Kyungpook Mathematical Journal 2018; 58(4): 789-799
Vaisman-Gray Manifold of Pointwise Holomorphic Sectional Conharmonic Tensor
Habeeb Mtashar Abood*, and Yasir Ahmed Abdulameer
Department of Mathematics,College of Education for Pure Sciences, University of Basrah, Basrah, Iraq
e-mail : iraqsafwan2006@gmail.com and yasirmath2017@gmail.com
*Corresponding Author.
Received: October 29, 2017; Revised: August 14, 2018; Accepted: August 20, 2018; Published online: December 23, 2018.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

The purpose of the present paper is to discuss the geometrical properties of the Vaisman-Gray manifold (VG-manifold) of a pointwise holomorphic sectional conharmonic tensor (PHT-tensor). Furthermore, the necessary and sufficient conditions required for the VG-manifold to admit such a PHT-tensor have been determined. In particular, under certain conditions, we have established that the aforementioned manifold was an Einstein manifold.

Keywords: Vaisman-Gray manifold, holomorphic sectional tensor, conharmonic tensor.
1. Introduction

The classification of the almost Hermitian structures was introduced by Gray and Hervella [4]. These structures have been categorized into sixteen different classes. Moreover, it has been found that the condition for each one of them depends on a Kozel’s operator method [15].

On the other hand, there is another significant classification method for the almost Hermitian structures that were introduced by Kirichenko. This method depends on the principle fibre bundle space of all complex frames of a smooth manifold M with the unitary structure group U(n). This space is called an adjoined G-structure space. For further information, refer to the following citations: [3], [8], [9], [10], and [11].

One of the interesting classes of almost Hermitian structures is a VG-manifold, which is denoted by W1W4, where W1 and W4 denote the nearly Kähler manifold and the locally conformal Kähler manifold, respectively.

It is a well-known fact that the harmonic function is one whose Laplacian vanishes. In general, it is not a conformal transformation harmonic function. With regard to this fact, Ishi [6] introduced a tensor that remained invariant under a conharmonic transformation for an n-dimensional Riemannian manifold. In addition, Khan [7] determined the properties of the conharmonically flat Sasakian manifolds. Moreover, it has been proved that a special weakly Ricci symmetric Sasakian manifold is an Einstein manifold. Subsequently, Shihab [13] went on to determine the geometrical properties of the conharmonic curvature tensor belonging to the nearly Kähler manifold. Furthermore, it has been established that a Kähler manifold with a dimension greater than four is a conharmonic parakähler manifold if, and only if, it has a flat Ricci tensor. On the other hand, Zengin and Tasci [19] studied a pseudo conharmonically symmetric manifold. In particular, they proved that the aforementioned manifold with a non-zero scalar curvature has a closed associated 1-form. Lastly, Abood and Abdulameer [1] considered the conharmonically flat VG-manifold, exclusively and identified the necessary and sufficient conditions required for the VG-manifold to be an Einstein manifold.

In this article, we have employed the adjoined G-structure space to study the geometry of the VG-manifold that corresponds with a PHT-tensor.

2. Preliminaries

Let M be a smooth manifold of even dimension, C(M) be an algebra of smooth functions on M, and X(M) be the module of smooth vector fields on M. An almost Hermitian manifold (AH-manifold) is a triple {M, J, g = 〈.,.〉}, where M is a smooth manifold, J is an almost complex structure, and g = 〈., .〉 is a Riemannian metric, such that the equality 〈JX, JY〉 = 〈X, Y〉 holds for X, YX(M).

Suppose that $Tpc(M)$ is the complexification of a tangent space Tp(M) at the point pM and {e1, …, en, Je1, …, Jen} is a real adapted basis of AH-manifold. Then, in the module $Tpc(M)$, there exists a basis given by {ɛ1, …, ɛn, ε̂1, …, ε̂n} which is called as an adapted basis, where, ɛa = σ(ea), ε̂a = σ̄(ea), and σ, σ̄ are two endomorphisms in the module Xc(M), which are given by $σ=12(id--1Jc)$ and $σ¯=-12(id+-1Jc)$, respectively, such that Xc(M) and Jc are the complexifications of X(M) and J, respectively. The corresponding frame of this basis is {p; ɛ1, … ɛn, ε̂1, …, ε̂n}. Suppose that the indexes i, j, k, and l are in the range 1, 2, …, 2n and the indexes a, b, c, d and f are in the range 1, 2, …, n. Moreover, â = a + n.

