On Norden-Walker 4-manifolds

On Norden-Walker 4-manifolds

Arif A. Salimov

Baku State University, Department of Algebra and Geometry, Baku, 370145, Azerbaijan and Ataturk University, Faculty of Science, Department of Mathematics,

Erzurum, Turkey

asalimov@atauni.edu.tr

Murat Iscan

Ataturk University, Faculty of Science, Department of Mathematics,

Erzurum, Turkey miscan@atauni.edu.tr

Received: 22.9.2008; accepted: 25.11.2009.

Abstract. A Walker 4-manifold is a semi-Riemannian manifold (M

, g) of neutral signature, which admits a field of parallel null 2-plane. The main purpose of the present paper is to study almost Norden structures on 4-dimensional Walker manifolds with respect to a proper and opposite almost complex structures. We discuss sequently the problem of integrability, K¨ ahler (holomorphic), isotropic K¨ ahler and quasi-K¨ ahler conditions for these structures. The curva- ture properties for Norden-Walker metrics is also investigated. Also, we give counterexamples to Goldberg’s conjecture in the case of neutral signature.

Keywords: Walker 4-manifolds, Proper almost complex structure, Opposite almost complex structure, Norden metrics, Holomorphic metrics, Goldberg conjecture

MSC 2000 classification: primary 53C50, secondary 53B30

1 Introduction

Let M

be a Riemannian manifold with neutral metric, i.e., with pseudo-Riemannian metric g of signature (n, n). We denote by ℑ

(M

) the set of all tensor fields of type (p, q) on M

. Manifolds, tensor fields and connections are always assumed to be differentiable and of class C

.

Let (M

, ϕ) be an almost complex manifold with almost complex structure ϕ. Such a structure is said to be integrable if the matrix ϕ = (ϕ

) is reduced to constant form in a certain holonomic natural frame in a neighborhood U

of every point x ∈ M

. In order that an almost complex structure ϕ be integrable, it is necessary and sufficient that there exists a torsion-free affine connection ∇ with respect to which the structure tensor ϕ is covariantly constant, i.e.,∇ϕ = 0. It is also know that the integrability of ϕ is equivalent to the vanishing

i

This work is supported by The Scientific and Technological Research Council of Turkey (TBAG-108T590).

http://siba-ese.unisalento.it/ c 2010 Universit` a del Salento

of the Nijenhuis tensor N

∈ ℑ

(M

). If ϕ is integrable, then ϕ is a complex structure and, moreover, M

is a C-holomorphic manifold X

(C) whose transition functions are holomorphic mappings.

1.1 Norden metrics

A metric g is a Norden metric [18] if

g(ϕX, ϕY ) = −g(X, Y ) or equivalently

g(ϕX, Y ) = g(X, ϕY )

for any X, Y ∈ ℑ

(M

). Metrics of this type have also been studied under the other names:

pure metrics, anti-Hermitian metrics and B-metrics (see [5], [6], [10], [17], [19], [23], [25]). If (M

, ϕ) is an almost complex manifold with Norden metric g, we say that (M

, ϕ, g) is an almost Norden manifold. If ϕ is integrable, we say that (M

, ϕ, g) is a Norden manifold.

1.2 Holomorphic (almost holomorphic) tensor fields

Let

t be a complex tensor field on a C-holomorphic manifold X

(C). The real model of such a tensor field is a tensor field on M

of the same order irrespective of whether its vector or covector arguments is subject to the action of the affinor structure ϕ. Such tensor fields are said to be pure with respect to ϕ. They were studied by many authors (see, e.g., [10], [20], [21], [23], [24], [25], [27]). In particular, for a (0, q)-tensor field ω, the purity means that for any X

, ..., X

∈ ℑ

(M

), the following conditions should hold:

ω(ϕX

, X

, ..., X

) = ω(X

, ϕX

, ..., X

) = ... = ω(X

, X

, ..., ϕX

).

We define an operator

Φ

: ℑ

(M

) → ℑ

(M

) applied to a pure tensor field ω by (see [27])

(Φ

ω)(X, Y

, Y

, ..., Y

) = (ϕX)(ω(Y

, Y

, ..., Y

)) − X(ω(ϕY

, Y

, ..., Y

)) +ω((L

ϕ)X, Y

, ..., Y

) + ... + ω(Y

, Y

, ..., (L

ϕ)X), where L

denotes the Lie differentiation with respect to Y .

