Y}H ₋_XH_{, φY}H _{+ φ}_XH_{, Y}H _{− φ}_φXH_{, φY}H

−φYH_ηH_XH_ξH_{+ φ}_YH_ηH_XH_ξH_{+ φX}H_ηH_YH_ξH_. (3.7)

ApplyingηH_{to (3.7), we get N}(2)_XH_{, Y}H_{= 0 . Similarly,∀X}V_{, Y}V_{, ξ}V _{∈ V}v

u, if N(1)



XV_{, Y}V_{= 0 ,}

then N(2)_XV_{, Y}V_{= 0 , N}(3)_XV_{= 0 , N}(4)_XV_{= 0 .}

If N(1)_XV_{, Y}H_{= 0 , from (3.3.a) we obtain}

N(1)XV, ξH=ξH, XV − φφXV, ξH − ξHηV XVξV = 0. (3.8) Applying ηV _{and η}H _{to (3.8), we get (3.9):}

Using (3.9) in (3.3.c), we obtain

N(4)XV=LH_ξ ηV XV= ξHηV XV− ηV ξH, XV = 0. Applying φ to (3.8), we get

N(3)XV= LH_ξ φXV = ξH, φXV + φXV, ξH = 0. On the other hand, replacing X by ξ in (3.3.a), we obtain



YH, ξV − φξV, φYH + ξV ηHYHξH = 0. (3.10) Applying ηH _{and η}V _{to (3.10), we get}

ηH_ξV_{, Y}H _{= ξ}V _ηH_YH_{, η}V _ξV_{, Y}H _{= 0. (3.11)}

Using (3.11) in (3.3.d), we obtain

N(4)YH=LV_ξ ηH YH= ξVηHYH− ηHξV, YH = 0. Applying φ to (3.10) and by using (3.11), we obtain

N(3)_YH₌_LV ξ φ

 

YH₌_ξV_{, φY}H _{+ φ}_YH_{, ξ}V _{= 0.}

By using (3.11), from (3.3.a), we calculate 0 = Nφ



φXV_{, Y}H_{+ dη}V _φXV_{, Y}H_ξV _{+ dη}H_φXV_{, Y}H_ξH

=YH_{, φX}V ₊_φYH_{, X}V _{+ φ}_XV_{, Y}H _{− φ}_φXV_{, φY}H _{− φY}H_ηV _XV_ξV _{+ φX}V _ηH_YH_ξH_.

(3.12) Applying ηV _{to (3.12), from (3.3.b), we obtain}

0 = N(2)_XV_{, Y}H_{= φX}V _ηH

YH_{− φY}H_ηV _XV_{− η}V _φXV_{, Y}H _{+ η}H_φYH_{, X}V

+ηV _φYH_{, X}V _{− η}H_φXV_{, Y}H _.

2

Proposition 3.1 The almost contact Finsler structure on V is normal if and only if

Nφ+ dηH⊗ ξH+ dηV ⊗ ξV = 0. (3.13)

Let (φ, η, ξ, G) be almost metrical Finsler structure on V with contact metric. If the structure vector field ξ is a Killing vector field with respect to G, the contact structure on V is called a K-contact Finsler structure and V is called a K-contact Finsler manifold.

Lemma 3.2 Let (φ, η, ξ, G) be a contact metrical Finsler structure on V. Then N(2) _{and N}(4) _vanish. Moreover, N(3) _{vanishes if and only if ξ is a Killing vector field with respect to G.}

Proof We have

dηH_φXH_{, φY}H_{= Ω}_φXH_{, φY}H_{= G}_φXH_{, φ}2_YH_=−G_XH_{, φ}3_YH_{= G}_XH_{, φY}H_{= dη}H_XH_{, Y}H

from which dηH_φXH_{, Y}H_{+ dη}H_XH_{, φY}H_{= 0 .This is equivalent to N}(2)_XH_{, Y}H_{= 0 .}

On the other hand, we have 0 = GXH_{, φξ}H _{= dη}H_XH_{, ξ}H _{= X}H_ηH_ξH_{− ξ}H_ηH_XH₋ ηH_XH_{, ξ}H _{. Thus we obtain ξ}H_ηH_XH_{− η}H_ξH_{, X}H  _{= 0 . Therefore, we have L}H

ξ ηH = 0 hence N(4)_XH_{= 0 .} We mention that  LH ξ G   XH_{, ξ}H_{= ξ}H_ηH_XH_{− η}H_ξH_{, X}H ₌_LH ξ ηH  XH _{= 0. Simply, it is} clear that LH ξ dηH= 0 and consequently,  LH ξ dηH   XH_{, Y}H₌_LH ξ Ω   XH_{, Y}H_{= 0 from which} 0 = ξH_G_XH_{, φY}H_{− G}_ξH_{, X}H _{, φY}H_{− G}_XH_{, φ}_ξH_{, Y}H  =LH ξ G   XH_{, φY}H_{+ G}_XH_,_LH ξ φ  YH₌_LH ξ G   XH_{, φY}H_{+ G}_XH_{, N}(3)_YH_.

Thus ξH _{is a Killing vector field if and only if N}(3)_YH_{= 0 . Similarly, we consider that N}(2)_XV_{, Y}V₌

0 and N(4)_XV_{= 0 . Moreover, N}(3)_XV_{= 0 if and only if ξ}V _{is a Killing vector field with respect to G}V_. 2

Lemma 3.3 For an almost contact metric Finsler structure (φ, η, ξ, G) on V, we have

2G ((∇Xφ) Y, Z) = dΩ (X, φY, φZ)− dΩ (X, Y, Z) + G



N(1)_{(Y, Z) , φX}

+N(2)_{(Y, Z) η (X) + dη (φY, X) η (Z)− dη (φZ, X) η (Y ) .} (3.14)

Proof The Finsler connection ∇ with respect to G is given by 2GH_∇H XYH, ZH  = XH_GH_YH_{, Z}H_{+ Y}H_GH_XH_{, Z}H_{− Z}H_G_XH_{, Y}H_{+ G}H_XH_{, Y}H _{, Z}H +GH_ZH_{, X}H _{, Y}H_{− G}H_YH_{, Z}H _{, X}H_, (3.15) 2GV _∇V XYV, ZV  = XV_GV_YV_{, Z}V_{+ Y}V_GV _XV_{, Z}V_{− Z}V_G_XV_{, Y}V_{+ G}V _XV_{, Y}V _{, Z}V +GV _ZV_{, X}V _{, Y}V_{− G}V _YV_{, Z}V _{, X}V (3.16) 2GH∇V_XYH, ZH= XVGHYH, ZH+ GHXV, YH H, ZH+ GHZH, XV H, YH (3.17) 2GV _∇H XYV, ZV  = XH_GV _YV_{, Z}V_{+ G}V _XH_{, Y}V V _{, Z}V_{+ G}V _ZV_{, X}H V _{, Y}V_{. (3.18)} Furthermore, we have dΩXH_{, Y}H_{, Z}H_{= X}H_Ω_YH_{, Z}H_{+ Y}H_Ω_ZH_{, X}H_{+ Z}H_Ω_XH_{, Y}H −ΩXH_{, Y}H _{, Z}H_{− Ω}_ZH_{, X}H _{, Y}H_{− Ω}_YH_{, Z}H _{, X}H_, (3.19) dΩXV_{, Y}V_{, Z}V_{= X}V_Ω_YV_{, Z}V_{+ Y}V_Ω_ZV_{, X}V_{+ Z}V_Ω_XV_{, Y}V −ΩXV_{, Y}V _{, Z}V_{− Ω}_ZV_{, X}V _{, Y}V_{− Ω}_YV_{, Z}V _{, X}V_, (3.20)

dΩXV, YH, ZH= XVΩYH, ZH− ΩXV, YH H, ZH− ΩZH, XV H, YH, (3.21) dΩXV, YV, ZH= ZHΩXV, YV− ΩZH, XV V , YV  − ΩYV, ZH V , XV  , (3.22) dΩXH, YV, ZH= YVΩZH, XH− ΩXH, YV H, ZH  − ΩYV, ZH H, XH  , (3.23) dΩXH_{, Y}V_{, Z}V_{= X}H_Ω_YV_{, Z}V_{− Ω}_XH_{, Y}V V _{, Z}V_{− Ω}_ZV_{, X}H V _{, Y}V_{, (3.24)} dΩXV, YH, ZV= YHΩZV, XV− ΩXV, YH V , ZV− ΩYH, ZV V , XV, (3.25) dΩXH, YH, ZV= ZVΩXH, YH− ΩZV, XH H, YH  − ΩYH, ZV H, XH  . (3.26) By using (2.9), from (3.15) we get

2GH_∇H Xφ  YH_{, Z}H_{= φY}H_G_XH_{, Z}H_{− Z}H_Ω_XH_{, Y}H_{+ G}H_XH_{, φY}H _{, Z}H +ΩZH_{, X}H _{, Y}H_{− G}H_φYH_{, Z}H _{, X}H_{+ Y}H_Ω_XH_{, Z}H_{− φZ}H_G_XH_{, Y}H +ΩXH_{, Y}H _{, Z}H_G_φZH_{, X}H _{, Y}H_{− G}H_YH_{, φZ}H _{, X}H_.

