The Relation Between the Associate Almost Complex Structure to $HM'$ and $(HM',S,T)$-Cartan Connections

In the present paper, the $(HM',S,T)$-Cartan connections on pseudo-Finsler manifolds, introduced by A. Bejancu and H.R. Farran, are obtained by the natural almost complex structure arising from the nonlinear connection $HM'$. We prove that the natural almost complex linear connection associated to a $(HM',S,T)$-Cartan connection is a metric linear connection with respect to the Sasaki metric $G$. Finally we give some conditions for $(M', J, G)$ to be a K\"ahler manifold.


Introduction
Almost complex structures are important structures in differential geometry [8,9,11]. These structures have found many applications in physics. H.E. Brandt has shown that the spacetime tangent bundle, in the case of Finsler spacetime manifold, is almost complex [4,5,6]. Also he demonstrated that in this case the spacetime tangent bundle is complex provided that the gauge curvature field vanishes [3]. In [1,2], for a pseudo-Finsler manifold F m = (M, M ′ , F * ) with a nonlinear connection HM ′ and any two skew-symmetric Finsler tensor fields of type (1, 2) on F m , A. Bejancu and H.R. Farran introduced a notion of Finsler connections which named "(HM ′ , S, T )-Cartan connections". If, in particular, HM ′ is the canonical nonlinear connection GM ′ of F m and S = T = 0, the Finsler connection is called the Cartan connection and it is denoted by F C * = (GM ′ , ∇ * ) (see [1]). They showed that ∇ * is the projection of the Levi-Civita connection of the Sasaki metric G on the vertical vector bundle. Also they proved that the associate linear connection D * to the Cartan connection F C * is a metric linear connection with respect to G [1]. In this paper we obtain the (HM ′ , S, T )-Cartan connections by using the natural almost complex structure arising from the nonlinear connection HM ′ , then the natural almost complex linear connection associated to a (HM ′ , S, T )-Cartan connection is defined. We prove that the natural almost complex linear connection associated to a (HM ′ , S, T )-Cartan connection is a metric linear connection with respect to the Sasaki metric G. Kähler and para-Kähler structures associated with Finsler spaces and their relations with flag curvature were studied by M. Crampin and B.Y. Wu (see [7,12]). They have found some interesting results on this matter. In [12], B.Y. Wu gives some equivalent statements to the Kählerity of (M ′ , G, J). In the present paper we give other conditions for the Kählerity of (M ′ , G, J), which extend the previous results. • F * is positively homogeneous of degree two with respect to (y 1 , . . . , y m ), i.e., we have F * (x 1 , . . . , x m , ky 1 , . . . , ky m ) = k 2 F * (x 1 , . . . , x m , y 1 , . . . , y m ) for any point (x, y) ∈ (Φ ′ , U ′ ) and k > 0. A nonlinear connection HM ′ enables us to define an almost complex structure on M ′ as follows: is assumed as a local frame field of T M ′ and Γ(T M ′ ) is the space of smooth sections of the vector bundle T M ′ . We call J the associate almost complex structure to HM ′ . Obviously we have J 2 = −Id T M ′ , also we can assume the conjugate of J, J ′ = −J, as an almost complex structure. Now we give the following proposition which was proved by B.Y. Wu [12].
Proposition 1. Let F m = (M, M ′ , F ) be a Finsler manifold. Then the following statements are mutually equivalent: Corollary 1. Let the associate almost complex structure to J (or J ′ ) be a complex structure; then we have So in this case the horizontal distribution is integrable.
is a linear connection on M ′ . Also J is parallel with respect to D, that is We call D the natural almost complex linear connection associated to F C ∇ on M ′ .
Therefore D is a linear connection on M ′ . Also we have Note that the torsion of D is given by the following expression: Theorem 2. Let HM ′ be a nonlinear connection on M ′ and S and T be any two skew-symmetric Finsler tensor fields of type (1, 2) on F m . Then there exists a unique linear connection ∇ on V M ′ satisfying the conditions: (i) ∇ is a metric connection; (ii) T D , S and T satisfy Then for any X, Y, Z ∈ Γ(T M ′ ) we have The above computation shows that the connection ∇ defined by (2) and (3) is a metric connection.
∂y k . Now in (2) let X = ∂ ∂y j , Y = ∂ ∂y i and Z = ∂ ∂y l . After performing some computations we obtain the following expression for the coefficients C m ij : Also in (3) let X = δ δx j , Y = δ δx i and Z = δ δx l . Then we can obtain the following expression for the coefficients F m ij :

By using the relations
Suppose that X, Y ∈ Γ(T M ′ ) are two arbitrary vector fields on M ′ . In local coordinates, let X = X i δ δx i +X i ∂ ∂y i and Y = Y i δ δx i +Ỹ i ∂ ∂y i , after performing some computations we have: The relations (6) and (7) show that ∇ satisfies (ii) of Theorem 2. Now let∇ be another linear connection on V M ′ which satisfies (i) and (ii). By using the relations (i), (ii), (4) and (5)  The relations (8) and (9) show that∇ satisfies (2) and (3), respectively. Therefore ∇ =∇.
The Finsler connection F C = (HM ′ , ∇) where ∇ is given by Theorem 2 is called the (HM ′ , S, T )-Cartan connection (see [1,2]) which in this case is obtained by the associate almost complex structure to HM ′ . If, in particular, HM ′ is just the canonical nonlinear connection GM ′ of F m (for more details about GM ′ see [1]) and S = T = 0, the F C is called the Cartan connection and it is denoted by F C * = (GM ′ , ∇ * ).
By means of the pseudo-Riemannian metric g on V M ′ we consider a pseudo-Riemannian metric on the vector bundle T M ′ similar to the Sasaki one and denote it by G, that is where δy i = dy i + N i j (x, y)dx j . Denote by ∇ ′ the Levi-Civita connection on M ′ with respect to G. A. Bejancu and H.R. Farran showed ∇ * is the projection of the Levi-Civita connection ∇ ′ on the vertical vector bundle also they proved the following theorem (see [1]).  Proof . For any X, X 1 , X 2 ∈ Γ(T M ′ ) we have By (10) and this fact that S and T are skew-symmetric we have: Therefore D X G = 0 for any X ∈ Γ(T M ′ ).
Let F m = (M, M ′ , F ) be a Finsler manifold. We can easily check that the pair (J, G) defines an almost Hermitian metric on M ′ . In the following theorem we give a sufficient condition for Finsler tensor fields S and T such that D be the Levi-Civita connection arising from G.
Proof . By Theorem 4, D is a metric linear connection with respect to G. Therefore if T D = 0 then D is the Levi-Civita connection. In local coordinates we have Therefore the proof is completed.
Proof . If T D = 0 then D is the Levi-Civita connection of G. Also J is parallel with respect to D. Therefore D (the Levi-Civita connection of G) is almost complex. Consequently by using Theorem 4.3 of [10], (M ′ , J, G) is a Kähler manifold.
We know that the almost Hermitian manifold (M ′ , J, G) is an almost Kähler manifold if and only if the fundamental 2-form Φ is closed (Φ is defined by Φ(X, Y ) = G(X, JY ) for all X, Y ∈ Γ(T M ′ )). Therefore we can give the following theorem.
and R h ij g hk − R h ik g hj + R h jk g hi = 0.