For a manifoldM, it is a well known that the given AH-structure is equivalent to the given G-structure space in the principle fibre bundle of all complex frames of M with the unitary structure group U(n). Whereas, in the adjoined G-structure space, the components matrices of the almost complex structure J and the Riemannian metric g are given as follows:

$(Jji)=(-1In00--1In), (gij)=(0-InIn0)$

where In is the identity matrix of order n.

### Definition 2.1.([14])

A Riemannian curvature tensor R of a smooth manifold M is a 4-covariant tensor R: Tp(M) × Tp(M) × Tp(M) × Tp(M) → ℝ which is given by

$R(X,Y,Z,W)=g(R(X,Y)Z,W)$

Moreover, the following properties:

R(X, Y, Z, W) = −R(Y, X, Z, W);

R(X, Y, Z, W) = −R(X, Y, W, Z);

R(X, Y, Z, W) = R(Z, W, X, Y);

R(X, Y, Z, W) + R(X, Z, W, Y) + R(X, W, Y, Z) = 0

hold, where R(X, Y)Z = ([∇X, ∇Y] − ∇[X,Y])Z; X, Y, Z, WTp(M).

### Definition 2.2.([17])

A Ricci tensor is a tensor of type (2, 0) which is a contracting of the Riemannian curvature tensor R, that is

$rij=Rijkk=gklRkijl.$

### Definition 2.3.([6])

A conharmonic tensor of an AH-manifold is a tensor T of type (4, 0) which is given by the following form:

$Tijkl=Rijkl-12(n-1)(rilgjk-rjlgik+rjkgil-rikgjl)$

where r, R and g are respectively the Ricci tensor, the Riemannian curvature tensor, and the Riemannian metric. Similar to the property of Riemannian curvature tensor, the conharmonic tensor has the following property:

$Tijkl=-Tjikl=-Tijlk=Tklij.$

### Definition 2.4

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

### Definition 2.5.([3])

In the adjoined G-structure space, an AH-manifold {M, J, g = 〈., .〉} is called a Vaisman-Gray manifold (VG-manifold) if Babc = −Bbac, $Bcab=α[aδcb]$; a locally conformal Kähler manifold (LCK-manifold) if Babc = 0 and $Bcab=α[aδcb]$; and a nearly Kähler manifold (NK-manifold) if Babc = −Bbac and $Bcab=0$, where $Babc=-12J[b^,c^]a,Bcab=-12Jb^,c]a,α=1(n-1)δF∘J$ is a Lie form, F is a Kähler form which is given by F(X, Y) = 〈JX, Y〉, δ is a codrivative; X, YX(M) and the bracket [ ] denotes the antisymmetric operation.

### Theorem 2.6.([3])

In the adjoined G-structure space, the components of the Riemannian curvature tensor of the VG-manifold are given by the following forms:

Rabcd = 2(Bab[cd] + α[aBb]cd);

$Ra^bcd=2Abcda$;

$Ra^b^cd=2(-BabhBhcd+α[c[aδd]b])$;

$Ra^bcd^=Abcad+BadhBhbc-BcahBhbd$,

where, {$Abcda$ } are some functions on the adjoined G-structure space, {$Abcad$ } are a system of functions in the adjoined G-structure space that are symmetric by the lower and upper indices, which are called the components of the holomorphic sectional curvature tensor.

The functions {$αba,αab$ } are the components of the covariant differential structure tensor of the first and second type, and {αab, αab} are the components of the Lie form on the adjoined G-structure space such that:

$dαa+αbωab=αabωb+αabωb and dαa-αbωba=αbaωb+αabωb,$

where, {ωa, ωa} are the components of mixture form and {$ωba$ } are the components of the Riemannian connection of the metric g. Other components of the Riemannian curvature tensor R can be obtained by the property of symmetry for R.