Also from (3.19), we calculate

dΩXH_{, φY}H_{, φZ}H_{= X}H_Ω_YH_{, Z}H_{+ φY}H_G_ZH_{, X}H_{− φY}H_ηH_ZH_ηH_XH −φZH_G_XH_{, Y}H_{+ φZ}H_ηH_XH_ηH_YH _{+ G}_XH_{, φY}H _{, Z}H

−ηH_XH_{, φY}H _ηH_ZH_{+ G}_φZH_{, X}H _{, Y}H_{− η}H_φZH_{, X}H _ηH_YH −ΩφYH_{, φZ}H _{, X}H_.

(3.27)

Also from (3.1.a) by using (2.9), we obtain

GN(1)_YH_{, Z}H_{, φX}H_=−Ω_YH_{, Z}H _{, X}H_{+ Ω}_φYH_{, φZ}H _{, X}H_{− G}_φYH_{, Z}H _{, X}H

+ηH_φYH_{, Z}H _ηH_XH_{− G}_YH_{, φZ}H _{, X}H_{+ η}H_YH_{, φZ}H _ηH_XH_. (3.28)

From (3.1.b), we have

N(2)_YH_{, Z}H_ηH_XH_{= φY}H_ηH_YH_ηH_XH_{− φZ}H_ηH_YH_ηH_XH

−ηH_φYH_{, Z}H _ηH_XH_{− η}H_YH_{, φZ}H _ηH_XH_. (3.29)

By using (3.27), (3.28) and (3.29), we have the equation.

Similarly by using (3.2.a), (3.2.b), (2.9), (3.16) and (3.20), we get 2G∇V

Xφ



YV_{, Z}V_{= dΩ}_XV_{, φY}V_{, φZ}V_{− dΩ}_XV_{, Y}V_{, Z}V_{+ G}_N(1)_YV_{, Z}V_{, φX}V

+N(2)_YV_{, Z}V_ηV _XV_{+ dη}V _φYV_{, X}V_ηV_ZV_{− dη}V _φZV_{, X}V_ηV _YV_.

By using (2.9), (3.1.a), (3.1.b), (3.17) and (3.21), we calculate

dΩXV_{, φY}H_{, φZ}H_{− Ω}_XV_{, Y}H_{, Z}H_{+ dη}H_φYH_{, X}V_ηH_ZH_{− dη}H_φZH_{, X}V_ηH_YH = GH_XV_{, φY}H H_{, Z}H_{+ G}H_φZH_{, X}V H_{, Y}H_{+ Ω}_XV_{, Y}H H_{, Z}H_{+ Ω}_ZH_{, X}V H_{, Y}H = 2G∇V Xφ  YH_{, Z}H_.

By using (2.9) and (3.18), (3.24), (3.2.a) and (3.2.b), we obtain dΩXH_{, φY}V_{, φZ}V_{− dΩ}_XH_{, Y}V_{, Z}V_{+ G}V _N(1)_YV_{, Z}V_{, φX}H +N(2)_YV_{, Z}V_ηV _XH_{+ dη}V_φYV_{, X}H_ηV _ZV_{− dη}V _φZV_{, X}H_ηV _YV +dηH_φYV_{, X}H_ηH_ZV_{− dη}H_φZV_{, X}H_ηH_YV = GXH_{, φY}V V_{, Z}V_{− η}V _ZV_ηV _XH_{, φY}V V _{+ G}_φZV_{, X}H V _{, Y}V −ηV _YV_ηV _φZV_{, X}H V _{+ Ω}_XH_{, Y}V V _{, Z}V_{+ Ω}_ZV_{, X}H V _{, Y}V −ηV _φYV_{, X}H V _ηV _ZV_{+ η}V _φZV_{, X}H V _ηV _YV = 2GV _∇H Xφ  YV_{, Z}V_. 2

Lemma 3.4 For a contact metric Finsler structure (φ, η, ξ, G) of V with Ω = dη and N(2) _{= 0 , we} get 2G ((∇Xφ) Y, Z) = G



N(1)_{(Y, Z) , φX}_{+ dη (φY, X) η (Z)−dη (φZ, X) η (Y ). Especially we have ∇}

ξφ = 0.

Proof The first equation is trivial by the assumption. We prove that ∇ξφ = 0.

From N(2)_{= 0 we have dη}H_XH_{, ξ}H_{= 0 and dη}V _XV_{, ξ}V_{= 0 . Thus the first equation implies that} ∇H

ξ φ = 0 and ∇Vξφ = 0 . 2

Proposition 3.2 Let (φ, η, ξ, G) be a contact metrical Finsler structure on V. Then (φ, η, ξ, G) is a

K-contact Finsler structure if and only if N(3) _vanishes.

