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
represented as projection). In  , 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
. 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 complex Finsler spaces, with respect to Chern (c.l.c) on
(definition (2.4) and (2.5)). We shall see that the fundamental metric tensor
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)).
This section, includes the basic notions of Complex Finsler spaces.
-Complex Finsler metric on M is continuous function
is a smooth on
, the equality holds if and only if
, for all
Let M be a complex manifold,
the holomorphic tangent bundle in which as a complex manifold the local coordinates will be denoted by
. The complexified tangent bundle of
is decomposed in
, where operator
becomes direct sum.
Considering the restriction of the projection to
, for pulling back of the holomorphic tangent bundle
then it obtain a holomorphic tangent bundle
, called the pull-back tangent bundle over
. We denote by
, the local frame and by
the local frame and its dual.
be the vertical bundle, spanned locally by
. A complex nonlinear connection, briefly (c.n.c), determines a supplementary complex subbundle to
, that is
The adapted frames is
are the coefficients of the (c.n.c). Further we shall use the abbreviations
, and their conjugates    .
be the fundamental metric tensor of a complex Finsler space
The isomorphism between
induces an isomorphism of
defines an Hermitian metric structure
, with respect to the natural complex structure. Further, the Hermitian metric structure G on
induces a Hermitian inner product
and the angle
the sections on
(for details see in  ).
On the other hand,
are isomorphic. Therefore, the structures on
can be pulled-back to
. By this isomorphism the natural co-basis
is identified with
. In view of this constructions the pull-back tangent bundle
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
determines the Chern-Finsler (c.n.c) on (
), with the coefficients
, and its local coefficients of torsion and curvature are
The Riemann type tensor
According to  the complex Finsler space
is strongly Kähler if and only if
, Kähler if and only if
and weakly Kähler if and only if
. Note that for a complex Finsler metric which comes from a Hermitian metric on M, so-called purely Hermitian metric. That is
, the three nuances of Kähler spaces consider, in  .
The holomorphic curvature of F in direction
, with respect to the Chern (c.l.c) is,
is viewed as local section of
, that is
. Further on,
we shall simply call it holomorphic curvature. It depends both on the position
and the direction
Definition 2.1.  The complex Finsler space
is called generalized Einstein if
is proportional to
, that is if there exists a real valued function
, such that
By finding the Chern (c.l.c) on
determines the Chern-Finsler on
, with the coefficient
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
-homogeneous with respect to the real scalar
and is proved that the following identities are fulfilled in  .
Here, to find the inverse of fundamental metric tensor
we use the following proposition.
Proposition 2.1. Suppose:
is a non-singular
complex matrix with inverse
are complex numbers;
and its conjugates;
, the matrix
is invertible and in this case its inverse is
3. Notation of Complex Square Metrics
-complex Finsler space produce the tensor fields
. The tensor field must
be invertible in Hermitian geometry. These problems are about to Hermitian
-complex Finsler spaces, if
-complex Finsler spaces, if
. In this section, we determine the fundamental tensor of complex Square metric and inverse also.
-complex Finsler space with Square metric,
then it follows that
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:
Then, we can find,
with respect to
respectively, which yields:
By direct computation using (3.11), (3.12), (3.13), we obtain the invariants of
-complex Finsler space with Square metric:
are given below:
Fundamental Metric Tensor of
-Complex Finsler Space with Square Metric
The fundamental metric tensors of
-complex Finsler space with
metric are given by  :
By using the Equations (3.14) to (3.18) in (3.19) we have
Next to determine the determinant and inverse of the tensor field
through the theorem below by using Proposition (2.1). The solution of the non-Hermitian metric
Theorem 3.2. For a non-Hermitian
-Complex Finsler space with Square metric
, then they have the following:
1) The contravariant tensor
of the fundamental tensor
Proof. We prove this theorem by following three steps:
Step 1: We write
from (3.21) in the form.
. By applying the proposition 2.1 we obtain
So, the matrix
, is invertible with
Step 2: Now, we consider
By applying the proposition 2.1 we have
It results that the inverse of
exists and it is
Step 3: We put
clearly observe that and obtain
clearly, the matrix
Again by applying Proposition (2.1) we obtain the inverse of
from last step. Thus
Therefore, from Equation (3.38) in Equation (3.40) and the Equation (3.39), then we obtained claims 1) and 2) are desired.
4. Holomorphic Curvature of Complex Square Metric
The holomorphic curvature is the correspondent of the holomorphic sectional curvature in Complex Finsler geometry. Our goal is to find a notation of Complex Finsler spaces with square metric. By analogy with the naming from the real case  , we shall call it the holomorphic flag curvature and we shall introduce it with respect to Chern-Finsler connection (c.n.c).
The holomorphic curvature
depends on the position
alone. In view of definition (2.1) we obtain the holomorphic curvature of Complex Finsler space with square metric if
is the Chern-Finsler connection coefficients.
To find Riemannian curvature
, we need the Chern Finsler connection (c.n.c) coefficients. Now, by direct computations, we get the Chern-Finsler (c.n.c) connection coefficients;
Observed that the Equation (4.1) can be expressed by the identity as:
Now using Equation (4.1) and (
(see definition (2.1)) we get the Riemann curvature tensor
Notice that, on contracting with
. We get the above coefficients D-tensor.
Again, by using Chern-Finsler connection coefficients, we get the coefficients of torsion.
Theorem 4.3. The holomorphic flag curvature of Complex Square metric
is given by,
Proof. From Equation (4.2) plugging into (2.1), it yields.
is in Equation (4.1).
Then, comparing (4.6) with (4.2) we get (4.5) as desired.
Proposition 4.4. If
be Complex Square metric of dimension
, then it is not a Kähler and not a weakly Kähler.
Proof. Observing Equation (4.3)
is non zero and since by definition it is not a Kähler. Further, on contracting
, it yields
Therefore, it is not a weakly Kähler.
The authors would like to thank the referees for their very detailed reports and many valuable suggestions on this paper.