ON COMPLEX FINSLER SPACE WITH INFINITE SERIES OF (α,β )-METRIC

The purpose of the present paper is to investigate the complex Finsler space with infinite series of (α,β )-metric F = |β | 2 |β |−α . Next we determine the fundamental metric tensor, angular metric tensor and ChernFinsler connection coefficients. Further, we discussed the special approach to Kähler-Randers change of infinite series (α,β )-metric using some examples.


INTRODUCTION
The real Randers metric were first introduced by G. Randers in the context of general relativity and they were applied to the theory of the electron microscope by R. S. Ingarden [16].
The importance of real Randers spaces is also pointed out in [7] and the obtained results are remarkable. Recently, it was shown that the real Randers metrics are solutions to Zermeto's navigation problem [9] and the classification of real Randers metrics of constant flag curvature was finally completed ( [7], [9], [24]).
The correct notion of complex Finsler metrics was probably proposed firstly by Rizza [14], and then developed by Rund [21] by defining the connection and geodesics. The famous complex Finsler metrics are the Kobayashi and Caratheodary metrics ( [11], [17]), which play important roles in the theory of the moduli space of Riemann surface [13]. In recent years complex Finsler geometry has attracted renewed interest as examples of such metrics appear in a natural way in the geometric theory of several complex variables.
The current paper is organized in two sections. The first section is to introduce the complex Finsler spaces with infinite series of (α, β )-metrics, i.e., complex metrics constructed from just two pieces of familiar data: a purely Hermitian metric and a differential (1, 0)-form, both globally defined on an underlying complex manifold. We determine the fundamental metric tensor of a complex Finsler spaces with infinite series of (α, β )-metric, its inverse and determinant.
Moreover, a complex Randers change of infinite series also produces another invertible d-tensor (Theorem 5). By deformation of some purely Hermitian metrics we obtain some example of complex Randers change of infinite series.
In the second section, we find the conditions such that a complex Randers change of infinite series is weakly Kähler or strongly Kähler. A special attention is devoted to a class of complex Randers change of infinite series with some additional assumption, and finaly our results are applied to some examples.

PRELIMINARIES
Let M be a complex manifold of dim C M = n. The complexified of the real tangent bundle T C M splits in to the sum of holomorphic tangent bundle T M and its conjugate T M. The bundle T M is in its turn a complex manifold, The local coordinates in a chart will be denoted by u = (z k , η k ) and these are changed by the rules: which have changes by the rules obtained with jacobi matrix of above transformations. Note that the change rules obtained with Jacobi matrix of above transformations. Note that the change rule of ∂ ∂ z k contains the second order partial derivatives. Let V (T M) = kerπ * ⊂ T (T M) be the vertical bundle, spanned locally by { ∂ ∂ η k }. A complex nonlinear connection, briefly (c.n.c), determines a supplementary complex subbundle to where N j k are the coefficients of (c.n.c), [2]. (ii) F(z, η) ≥ 0, the equality holds if and only if η = 0; (iii) F(z, λ η)=|λ |F(z, η); for λ ∈ C; (iv) the Hermitian matrix (g ij (z, η)), with g ij = ∂ 2 L ∂ η i ∂η j , called the fundamental metric tensor, is positive definite.
Let us write L = F 2 , the pair (M, F) is called a complex Finsler space. The (iv)-th assumption involves the strongly pseudoconvexity of the Finsler metric F on complex indicatrix, Further, in a complex Finsler space a Hermitian connection of (1, 0)−type has a special meaning, named in [1] the Chern-Finsler Connection. In notations from [19] Now, let us recall that in [1]'s terminology, the complex Finsler space (M, F) is strongly Kähler if and only if L i jk − L i k j = 0, Kähler if and only if (L i jk − L i k j )η jη l = 0. In the particular case of purley Hermitian metrics, that is g ij = g ij (z), those three nuances Kähler coincide [22]. For the vertical £ = η k∂ k with∂ k :=∂ ∂ η k , called the Liouville complex field (or the vertical radial vector field in [1]), we consider its horizontal lift χ := η k δ k , (δ k := δ δ z k ). According to the holomorphic curvature of the complex Finsler space (M, F) in direction η is and locally it has the following expression ( [3]) And more informations can be found in ( [1], [2], [19]).