Proposition 3.3 Let (φ, η, ξ, G) be contact metrical Finsler structure on V. Then (φ, η, ξ, G) is a K-contact

structure if and only if

∇XξH =− 1 2φX H_,∇ XξV =− 1 2φX V_{. (3.30)}

Proof If the structure vector field ξ is a Killing vector field with respect to G, then we have LH

ξ GH= 0, LξVGV = 0. (3.31)

∀XH_{, Y}H_{, ξ}H_{∈ N}

u and ∀XV, YV, ξV ∈ Vuv from (3.31), we can get

G∇H_XξH, YH=−GXH,∇H_YξH, G∇V_XξV, YV=−GXV,∇V_YξV. (3.32) Replacing YH _{by ξ}H _{and Z}H _{by Y}H _{in (3.15), we have}

2G∇H XξH, YH



= XH_ηH_YH_{+ ξ}H_G_XH_{, Y}H_{− Y}H_ηH_XH

+G XH_{, ξ}H _{, Y}H_{− η}H_{[X, Y ]}H_{− G} _ξH_{, Y}H _{, X}H_. (3.33)

Replacing YH _{by ξ}H _{, X}H _{by Y}H _{and Z}H _{by X}H _{in (3.15), we can get}

2G∇H YξH, XH



= YH_ηH_XH_{+ ξ}H_G_XH_{, Y}H_{− X}H_ηH_YH

Using (3.33) and (3.34), we get G∇H XξH, YH  − GXH_,∇H YξH  = dηH_XH_{, Y}H_.

Since ξH _{is a Killing vector field with respect to G}H_{, using (3.32), we obtain} dηH_XH_{, Y}H_{= 2G}_∇H XξH, YH  = GXH_{, φY}H_=−G_φXH_{, Y}H _{and ∇}H XξH=−12φXH. Similarly for XV_{, Y}V_{, ξ}V _{∈ V}v

u, from (3.16) and (3.32), we get ∇VXξV =−12φXV. 2

Example 3.1 Let V (M ) ={ V, π, M } be a vector bundle with the total space V = R6 _{is a 6-dimensional} C∞-manifold and the base space M = R3 _{is a 3-dimensional C}∞_{-manifold. Let x}i_{, 1 ≤ i ≤ 3 and} ya_{, 1≤ a ≤ 3 be the coordinates of u = (x, y) ∈ V that is u =}_x1_{, x}2_{, x}3_{, y}1_{, y}2_{, y}3_{∈ V .}

The local base of Nu is

 _δ δx1,_δxδ2,_δxδ3  and that of Vv u is  ∂

∂y1,_∂y∂2,_∂y∂3

 . Let X = Xi δ

δxi + ˜Xa ∂_∂ya∀X ∈ Tu(V ). Then XH = X1 δ_δx1 + X2 δ_δx2 + X3 δ_δx3, XV = ˜X1 ∂_∂y1 + ˜X2 ∂_∂y2 +

˜ X3 ∂

∂y3 where XH ∈ Nu and XV ∈ Vuv. Similarly Y can be written as

YH = Y1 δ δx1+ Y 2 δ δx2+ Y 3 δ δx3, Y V _{= ˜Y}1 ∂ ∂y1 + ˜Y 2 ∂ ∂y2 + ˜Y 3 ∂ ∂y3.

Let η be a 1-form, η = ηidxi+ ˜ηaδya then ηH = η1dx1+ η2dx2+ η3dx3 and ηV = ˜η1δy1+ ˜η2δy2+ ˜η3δy3 where η = ηH _{+ η}V _{and η}H_XV _{= 0 and η}V _XH _{= 0 . We put η}H ₌ 1

2  dx3_{− x}2_dx1 _{and η}V ₌ 1 2 

δy3_{− y}2_δy1_{. Then the structure vector field ξ is given by ξ = 2} δ δx3 +_∂y∂3



and ξ is decomposed as ξH_{= 2} δ

δx3 and ξV = 2_∂y∂3. The tensor field φH of type (1, 1) and φV of type (1, 1) by a matrix form is given

by φH = ⎡ ⎢ ⎣ 0 1 0 −1 0 0 0 x2 ₀ ⎤ ⎥ ⎦ , φV ₌ ⎡ ⎢ ⎣ 0 1 0 −1 0 0 0 y2 ₀ ⎤ ⎥ ⎦ . The Riemann metric tensor field G = GH_{+ G}V _{is given by}

GH ₌ 1 4  dx1⊗ dx1+ dx2⊗ dx2+ ηH_{⊗ η}H₌1 4  1 +x22 dx12+dx22+dx32− 2x2dx1 dx3 GV = 1 4 