When ϕ is a complex structure on M

and the tensor field Φ

ω vanishes, the complex tensor field ω on X

(C) is said to be holomorphic (see [10], [23], [27]). Thus, a holomorphic tensor field ω on X

(C) is realized on M

in the form of a pure tensor field ω, such that

(Φ

ω)(X, Y

, Y

, ..., Y

) = 0

for any X, Y

, ..., Y

∈ ℑ

(M

). Such a tensor field ω on M

is also called holomorphic tensor field. When ϕ is an almost complex structure on M

, a tensor field ω satisfying Φ

ω = 0 is said to be almost holomorphic.

1.3 Holomorphic Norden (K¨ ahler-Norden or anti-K¨ ahler) met- rics

On a Norden manifold, a Norden metric g is called a holomorphic if

(Φ

g)(X, Y, Z) = −g((∇

ϕ)Y, Z) + g((∇

ϕ)Z, X) + g((∇

ϕ)X, Y ) = 0 (1)

for any X, Y, Z ∈ ℑ

(M

).

By setting X = ∂

, Y = ∂

, Z = ∂

in equation (1), we see that the components (Φ

g)

of Φ

g with respect to a local coordinate system x

, ..., x

can be expressed as follows:

(Φ

g)

= ϕ

∂

g

− ϕ

∂

g

+ g

(∂

ϕ

− ∂

ϕ

) + g

∂

ϕ

.

If (M

, ϕ, g) is a Norden manifold with holomorphic Norden metric, we say that (M

, ϕ, g) is a holomorphic Norden manifold.

In some aspects, holomorphic Norden manifolds are similar to K¨ ahler manifolds. The following theorem is an analogue to the next known result: an almost Hermitian manifold is K¨ ahler if and only if the almost complex structure is parallel with respect to the Levi-Civita connection.

Theorem 1. [8] (For a paracomplex version see [22]) For an almost complex manifold with Norden metric g , the condition Φ

g = 0 is equivalent to ∇ϕ = 0, where ∇ is the Levi- Civita connection of g.

A K¨ ahler-Norden manifold can be defined as a triple (M

, ϕ, g) which consists of a man- ifold M

endowed with an almost complex structure ϕ and a pseudo-Riemannian metric g such that ∇ϕ = 0, where ∇ is the Levi-Civita connection of g and the metric g is assumed to be a Norden one. Therefore, there exists a one-to-one correspondence between K¨ ahler-Norden manifolds and Norden manifolds with holomorphic metric. Recall that the Riemannian cur- vature tensor of such a manifold is pure and holomorphic, and the scalar curvature is locally holomorphic function (see [8], [19]).

Remark 1. We know that the integrability of an almost complex structure ϕ is equivalent to the existence of a torsion-free affine connection with respect to which the equation ∇ϕ = 0 holds. Since the Levi-Civita connection ∇ of g is a torsion-free affine connection, we have: if Φ

g = 0, then ϕ is integrable. Thus, almost Norden manifold with conditions Φ

g = 0 and N

6= 0, i.e., almost holomorphic Norden manifolds (analogues of almost K¨ ahler manifolds with closed K¨ ahler form) do not exist.

1.4 Quasi-K¨ ahler manifolds

The basis class of non-integrable almost complex manifolds with Norden metric is the class of the quasi-K¨ ahler manifolds. An almost Norden manifold (M

, ϕ, g) is called a quasi- K¨ ahler [17], if

σ

X,Y,Z

g((∇

ϕ)Y, Z) = 0, where σ is the cyclic sum by three arguments.

From (1) and the last equation we have

(Φ

g)(X, Y, Z) + 2g((∇

ϕ)Y, Z) = σ

X,Y,Z

g((∇

ϕ)Y, Z) = 0, which is satisfied by the Norden metric in the quasi-K¨ ahler manifold.

1.5 Twin Norden metrics

Let (M

, ϕ, g) be an almost Norden manifold. The associated Norden metric of almost Norden manifold is defined by

G(X, Y ) = (g ◦ ϕ)(X, Y )

for all vector fields X and Y on M

. One can easily prove that G is a new Norden metric,

which is also called the twin(or dual) Norden metric of g.

We denote by ∇

the covariant differentiation of the Levi-Civita connection of Norden metric g. Then, we have

∇

G = (∇

g) ◦ ϕ + g ◦ (∇

ϕ) = g ◦ (∇

ϕ),

which implies ∇

G = 0 by virtue of Theorem 1. Therefore we have: the Levi-Civita connection of K¨ ahler-Norden metric g coincides with the Levi-Civita connection of twin metric G ( i.e.

nonuniqueness of the metric for the Levi-Civita connection in K¨ ahler-Norden manifolds).

2 Norden-Walker metrics

In the present paper, we shall focus our attention to Norden manifolds of dimension four.

Using a Walker metric we construct new Norden-Walker metrics together with a proper and opposite almost complex structures.

