1. Introduction

One of the representative work of differential geometry is an almost Hermitian structure. Gray and Hervalla [1] found that the action of the unitary group on the space of all tensors of type (3,0) decomposed this space into sixteen classes. The conditions that determined each one of these classes belongs to the type of almost Hermitian structure have been identified. These conditions were formulated by using the method of Kozel's operator [2].

The Russian researcher Kirichenko found an interesting method to study the different classes of almost Hermitian manifold. This method depending on the space of the principal fiber bundle of all complex frames of manifold with structure group is the unitary group This space is called an adjoined -structure space, more details about this space can be found in [3-6].

One of the most important classes of almost Hermitian structures is denoted by where and respectively denoted to the nearly manifold and local conformal manifold.

A harmonic function is a function whose Laplacian vanishes. Related to this fact, Y. Ishi [7] has studied conharmonic transformation which is a conformal transformation that preserves the harmonicity of a certain function. Agaoka, et al. [8] studied the twisted product manifold with vanishing conharmonic curvature tensor. Agaoka, et al. [9] studied the fibred Riemannian space with flat conharmonic curvature tensor, in particular, they proved that a conharmonically flat manifold is locally the product manifold of two spaces of constant curvature tensor with constant scalar curvatures. Siddiqui and Ahsan [10] gave an interesting application when they studied the conharmonic curvature tensor on the four dimensional space-time that satisfy the Einstein field equations. Abood and Lafta [11] studied the conharmonic curvature tensor of nearly and almost manifolds. The present work devoted to study the flatness of conharmonic curvature tensor of Vaisman-Gray manifold by using the methodology of an adjoined space.

2. Preliminaries

Suppose that is -dimensional smooth manifold, is a set of all smooth functions on is the module of smooth vector fields on An almost Hermitian manifold is the set where is a smooth manifold, and is an almost complex structure, and is a Riemannian metric, such that

Suppose that is the complexification of tangent space at the point and is a real adapted basis of -manifold. Then in the module there exists a basis given by which is called adapted basis, where, and and are two endomorphisms in the module which are defined by and such that, and are the complexifications of and respectively. The corresponding frame of this basis is Suppose that the indexes and are in the range and the indexes and are in the range And

The -structure space is the principal fiber bundle of all complex frames of manifold with structure group is the unitary group This space is called an adjoined -structure space.

In the adjoined -structure space, the components matrices of complex structure and Riemannian metric are given by the following:

(2.1)

where is the identity matrix of order

Definition 2.1 [12] The Riemannian curvature tensor of a smooth manifold is an 4-covariant tensor

which is defined by:

where and satisfies the following properties:

Definition 2.2 [13] The Ricci tensor is a tensor of type (2,0) which is defined as follows:

Definition 2.3 [7] The conharmonic tensor of an -manifold is a tensor of type (4,0) which is defined as the form:

where and are respectively Ricci tensor, Riemannian curvature tensor and Riemannian metric. Similar to the properties of Riemannian curvature tensor, the conharmonic tensor has the following properties:

Definition 2.4. An -manifold is called a conharmonically flat if the conharmonic tensor vanishes.

Definition 2.5 [14] In the adjoined -structure space, an -manifold

is called a Vaisman-Gray manifold if

is called a locally conformal manifold if and

and is called a nearly manifold if and , where and is a Lie form; is a form which is defined by is a coderivative and and the bracket [ ] denote to the antisymmetric operation.

Theorem 2.6 [15] In the adjoined space, the components of Riemannian curvature tensor of are given by the following forms:

where, are some functions on adjoined - structure space and are system of fuctions in the adjoined -structure space which are symmetric by the lower and upper indices which are called components of holomorphic sectional curvature tensor.

and are the components of the covariant differential structure tensor of first and second type and are the components of the Lee form on adjoint -structure space such that:

where, are the components of mixture form, are the components of Riema-nnian connection of metric

The other components of Riemannian curvature tensor can be obtained by the property of symmetry for

There are three special classes of almost Hermitian manifold depending on the components of the Riemannian curvature tensor. Their conditions are embodied in the following definition:

Definition 2.7. [16] In the adjoined -structure space, an -manifold is a manifold of class:

1) if and only if,

2) if and only if,

3) if and only if,

It easy to see that

Theorem 2.8 [15] In the adjoined -structure space, the components of Ricci tensor of -manifold are given by the following forms:

and the others are conjugate to the above components.