### Theorem 2.7.([5])

In the adjoined G-structure space, the components of Ricci tensor of the VG-manifold are given by the following forms:

$rab=1-n2 (αab+αba+αaαb)$;

$ra^b=3BcahBcbh-Abcca+n-12 (αaαb-αhαh)-12αhhδba+(n-2)αba$.

Whereas, the other components are conjugate to the above components.

The next theorem gives the components of the conharmonic tensor of the VG-manifold in the adjoined G-structure space.

### Theorem 2.8.([1])

In the adjoined G-structure space, the components of the conharmonic tensor of the VG-manifold are given by the following forms:

Tabcd = 2(Bab[cd] + α[aBb]cd);

$Ta^bcd=2Abcda+12(n-1)(rbdδca-rbcδda)$;

$Ta^b^cd=2(-BabhBhcd+α[c[aδd]b])-1n-1(rd[aδcb]+rc[bδda])$;

$Ta^bcd^=Abcad+BadhBhbc-BcahBhbd+1n-1(r(c(aδb)d))$,

Whereas, the other components can be obtained by the conjugate operation regarding the above components.

### Definition 2.9.([16])

A Riemannian manifold is called an Einstein manifold if the Ricci tensor satisfies the equation rij = Cgij, where C is an Einstein constant.

### Definition 2.10.([12])

An AH-manifold has a J-invariant Ricci tensor if Jr = rJ.

The following Lemma shows the invariant Ricci tensor in the adjoined G-structure space.

### Lemma 2.11.([18])

An AH-manifold has a J-invariant Ricci tensor if, and only if, the equality$rba^=rab=0$holds.

### Definition 2.12.([8])

Define two endomorphisms on $τr0(V)$ as follows:

Symmetric mapping Sym: $τr0(V)→τr0(V)$ by:

$sym(t) (v1,…,vr)=1r!∑σ∈Srt(vσ(1),…,vσ(r)).$

Antisymmetric mapping Alt: $τr0(V)→τr0(V)$by:

$Alt(t) (v1,…,vr)=1r!∑σ∈Sr=ɛ(σ)t(vσ(1),…,vσ(r)).$

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

3. The main results

### Definition 3.1

Let M be an AH-manifold. A holomorphic sectional conharmonic (HT-tensor) of a manifold M in the direction XX(M), X ≠ 0 is a function h(X), which is given by

$〈T(X,JX,X,JX,)〉=h(X)‖X‖4; ‖X2‖=〈X,X〉.$

### Definition 3.2

A manifold M has a pointwise holomorphic sectional conharmonic (PHT- tensor) if h does not depend on X, then this means

$〈T(X,JX,X,JX,)〉=h‖X‖4; X∈X(M), h∈C∞(M).$

### Lemma 3.3.([2])

If M is an AH-manifold of PHT- tensor, then the equation$‖X‖4=2δadb˜cXaXdXbXc$holds, where$δ˜adbc=δabδdc+δdbδac$is a Kroneker delta of the second type.

The necessary condition for a VG-manifold to be a PHT-tensor is summarized in the following theorem.

### Theorem 3.4

Suppose that M is a VG-manifold of the conharmonic tensor and the J-invariant Ricci tenor. Then, the necessary condition for a VG-manifold to be a PHT- tensor is for the components of the HT-curvature tensor to satisfy the following condition:

$Aadbc=c2δ˜adbc+BachBhdb-1n-1(r(d(bδa)c)).$
Proof

Suppose that M is a VG-manifold of the PHT-tensor.