δy1⊗ δy1+ δy2⊗ δy2+ ηV ⊗ ηV=1 4



1 +y22 δy12+δy22+δy32− 2y2δy1 δy3. Thus we give a metric tensor field G by a matrix form

GH=1 4 ⎡ ⎢ ⎢ ⎣ 1 +x22 _{0 −x}2 0 1 0 −x2 _{0 1} ⎤ ⎥ ⎥ ⎦ , GV = 1₄ ⎡ ⎢ ⎢ ⎣ 1 +y22 _{0 −y}2 0 1 0 −y2 _{0 1} ⎤ ⎥ ⎥ ⎦ . We analyze that ηHξH= 1 , ηV ξV= 1 , φHξH= 0 , ηHξV= 0 , φV ξV= 0 , ηV ξH= 0 , 

φH2_XH _=−XH_+ηH_XH_ξH _and _φV2_XV _=−XV_+ηV _XV_ξV_{, hence ( φ, ξ, η ) is an almost contact} Finsler structure on R6_.

On the other hand, we formulize that ηHXH= GHXH, ξH, ηV XV= GV XV, ξV GHφXH, φYH= GHXH, YH−ηHXHηHYH, GVφXV, φYV= GVXV, YV−ηV XVηV YV ηHXH=1 2  dx3− x2_dx1 _X1 δ δx1+ X 2 δ δx2 + X 3 δ δx3  = 1 2  X3− X1_x2 _(3.35) GH_XH_{, ξ}H₌ 1 4  X1 _X2 _X3 ⎡ ⎢ ⎢ ⎣ 1 +x22 _{0 −x}2 0 1 0 −x2 _{0 1} ⎤ ⎥ ⎥ ⎦ ⎡ ⎢ ⎣ 0 0 2 ⎤ ⎥ ⎦ = 1 4  X1 _X2 _X3 ⎡ ⎢ ⎣ −2x2 0 2 ⎤ ⎥ ⎦ =1₄−2X1_x2_{+ 2X}3₌1 2  X3_{− X}1_x2_. (3.36) From (3.35) and (3.36) we get ηH_XH_{= G}H_XH_{, ξ}H_{. Similarly, we have}

ηV XV= 1 2  δy3− y2δy1 X˜1 ∂ ∂y1 + ˜X 2 ∂ ∂y2 + ˜X 3 ∂ ∂y3  =1 2  ˜ X3− ˜X1y2  ηV = GVXV, ξV, φHXH=X2,−X1, X2x2, φHYH=Y2,−Y1, Y2x2, φV XV=  ˜ X2,− ˜X1, ˜X2y2  , φVYV=  ˜ Y2,− ˜Y1, ˜Y2y2  , GHφHXH, φHYH=1 4  X1Y1+ X2Y2, GVφVXV, φVYV= 1 4  ˜ X1Y˜1, + ˜X2Y˜2, GHXH, YH=1 4  Y1  1 +x22  − Y3_x2_X1_{+ X}2_Y2_{+ X}3_Y3_{− Y}1_x2_, GV XV, YV= 1 4  ˜ Y1  1 +y22  − ˜Y3_y2_X˜1_{+ ˜X}2_Y˜2_{+ ˜X}3_Y˜3_{− ˜Y}1_y2_, ηH_XH_ηH_YH₌1 4  X3_Y3_{+ X}1_Y1_x22_{− X}1_Y3_x2_{− X}3_Y1_x2_, ηV _XV_ηV_YV₌1 4  ˜ X3_Y˜3_{+ ˜X}1_Y˜1_y22_{− ˜X}1_Y˜3_y2_{− ˜X}3_Y˜1_y2_.

Thus, we get GH_φXH_{, φY}H_{= G}H_XH_{, Y}H_−ηH_XH_ηH_YH_{, G}V _φXV_{, φY}V_{= G}V _XV_{, Y}V₋ ηV _XV_ηV _YV _{and hence ( φ, ξ, η, G ) is an almost contact Finsler metric structure.}

GHXH, φYH= 1 4  X1 _X2 _X3 ⎡ ⎢ ⎢ ⎣ 1 +x22 _{0 −x}2 0 1 0 −x2 _{0 1} ⎤ ⎥ ⎥ ⎦ ⎡ ⎢ ⎣ Y2 −Y1 Y2_x2 ⎤ ⎥ ⎦ =1₄X1Y2− X2Y1.