COMPLEX FINSLER SPACE WITH INFINITE SERIES OF (α, β )-METRIC
In the r-th series (α, β )-metric F(α, β ) = |β | r ∑ k=0 ( α |β | ) n , where we assume α < |β |. If r = ∞, then this metric is expressed in the form F(α, β ) = |β | 2 |β |−α and is called an infinite series (α, β )metric. Interestingly this metric is the difference between a Randers metric and a Matsumoto metric. Following the ideas from real case ( [7], [9], [24]), we shall introduce a new class of complex Finsler metrics. We consider z ∈ M and η ∈ T z M, η = η i ∂ ∂ z i . On M let • a := a ij (z)dz i ⊗ dz j be a purely Hermitian positive metric and By these objects we define the function F on T M where By analogy with the real case the function from (4) the complex infinite series of (α, β )-metric and the pair (M, |β | 2 |β |−α ) a complex Finsler space with infinite series of (α, β )-metric such a metric is only quoted as an example of complex Finsler metric in [12].
Our goal in the sequel is to find the circumstances in which the function (4) is a complex Finsler metric. Some remarks are immediate due to the presence of |β | be the complex Finsler space with infinite series of (α, β )-metric F(α, β ) := |β | 2 |β |−α is positive and smooth on T M \{0}. The complex Finsler space with infinite series of (α, β )-metric is purely Hermitian if and only if β vanishes identically.
Theorem 1. The fundamental metric tensor of the complex Finsler space with infinite series of (α, β )-metric F = |β | 2 |β |−α is given by Proof. Indeed, from Using (7) and (8) we have its equivalent form is given by [10] and also we simplify that where The next goal is to find the formulas for the inverse and the determinant of the fundamental metric tensor g ij . For this purpose we use the proposition proved in [5].
Proposition 2. Suppose: • (Q ij ) is a non-singular n × n complex matrix with inverse (Q¯j i ).
• C i := Q¯j i C¯j and its conjugates; C 2 := C i C i =C i C¯i; H ij := Q ij ±C i C¯j.
(2) whenever (1 ± C 2 ) = 0, the matrix (H ij ) is invertible and in this case its inverse is Theorem 3. For the complex Finsler space with infinite series of (α, β )-metric F = |β | 2 |β |−α we have Proof. To prove the claims we apply the above proposition in a recursive algorithm in three steps. We write g ij from (8) in the form Step-1: We set Q ij := a ij and C i := 1 α K 2V l i . By applying the proposition (2) we obtain So the matrix H ij = a ij + K 2α 2 V l i l¯j is invertible with H¯j i = a¯j i + K α 2 M η iη j and det(a ij + K 2α 2 V l i l¯j) = M 2V det(a ij ).

CHERN FINSLER CONNECTION COEFFICIENTS AND CARTAN TENSOR FOR INFI-
The Chern-Finsler connection coefficients (c.n.c) of a complex Finsler space (M, F) with (α, β )-metric is defined by Once obtained the metric tensor of a complex Finsler space with infinite series, it is a technical computation to get the expression of Chern-Finsler connection by using (1). The First computation refers to the coefficients of the Chern-Finsler connection (c.n.c). A simplified writing for them is Theorem 6. The coefficients of Chern Finsler connection and Cartan tensor for an complex Finsler space with infinite series of (α, β )-metric is given in (25) and (30) respectively.
We remark that a complex Finsler metric is purely Hermitian it leads to F = α(1 + ||b||) and a ij ||b|| 2 = b i b¯j. thus, we have proved.

KÄHLER-RANDERS CHANGE OF INFINITE SERIES OF (α, β )-METRICS
When trying to show more geometrical properties of complex Finsler space with infinite series of (α, β )-metrics, we face the fact that there are so many computationas. Certainly, one should not infer that this class of complex Finsler metrics is less significant. On the contrary, beyond the computations in the sequel we show that there are interesting results.