According to the Definition 3.2, we have

$〈T(X,JX,X,JX,)〉=c‖X‖4.$

By using the Lemma 3.3, it follows that

$〈T(X,JX,X,JX,)〉=2cδ˜adbcXaXdXbXc.$

In the adjoined G-structure space, we have

$TijklXi(JX)jXk(JX)l=TabcdXa(JX)bXc(JX)d+Tabc^dXa(JX)bXc^(JX)d+Tabcd^Xa(JX)bXc(JX)d^+Tabc^d^Xa(JX)bXc^(JX)d^+Tab^cdXa(JX)b^Xc(JX)d+Tab^c^dXa(JX)b^Xc^(JX)d+Tab^cd^Xa(JX)b^Xc(JX)d^+Tab^c^d^Xa(JX)b^Xc^(JX)d^+Ta^bcdXa^(JX)bXc(JX)d+Ta^bc^dXa^(JX)bXc^(JX)d+Ta^bcd^Xa^(JX)bXc(JX)d^+Ta^bc^d^Xa^(JX)bXc^(JX)d^+Ta^b^cdXa^(JX)b^Xc(JX)d+Ta^b^c^dXa^(JX)b^Xc^(JX)d+Ta^b^cd^Xa^(JX)b^Xc(JX)d^+Ta^b^c^d^Xa^(JX)b^Xc^(JX)d^.$

According to the properties $(JX)a=-1Xa$ and $(JX)a^=--1Xa^$, it follows that

$TijklXi(JX)jXk(JX)l=-TabcdXaXbXcXd-Tabc^dXaXbXc^Xd+Tabcd^XaXbXcXd^+Tabc^d^XaXbXc^Xd^+Tab^cdXa(X)b^XcXd^+Tab^c^dXaXb^Xc^Xd-Tab^cd^XaXb^XcXd^-Tab^c^d^XaXb^Xc^Xd^-Ta^bcdXa^XbXcXd-Ta^bc^dXa^XbXc^Xd+Ta^bcd^Xa^XbXcXd^+Ta^bc^d^Xa^XbXc^Xd^+Ta^b^cdXa^Xb^XcXd+Ta^b^c^dXa^Xb^Xc^Xd+Ta^b^cd^Xa^Xb^XcXd^-Ta^b^c^d^Xa^Xb^Xc^X)d^.$

By using the properties of the conharmonic tensor, we get the following:

$TijklXi(JX)jXk(JX)l=-TabcdXaXbXcXd-4Tab^c^d^XaXb^Xc^Xd^-4Ta^bcdXa^XbXcXd+4Tab^c^dXaXb^Xc^Xd-Ta^b^c^d^Xa^Xb^Xc^X)d^+2Ta^b^cdXa^Xb^XcXd=2cδ˜adbcXaXdXbXc.$

Making use of the Theorem 2.3, we obtain

$-2(Bab[cd]+α[aBb]cd)XaXbXcXd-4(2Aabcd+12(n-1)(rb^[dδac]))XaXb^Xc^Xd^-4(2Abcda+12(n-1)(rbdδca-rbcδda))Xa^XbXcXd+4(Aadbc+BbchBhad-BachBhdb+1n-1(r(d(bδa)c)))XaXb^Xc^Xd-2(Bab[cd]+α[aBb]cd)Xa^Xb^Xc^Xd^+2(2(-BabhBhcd+α[c[aδd]b])-1n-1(rd[aδcb]+rd[bδda]))Xa^Xb^XcXd=2cδ˜adbcXaXdXbXc.$

Symmetrizing and antisymmetrizing the last equation by the indices (c, d), we have

$-4(12(n-1))(rbdδca-rbcδda))Xa^XbXcXd+4(Aadbc+BbchBhad-BachBhdb+1n-1(r(d(bδa)c)))XaXb^Xc^Xd=2cδ˜adbcXaXdXbXc.$

Since M has a J-invariant Ricci tensor, then

$6(Aadbc+BbchBhad-BachBhdb+1n-1(r(d(bδa)c)))XaXb^Xc^Xd=2cδ˜adbcXaXdXbXc.$

Symmetrizing by the indices (b, c), we deduce

$4(Aadbc+12(BbchBhad+BcbhBhad)-BachBhdb+1n-1(r(d(bδa)c)))XaXb^Xc^Xd=2cδ˜adbcXaXdXbXc,4(Aadbc-BachBhdb+1n-1(r(d(bδa)c)))XaXb^Xc^Xd=2cδ˜adbcXaXdXbXc,4(Aadbc-BachBhdb+1n-1(r(d(bδa)c)))XaXdXbXc=2cδ˜adbcXaXdXbXc,$$Aadbc-BachBhdb+1n-1(r(d(bδa)c))=c2δ˜adbc.$

Therefore,

$Aadbc=c2δ˜adbc+BachBhdb-1n-1(r(d(bδa)c)).$

### Theorem 3.5

If M is a VG-manifold of the PHT-tensor, then M is an NK-manifold.