Also, we know that dηH₌ 1 2 

dx1_{∧ dx}2_{. By using this equality, we obtain} dηH_XH_{, Y}H_{= G}H_XH_{, φY}H_{. Similarly we get}

GV XV, φYV=1 4  _˜ X1 _X˜2 _X˜3 ⎡ ⎢ ⎢ ⎣ 1 +y22 _{0 −y}2 0 1 0 −y2 _{0 1} ⎤ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎣ ˜ Y2 − ˜Y1 ˜ Y2_y2 ⎤ ⎥ ⎥ ⎦ = 1₄  ˜ X1Y˜2− ˜X2Y˜1  .

By using dηV ₌ 1 2 

δy1_{∧ δy}2_{, we derived dη}V _XV_{, Y}V _{= G}V_XV_{, φY}V_{. As a result we come up} with the following equation:

dηHXH, YH= GHXH, φYH= ΩXH, YH, dηV XV, YV= GV XV, φYV= ΩXV, YV. (3.37) Then the almost contact metrical Finsler structure ( φ, ξ, η, G ) is called almost Sasakian Finsler struc-ture.

Because of η ∧ (dη) = 0, ( φ, ξ, η, G ) is a contact metrical Finsler structure. The vector fields X1 = 2  δ δx2 +_∂y∂2  , X2 = 2  δ δx1 + x2 δ_δx3 +_∂y∂1+ y2 ∂_∂y3  and ξ = 2  δ δx3 +_∂y∂3 

form a φ -basis for the contact metrical Finsler structure, where these are decomposed as

X₁H = 2  δ δx2  , X₁V = 2  ∂ ∂y2  , X₂H= 2  δ δx1 + x 2 δ δx3  , X₂V = 2  ∂ ∂y1 + y 2 ∂ ∂y3  , ξH = 2  δ δx3  , ξV = 2  ∂ ∂y3  .

On the other hand, we can see thatNφ+ dη⊗ξ = 0, that is NφH+ dηH⊗ξH = 0 and NφV+ dηV⊗ξV = 0 . Hence the contact metrical Finsler structure is normal.

4. The curvature of a Finsler connection

The curvature of a Finsler connection ∇ is given by:

R (X, Y ) Z =∇X∇YZ− ∇Y∇XZ− ∇[X,Y ]Z,∀X, Y, Z ∈ Tu(V ) . (4.1)

As ∇ preserves by parallelism the horizontal and the vertical distributions, from (4.1) we have that the operator R (X, Y ) carries horizontal vector fields into horizontal vector fields and vertical vector fields into verticals. Consequently,

R (X, Y ) Z = RH_{(X, Y ) Z}H_{+ R}V_{(X, Y ) Z}V_{,∀X, Y, Z ∈ T}

u(V ) . (4.2)

Noting that the operator R (X, Y ) is skew-symmetric with respect to X and Y, a theorem follows [1]:

Theorem 4.1 The curvature of a Finsler connection ∇ on the tangent space Tu(V ) is completely determined by the following six Finsler tensor fields:

RXH_{, Y}H_ZH _=∇H X∇HYZH− ∇HY∇XHZH− ∇[XH_,YH_]ZH, RXH_{, Y}H_ZV _=∇H X∇HYZV − ∇HY∇XHZV − ∇[XH_,YH_]ZV, RXV_{, Y}H_ZH _=∇V X∇HYZH− ∇HY∇XVZH− ∇[XV_,YH_]ZH, RXV_{, Y}H_ZV _=∇V X∇HYZV − ∇HY∇XVZV − ∇[XV_,YH_]ZV, RXV_{, Y}V_ZH_=∇V X∇VYZH− ∇VY∇XVZH− ∇[XV_,YV_]ZH, RXV_{, Y}V_ZV _=∇V X∇VYZV − ∇VY∇XVZV − ∇[XV_,YV_]ZV. (4.3)

Then the curvature tensor of a Finsler connection ∇ has only three different components with respect to the Berwald basis. These are given by:

R  δ δxk, δ δxj  δ δxh = R i h jk δ δxi, R  ∂ ∂yk, δ δxj  δ δxh = P i h jk δ δxi, R  ∂ ∂yk, ∂ ∂xj  δ δxh = S i h jk δ δxi. (4.4)

These three components are the first, third and fifth Finsler tensors from (4.3). The other three Finsler tensors from (4.3) have the same local componentsRi

h jk, Ph jki , and Sh jki .  δ δxk, δ δxj  ∂ ∂yh = R i h jk ∂ ∂yi, R  ∂ ∂yk, δ δxj  ∂ ∂yh = P i h jk ∂ ∂yi, R  ∂ ∂yk, ∂ ∂xj  ∂ ∂yh = S i h jk ∂ ∂yi. (4.5) (4.5)

