On Holomorphic Curvature of Complex Finsler Square Metric

The notion of the holomorphic curvature for a Complex Finsler space ( ) , M F is defined with respect to the Chern complex linear connection on the pull-back tangent bundle. This paper is about the fundamental metric tensor, inverse tensor and as a special approach of the pull-back bundle is devoted to obtaining the holomorphic curvature of Complex Finsler Square metrics. Further, it proved that it is not a weakly Kähler.


Introduction
The notion of holomorphic curvature of a complex Finsler space is defined with respect to the Chern complex linear connection in briefly Chern (c.l.c) as a connection in the holomorphic pull back tangent bundle ( ) (here π represented as projection). In [1], Nicolta Aldea has obtained the characterization of the holomorphic bisectional curvature and gave the generalization of the holomorphic curvature of the complex Finsler spaces which are called holomorphic flag curvature. After that in (2006) he devoted to obtaining the characterization of holomorphic flag curvature.
In complex Finsler geometry, it is systematically used the concept of holomorphic curvature in direction η . But, the holomorphic curvature is not an analogue of the flag curvature from real Finsler geometry.
This problem sets up the subject of the present paper. Our goal is to determine the conditions in which complex Finsler spaces with square metric of holomorphic curvature. As per our claim, we shall use the holomorphic curvature of ′ (definition (2.4) and (2.5)). We shall see that the fundamental metric tensor ij g and its inverse are obtained (see in Section-3). Moreover, we determine the holomorphic curvature of complex square metric (theorem (4.3)) and some special properties of holomorphic curvature are obtained (proposition (4.4)).

Preliminaries
This section, includes the basic notions of Complex Finsler spaces.
An  -Complex Finsler metric on M is continuous function The adapted frames is admits a unique complex linear connection ∇ , called the Chern (c.l.c), which is metric with respect to G and of ( ) The Chern (c.l.c) on ( ) with the coefficients , and its local coefficients of torsion and curvature are : ; the three nuances of Kähler spaces consider, in [6].
The holomorphic curvature of F in direction η , with respect to the Chern (c.l.c) is, where η is viewed as local section of ( ) By finding the Chern (c.l.c) on ( ) determines, we need to find the fundamental metric tensor followed by the invariants are given below: Now, from definition of Complex Finsler metric follows that L is ( ) 2, 0 -homogeneous with respect to the real scalar λ and is proved that the following identities are fulfilled in [8].
Here, to find the inverse of fundamental metric tensor ij g we use the following proposition.
Proposition 2.1. Suppose: Then, 2) whenever ( ) is invertible and in this case its inverse is

Notation of Complex Square Metrics
The  -complex Finsler space produce the tensor fields ij g and ij g . The tensor field must ij g be invertible in Hermitian geometry. These problems are about Now, we find the following quantities of F.
From the equalities (2.6) and (2.7) with metric (3.1), we have We propose to determine the metric tensors of an  -complex Finsler space using the following equalities: Each of these being of interest in the following: Consider, Then, we can find, We denote: Differentiating 0 ρ and 1 ρ with respect to j η and j η respectively, which yields: By direct computation using (3.11), (3.12), (3.13), we obtain the invariants of  -complex Finsler space with Square metric:

Fundamental Metric Tensor of  -Complex Finsler Space with Square Metric
The fundamental metric tensors of  -complex Finsler space with ( ) , α β metric are given by [9]: By using the Equations (3.14) to (3.18) Or, equivalently, , where, 1) The contravariant tensor ij g of the fundamental tensor ij g is: Proof. We prove this theorem by following three steps: Step 1: We write ij g from (3.21) in the form. .
It results that the inverse of Step clearly, the matrix ij H is invertible.
Again by applying Proposition (2.1) we obtain the inverse of ij H as: