1. 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
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
. 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)).
2. Preliminaries
This section, includes the basic notions of Complex Finsler spaces.
An
-Complex Finsler metric on M is continuous function
satisfying:
1)
is a smooth on
;
2)
, the equality holds if and only if
;
3)
, for all
.
Let M be a complex manifold,
and
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
the slit
. We denote by
, the local frame and by
the local frame and its dual.
Let
be the vertical bundle, spanned locally by
. A complex nonlinear connection, briefly (c.n.c), determines a supplementary complex subbundle to
in
, that is
.
The adapted frames is
, where
are the coefficients of the (c.n.c). Further we shall use the abbreviations
,
,
,
, and their conjugates [2] [3] [4] .
On
, let
be the fundamental metric tensor of a complex Finsler space
.
The isomorphism between
and
induces an isomorphism of
and
. Thus,
defines an Hermitian metric structure
on
, with respect to the natural complex structure. Further, the Hermitian metric structure G on
induces a Hermitian inner product
and the angle
,
for any
,
the sections on
, where
(for details see in [5] ).
On the other hand,
and
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
-type.
(2.1)
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
(2.2)
The Riemann type tensor
has properties:
(2.3)
According to [2] 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 [6] .
The holomorphic curvature of F in direction
, with respect to the Chern (c.l.c) is,
(2.4)
where
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. [7] The complex Finsler space
is called generalized Einstein if
is proportional to
, that is if there exists a real valued function
, such that
(2.5)
where
,
,
,
.
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 [8] .
(2.6)
(2.7)
(2.8)
where,
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
;
·
and
are complex numbers;
·
and its conjugates;
;
.
Then,
1)
(Here,
indicates determinant),
2) whenever
, the matrix
is invertible and in this case its inverse is
.
3. Notation of Complex Square Metrics
The
-complex Finsler space produce the tensor fields
and
. The tensor field must
be invertible in Hermitian geometry. These problems are about to Hermitian
-complex Finsler spaces, if
and non-Hermitian
-complex Finsler spaces, if
. In this section, we determine the fundamental tensor of complex Square metric and inverse also.
Consider
-complex Finsler space with Square metric,
(3.1)
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
(3.2)
where
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
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,
where,
Then, we can find,
We denote:
where
(3.11)
and
(3.12)
Differentiating
and
with respect to
and
respectively, which yields:
and
Similarly,
where,
(3.13)
By direct computation using (3.11), (3.12), (3.13), we obtain the invariants of
-complex Finsler space with Square metric:
,
,
,
are given below:
(3.14)
(3.15)
(3.16)
(3.17)
(3.18)
Fundamental Metric Tensor of
-Complex Finsler Space with Square Metric
The fundamental metric tensors of
-complex Finsler space with
metric are given by [9] :
(3.19)
By using the Equations (3.14) to (3.18) in (3.19) we have
(3.20)
(3.21)
Or, equivalently,
(3.22)
(3.23)
where,
(3.24)
(3.25)
(3.26)
(3.27)
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
as follows.
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
is:
(3.28)
where
(3.29)
2)
where,
.
Proof. We prove this theorem by following three steps:
Step 1: We write
from (3.21) in the form.
(3.30)
We take
and
. By applying the proposition 2.1 we obtain
,
, and
.
So, the matrix
, is invertible with
Step 2: Now, we consider
, and
,
By applying the proposition 2.1 we have
Therefore,
where,
,
.
It results that the inverse of
exists and it is
(3.31)
(3.32)
where,
and,
(3.33)
Step 3: We put
(3.34)
and
clearly observe that and obtain
(3.35)
and
, where
(3.36)
(3.37)
And
,
clearly, the matrix
is invertible.
and
where
Again by applying Proposition (2.1) we obtain the inverse of
as:
(3.38)
(3.39)
But
, with
from last step. Thus
(3.40)
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 [10] , 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
, where,
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;
(4.1)
where
Observed that the Equation (4.1) can be expressed by the identity as:
(4.2)
where,
.
Now using Equation (4.1) and (
) on
(see definition (2.1)) we get the Riemann curvature tensor
as,
(4.3)
where
Notice that, on contracting with
in
. We get the above coefficients D-tensor.
Again, by using Chern-Finsler connection coefficients, we get the coefficients of torsion.
(4.4)
where,
Theorem 4.3. The holomorphic flag curvature of Complex Square metric
is given by,
(4.5)
where,
Proof. From Equation (4.2) plugging into (2.1), it yields.
(4.6)
where,
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
with non-zero
, 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
by
, it yields
.
Therefore, it is not a weakly Kähler.
Acknowledgements
The authors would like to thank the referees for their very detailed reports and many valuable suggestions on this paper.