So, a Finsler connection ∇Γ = (Ni

j, Fjki , Cjki ) has only three local componentsRih jk, Ph jki , Sh jki [1]. For a Finsler connection ∇, consider the torsion T, defined as usual

T (X, Y ) =∇XY − ∇YX− [X, Y ] , ∀X, Y ∈ Tu(V ) . (4.6) Breaking T down into horizontal and vertical parts gives the torsion of a Finsler connection, ∇ on Tu(V ) is completely determined by the following Finsler tensor fields [1]:

TH_XH_{, Y}H_=∇H XYH− ∇HYXH−  XH_{, Y}H H_{, T}V _XH_{, Y}H₌₋_XH_{, Y}H V_, TH_XH_{, Y}V_=−∇V YXH−  XH_{, Y}V H_{, T}V _XH_{, Y}V_=∇H XYV −  XH_{, Y}V V_, TV _XV_{, Y}V_=∇V XYV − ∇VYXV −  XV_{, Y}V V _. (4.7)

Let ∇ be the torsion free Finsler connection, then we get  XH_{, Y}H H _=∇H XYH− ∇HYXH,  XH_{, Y}H V _{= 0,} _XH_{, Y}V H_=−∇V YXH,  XH_{, Y}V V _=∇H XYV,  XV_{, Y}V V _=∇V XYV − ∇VYXV. (4.8)

Theorem 4.2 In order for a (n+m)-dimensional Finsler manifold V to be K-contact, it is necessary and

sufficient that the following two conditions are satisfied: 1. V admits a unit Killing vector field ξ ;

2. The flag curvature for plane sections containing ξ are equal to 1₄ at every point of V.

Proof Let V be a K -contact manifold. From (4.3) and (3.30), we have GH_R_XH_{, ξ}H_ξH_{, X}H _{= G}H_∇H X∇Hξ ξH− ∇Hξ ∇XHξH− ∇H_{[XH ,ξH]}ξH, XH  = 1 4GH  XH_{, X}H₌1 4, GV  RXV_{, ξ}V_ξV_{, X}V = GV _∇V X∇VξξV − ∇Vξ∇VXξV − ∇V[X,ξ]ξV, XV  = 1₄GV _XV_{, X}V₌ 1 4, where XH _{is a unit vector field orthogonal to ξ}H _{and X}V _{is a unit vector field orthogonal to ξ}V_{. Hence}

G (R (X, ξ) ξ, X) = GH_RH_{(X, ξ) ξ}H_{, X}H_{+ G}V _RV _{(X, ξ) ξ}V_{, X}V = 1 4  GH_XH_{, X}H_{+ G}V_XV_{, X}V₌ 1 4G (X, X) =14. Thus we obtain K (X, ξ) = G(R(X,ξ)ξ,X)_G(X,X) =1 4.

Conversely, we suppose that M satisfies the conditions (1.1) and (1.2). Since ξ is a Killing vector field, we have dηXH_{, Y}H₌_GH_∇H XξH, YH  − GH_∇H YξH, XH  =−2G∇H YξH, XH  = GXH_{, φY}H_,

dηXV, YV= GXV, φYV. Consequently, (φ, η, ξ, G) is a K -contact Finsler structure on V.

Let (φ, η, ξ, G) be a contact metrical Finsler structure on V. If the metric structure of V is normal, then V is mentioned to have a Sasakian Finsler structure and V is called a Sasakian Finsler manifold. 2

Theorem 4.3 An almost contact metrical Finsler structure (φ, η, ξ, G) on V is a Sasakian Finsler structure

if and only if  ∇H Xφ  YH= 1 2  GHXH, YHξH− ηHYHXH , (4.9)  ∇V Xφ  YV = 1 2  GVXV, YVξV − ηV YVXV . (4.10)

Proof If the structure is normal, we have Ω = dη and N(1) _{= N}(2) _{= 0 . Thus, by using (3.14),} (3.18) and (3.19), we get 2GH_∇H Xφ  YH_{, ξ}H _{= −dΩ}_XH_{, Y}H_{, ξ}H_{+ dη}_φYH_{, X}H _{= G}H_YH_{, X}H₋ ηH_XH_ηH_YH_{. Thus we have}_∇H Xφ  YH ₌1 2  GH_XH_{, Y}H_ξH_{− η}H_YH_XH _.

Similarly, from Lemma 3.3, we have

2GV ∇V_XφYV, ξV= GV YV, XV− ηV XVηV YV. Thus we get ∇V Xφ  YV ₌ 1 2  GV _XV_{, Y}V_ξV _{− η}V _YV_XV _.