2.1 Walker metric g

A neutral metric g on a 4-manifold M

is said to be a Walker metric if there exists a 2-dimensional null distribution D on M

, which is parallel with respect to g. From Walker’s theorem [26], there is a system of coordinates (x, y, z, t) with respect to which g takes the following local canonical form

g = (g

) =



 



0 0 1 0

0 0 0 1

1 0 a c

0 1 c b



 

 , (2)

where a, b, c are smooth functions of the coordinates (x, y, z, t). The paralel null 2-plane D is spanned locally by {∂

, ∂

}, where ∂

=

, ∂

=

.

2.2 Almost Norden-Walker manifolds

Let F be an almost complex structure on a Walker manifold M

, which satisfies i) F

= −I,

ii) g(F X, Y ) = g(X, F Y ) (Nordenian property),

iii) F ∂

= ∂

, F ∂

= −∂

(F induces a positive

−rotation on D).

We easily see that these three properties define F non-uniquely, i.e.,

 

 

 



F ∂

= ∂

, F ∂

= −∂

,

F ∂

= α∂

+

(a + b)∂

− ∂

, F ∂

= −

(a + b)∂

+ α∂

+ ∂

and F has the local components

F = (F

) =



 



0 −1 α −

(a + b) 1 0

(a + b) α

0 0 0 1

0 0 −1 0



 



with respect to the natural frame {∂

, ∂

, ∂

, ∂

}, where α = α(x, y, z, t) is an arbitrary func- tion.

Therefore, we now put α = c. Then g defines a unique almost complex structure

ϕ = (ϕ

) =



 



0 −1 c −

(a + b) 1 0

(a + b) c

0 0 0 1

0 0 −1 0



 

 . (3)

The triple (M

, ϕ, g) is called almost Norden-Walker manifold. In conformity with the termi- nology of [3], [4], [14], [15] we call ϕ the proper almost complex structure.

We note that the typical examples of Norden-Walker metrics with proper almost complex structure

J = (J

) =



 



0 −1 −c

(a − b) 1 0

(a − b) c

0 0 0 −1

0 0 1 0



 



are studied in [2].

2.3 Isotropic K¨ ahler-Norden-Walker structures

A proper almost complex structure ϕ on Norden-Walker manifold (M

, ϕ, g) is said to be isotropic K¨ ahler if k∇ϕk

= 0, but ∇ϕ 6= 0. Examples of isotropic K¨ ahler structures were given first in [7] in dimension 4, subsequently in [1] in dimension 6 and in [3] in dimension 4. Our purpose in this section is to show that a proper almost complex structure on almost Norden-Walker manifold (M

, ϕ, g) is isotropic K¨ ahler as we will see Theorem 2.

The inverse of the metric tensor (2), g

= (g

), given by

g

=



 



−a −c 1 0

−c −b 0 1

1 0 0 0

0 1 0 0



 

 . (4)

For the covariant derivative ∇ϕ of the almost complex structure put (∇ϕ)

= ∇

ϕ

. Then, after some calculations we obtain

∇

ϕ

= ∇

ϕ

= c

, ∇

ϕ

= ∇

ϕ

= c

, (5)

∇

ϕ

= −∇

ϕ

= ∇

ϕ

= −∇

ϕ

= 1 2 a

+ 1

2 c

,

∇

ϕ

= ∇

ϕ

= ∇

ϕ

= ∇

ϕ

= − 1 2 a

+ 1

2 c

,

∇

ϕ

= 2c

+ ca

− a

− 1 2 cc

− 1

2 ac

+ 1 2 ba

,

∇

ϕ

= a

+ 1 4 ac

− 1

4 bc

+ ca

+ 3 4 aa

+ 1

4 ba

,

∇

ϕ

= 1 4 aa

− 1

4 ba

+ ca

+ 3

4 bc

+ cc

+ 1 4 ac

,

∇

ϕ

= a

+ 1 4 ac

− 1

4 bc

+ ca

+ 3 4 aa

+ 1

4 ba

,

∇

ϕ

= 1 4 aa

− 1

4 ba

+ ca

+ 3

4 bc

+ cc

+ 1

4 ac

,

∇

ϕ

= 2c

+ 1

2 cc

− a

+ 1 2 ba

+ 1

2 ca

− 1 2 ac

,

∇

ϕ

= −∇

ϕ

= ∇

ϕ

= −∇

ϕ

= 1 2 c

+ 1

2 b

,

∇

ϕ

= ∇

ϕ

= ∇

ϕ

= ∇

ϕ

= − 1 2 c

+ 1

2 b

,

∇

ϕ

= 3

2 cc

+ b

− 1 2 cb

− 1

2 ab

+ 1 2 bc

,

∇

ϕ

= 1 4 ab

− 1

4 bb

− 1 4 ac

+ 1

4 bc

,

∇

ϕ

= 1 4 ac

− 1

4 bc

+ cc

+ 1

4 bb

+ cb

− 1 4 ab

∇

ϕ

= 1

2 cb

+ b

+ 1 2 bc

+ 1

2 cc

− 1 2 ab

. Now a long but straightforward calculation shows that

k∇ϕk

= g

g

g

(∇ϕ)

(∇ϕ)

= 0.