Proof

Suppose that M is a manifold of the PHT-tensor. According to the Theorem 3.4, we get

$Aadbc-BachBhdd+1n-1(r(d(bδa)c))=c2δ˜adbc.$

Symmetrizing and antisymmetrizing the above equation by the indices (b, c) consequently, we get

$BachBhdb=0.$

Contracting the last equation by the indices (a, b) and (c, d), it follows that

$BachBhca=0⇔BachB¯ach=0⇔∑a,d,h∣Bach∣2=0⇔Bach=0.$

Therefore, according to the Definition 2.5, M is an NK-manifold.

### Theorem 3.6

Let M be a VG-manifold of the PHT-tensor with a flat holomorphic sectional curvature tensor and J-invariant Ricci tensor. Then, M is an Einstein manifold.

Proof

Suppose thatM is a VG-manifold of the PHT-curvature tensor. According to the Theorem 3.4, we get

$Aadbc-BachBhdd+1n-1(r(d(bδa)c))=c2δ˜adbc.$

Making use of the Theorem 3.5 consequently, we obtain

$Aadbc+1n-1(r(d(bδa)c))=c2δ˜adbc.$

Since M has flat holomorphic sectional tensor, then

$1n-1(r(d(bδa)c))=c2δ˜adbc,12(n-1)(rdbδac+racδdb)=c2(δabδdc+δdbδac).$

Contracting by the indices (d, c), we have

$12(n-1)(rdbδad+radδdb) =c2(δabδdd+δdbδad)12(n-1)(rab+rab) =c2(nδab+δab),1(n-1)rab =c2δab(n+1),rab =c(n2-1)2δab,rab =eδab.$

Since M has a J-invariant Ricci tensor, then M is an Einstein manifold.

### Theorem 3.7

Let M be a VG-manifold of the PHT-tensor with a J-invariant Ricci tensor, then M is an Einstein manifold if, and only if, $Aacbc=c1δab$, where c1is a constant.

Proof

Suppose that M is a VG-manifold of the PHT-tensor. According to the Theorem 3.4, we have

$Aadbc-BachBhdd+1n-1(r(d(bδa)c))=c2δ˜adbc.$

By using the Theorem 3.5, we obtain

$Aadbc+1n-1(r(d(bδa)c)) =c2δ˜adbc,Aadbc+12(n-1)(rdbδac+racδdb) =c2δ˜adbc,Aadbc+12(n-1)(rdbδac+racδdb) =c2(δabδdc+δdbδac).$

Contracting by the indices (c, d), it follows that

$Aacbc+12(n-1)(rcbδac+racδcb) =c2(δabδcc+δcbδac),Aacbc+12(n-1)(rab+rab) =c2(nδab+δab),Aacbc+1(n-1)rab =cδab2(n+1),Aacbc =cδab2(n+1)-1(n-1)rab.$

Since M is an Einstein manifold, then

$Aacbc=cδab2(n+1)-1(n-1)eδab,Aacbc={c2(n+1)-1(n-1)e}δab,Aacbc=c1δab.$

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

$Aacbc=cδab2(n+1)-1(n-1)rab.$

Since $Aacbc=c1δab$, then

$c1δab =cδab2(n+1)-1(n-1)rab,1(n-1)rab ={c2(n+1)-c1}δab,rab =(c(n+1)-3c1)(n-1)2δab,rab =eδab.$

Since M has a J-invariant Ricci tensor, then M is an Einstein manifold.

4. Conclusions

This article clearly aimed to study the geometrical properties of the VG-manifold of the pointwise holomorphic sectional curvature conharmonic tensor. We have found out the necessary conditions for the VG-manifold to be a manifold of the pointwise holomorphic sectional conharmonic tensor. Furthermore, we have formulated an interesting theoretical physical application. In particular, we have concluded the necessary and sufficient conditions for a VG-manifold to be an Einstein manifold.

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