Conversely, we suppose that the structure satisfies (4.9) and (4.10). Putting YH_{= ξ}H _{in (4.9) we have} −φ∇H

XξH=12 

ηH_XH_ξH_{− X}H_{, and putting Y}V _{= ξ}V _{in (4.10), we can get} −φ∇V

XξV = 12 

ηV _XV_ξV _{− X}V_{, and hence, applying φ to this, we obtain ∇}H

XξH = −12φXH and ∇V

YξV = −12φYV. Since ξ is skew-symmetric, we prove that ξH and ξV is a Killing vector field. Moreover, we obtain dηXH, YH= 1 2  ∇H Xη  YH−∇H_YηXH= GXH, φYH= ΩXH, YH, dηXV, YV= 1 2  ∇V Xη  YV −∇V_YηXV= GXV, φYV= ΩXV, YV. Thus the structure is a contact metric Sasakian structure.

If (φ, η, ξ, G) is a Sasakian Finsler structure on V, from (4.9) and (4.10) we obtain RXH_{, Y}H_ξH ₌ 1 4  ηH_YH_XH_{− η}H_XH_YH_{, (4.11)} RXV, YVξV = 1 4  ηV YVXV − ηV XVYV. (4.12) That is, we have

R (X, Y ) ξ = RH_{(X, Y ) ξ}H_{+ R}V _{(X, Y ) ξ}V _{= R}_XH_{, Y}H_ξH_{+ R}_XV_{, Y}V_ξV

= 1₄ηH_YH_XH_{+ η}V _YV_XV _{− η}H_XH_YH_{− η}V _XV_YV _. (4.13) 2

Theorem 4.4 Let V be a (n+m)-dimensional Finsler manifold admitting a unit Killing vector field ξ . Then

V is a Sasakian Finsler manifold if and only if R (X, ξ) Y = 1 4  −GH_{(X, Y ) ξ}H_{− G}V _{(X, Y ) ξ}V _{+ η}H_YH_XH_{+ η}V _YV_XV _{. (4.14)} Proof RH_{(X, ξ) Y}H _=∇H X∇Hξ YH− ∇Hξ ∇XHYH− ∇H[X,ξ]YH =−12  ∇H Xφ  YH =1 4  −GXH_{, Y}H_ξH_{+ η}H_YH_XH _, RV(X, ξ) YV =−1 2  ∇V Xφ  YV =−1 4  GXV, YVξV − ηV YVXV . From these equations mentioned above, we have the equation.

Let (.φ, η, ξ, G) be a Sasakian Finsler structure on V. From (4.9) and (4.10), we realize that RXH_{, Y}H_φZH_{= φR}_XH_{, Y}H_ZH₊1 4  GφXH_{, Z}H_YH _{− G}_YH_{, Z}H_φXH +GXH_{, Z}H_φYH_{− G}_φYH_{, Z}H_XH_, (4.15) RXV_{, Y}V_φZV _{= φR}_XV_{, Y}V_ZV ₊1 4  GφXV_{, Z}V_YV _{− G}_YV_{, Z}V_φXV +GXV_{, Z}V_φYV _{− G}_φYV_{, Z}V_XV_, (4.16) RXH, YHφZV = φRXH, YHZV, (4.17) RXH_{, Y}V_φZV _{= φR}_XH_{, Y}V_ZV ₋1 4  GYV_{, Z}V_φXH _{− G}_φYV_{, Z}V_XH_{, (4.18)} RXV, YHφZV = φRXV, YHZV +1 4  GφXV, ZVYH + GXV, ZVφYH, (4.19) RXV, YVφZH= φRXV, YVZH, (4.20) RXV, YHφZH = φRXV, YHZH−1 4  GYH, ZHφXV − GφYH, ZHXV, (4.21) RXH, YVφZH = φRXH, YVZH+1 4  GφXH, ZHYV + GXH, ZHφYV, (4.22) R (X, Y ) φZ = R (X, Y ) φZH+ R (X, Y ) φZV. (4.23) From (4.15), (4.16), (4.17), (4.18), (4.19), we also have the following equations:

RXH, YHZH= −φRXH, YHφZH+1₄GYH, ZHXH − GXH, ZHYH −GφYH_{, Z}H_φXH_{+ G}_φXH_{, Z}H_φYH_, (4.24) RXV_{, Y}V_ZV _{= −φR}_XV_{, Y}V_φZV ₊1 4  GYV_{, Z}V_XV _{− G}_XV_{, Z}V_YV −GφYV_{, Z}V_φXV _{+ G}_φXV_{, Z}V_φYV_, (4.25) RXH_{, Y}H_ZV _{= −φR}_XH_{, Y}H_φZV_{, (4.26)} RXH, YVZV = −φRXH, YVφZV +1 4  GYV, ZVXH− GφYV, ZVφXH, (4.27)