Definition 2.9 [17] A Riemannian manifold is called an Einstein manifold, if the Ricci tensor satisfies the equation where, is an Einstein constant.

Definition 2.10 [18] An -manifold has -invariant Ricci tensor, if

The following Lemma gives a fact about Ricci tensor in the adjoined -structure space.

Lemma 2.11 [16] An -manifold has -invariant Ricci tensor if and only if, we have

Definition 2.12 [3] Define two endomorphisms on as follows:

i) Symmetric mapping Sym: by:

ii) Antisymmetric mapping Alt: by:

The symbols ( ) and [ ] are usually used to denote the symmetric and antisymmetric respectively.

3. Main Results

Theorem 3.1. In the adjoined -structure space, the components of conharmonic tensor of -manifold are given by the following forms:

and the others are conjugate to the above components.

Proof:

1) Put and we have

According to the equation (2.1) we deduce that

2) Put and we get

3) Put and it follows that

4) Put and we obtain

Definition 3.2. In the adjoined -structure space, an almost Hermitian manifold is a manifold of class:

We call a conharmonic para manifold.

Theorem 3.3. Let be a -manifold of class with -invariant Ricci tensor, then is a manifold of class if and only if, is a manifold of flat Ricci tensor.

Proof:

To prove is a manifold of class we must prove that

Let be a manifold of class according to definition 3.2, we have

(3.1)

According to theorem 3.1, we have

(3.2)

By using the equation (3.1), we get

According to theorems 2.6 and 3.1, we deduce

Since, has -invariant Ricci tensor, then

(3.3)

Also, by the equation (3.1), we deduce

By using theorems 2.6 and 3.1, we have

Suppose that is a manifold of flat Ricci tensor, then

(3.4)

Hence, according to the equations, (3.2), (3.3) and (3.4), we get

Conversely, by using the equation (3.1), we have

By using theorem 3.1, it follows that

Let be a manifold of class then, according to theorem 2.6 and definition 2.7, we obtain

Symmetrizing by the indexes we get

Antisymmetrizing by the indexes we have

Symmetrizing by the indexes we deduce

Contracting by the indexes we have

Since has -invariant Ricci tensor, then

Theorem 3.4. Suppose that is -manifold of class with -invariant Ricci tensor, then is a manifold of class if and only if, is a manifold of flat Ricci tensor.

Proof:

Suppose that is a manifold of class According to definition 3.2., we have

By using the Theorem 3.1, we deduce

(3.5)

Let be a manifold of class then

Symmetrizing by the indexes we get

Antisymmetrizing by the indexes we have

Contracting by the indexes it follows that

Since has -invariant Ricci tensor, then

Conversely, by using the equation (3.5), we have

Suppose that is a manifold of flat Ricci tensor, then

Therefore, is a manifold of class

The following theorem gives the necessary and sufficient condition in which an -manifold is an Einstein manifold.

Theorem 3.5. Suppose that is conharmonically flat -manifold with -invariant Ricci tenor. Then the necessary and sufficient condition that an Einstein manifold, is where is a constant.

Proof:

Let be a conharmonically flat -manifold. According to the definition 2.4 and theorem 3.1, we have

Contracting by the indexes it follows that

Symmetrizing by the indexes we deduce

Antisymmetrizing by the indexes it follows that

(3.6)

Let be an Einstein manifold, then

Conversely, by using the equation (3.6), we have

Since, we deduce

where, represent an Einstein constant.

Since has -invariant Ricci tensor. Therefore, is an Einstein manifold.

4. Conclusions

The present work is devoted to study the flatness of conharmonic curvature tensor of Vaisman-Gray manifold. We found out an interesting application in theoretical physics. In particular, we found the necessary and sufficient condition that a conharmonically flat Vaisman-Gray manifold is an Einstein manifold.