Theorem 2. A proper almost complex structure on almost Norden-Walker manifold (M

, ϕ, g) is isotropic K¨ ahler.

2.4 Integrability of ϕ

We consider the general case.

The almost complex structure ϕ of an almost Norden-Walker manifold is integrable if and only if

(N

)

= ϕ

∂

ϕ

− ϕ

∂

ϕ

− ϕ

∂

ϕ

+ ϕ

∂

ϕ

= 0. (6) From (3) and (6) find the following integrability condition.

Theorem 3. The proper almost complex structure ϕ of an almost Norden-Walker mani- fold is integrable if and only if the following PDEs hold:

 a

+ b

+ 2c

= 0,

a

+ b

− 2c

= 0. (7)

From this theorem, we see that, in the case a = −b and c = 0, ϕ is integrable.

Let (M

, ϕ, g) be a Norden-Walker manifolds (N

= 0) and a = b. Then the equation (7)

reduces to 

a

= −c

,

a

= c

, (8)

from which follows

a

+ a

= 0,

c

+ c

= 0, (9)

e.g., the functions a and c are harmonic with respect to the arguments x and y.

Thus we have

Theorem 4. If the triple (M

, ϕ, g) is Norden-Walker and a = b, then a and c are all

harmonic with respect to the arguments x, y.

2.5 Example of Norden-Walker metric

We now apply the Theorem 4 to establish the existence of special types of Norden-Walker metrics. In our arguments, the harmonic function plays an important part.

Let a = b and h(x, y) be a harmonic function of variables x and y, for example h(x, y) = e

cos y. We put

a = a(x, y, z, t) = h(x, y) + α(z, t) = e

cos y + α(z, t)

where α is an arbitrary smooth function of z and t. Then, a is also hormonic with respect to x and y. We have

a

= e

cos y, a

= −e

sin y.

From (8), we have PDE’s for c to satisfy as

c

= a

= −e

sin x, c

= −a

= −e

cos y.

For these PDE’s, we have solutions

c = −e

sin y + β(z, t),

where β is arbitrary smooth function of z and t. Thus the Norden-Walker metric has compo- nents of the form

g = (g

) =



 



0 0 1 0

0 0 0 1

1 0 e

cos y + α(z, t) −e

sin y + β(z, t) 0 1 −e

sin y + β(z, t) e

cos y + α(z, t)



 

 .

3 Holomorphic Norden-Walker(K¨ ahler-Norden- Walker) and quasi-K¨ ahler-Norden-Walker metrics on (M 4 , ϕ, g)

Let (M

, ϕ, g) be an almost Norden-Walker manifold. If

(Φ

g)

= φ

∂

g

− φ

∂

g

+ g

(∂

φ

− ∂

φ

) + g

∂

φ

= 0, (10) then, by virtue of Theorem 1, ϕ is integrable and the triple (M

, ϕ, g) is called a holomorphic Norden-Walker or a K¨ ahler-Norden-Walker manifold. Taking into account Remark 1, we see that an almost K¨ ahler-Norden-Walker manifold with conditions Φ

g = 0 and N

6= 0 does not exist.

Substitute (2) and (3) into (10), we see that the non-vanishing components of (Φ

g)

are

(Φ

g)

= a

, (Φ

g)

= (Φ

g)

= 1

2 (b

− a

) + c

, (11) (Φ

g)

= b

− 2c

, (Φ

g)

= −a

,

(Φ

g)

= (Φ

g)

= 1

2 (b

− a

) − c

, (Φ

g)

= −b

− 2c

, (Φ

g)

= (Φ

g)

= (Φ

g)

= (Φ

g)

= c

,

(Φ

g)

= (Φ

g)

= − (Φ

g)

= − (Φ

g)

= 1

2 (a

+ b

), (Φ

g)

= (Φ

g)

= (Φ

g)

= (Φ

g)

= c

,

(Φ

g)

= (Φ

g)

= − (Φ

g)

= − (Φ

g)

= 1

2 (a

+ b

), (Φ

g)

= ca

− a

+ 2c

+ 1

2 (a + b)a

, (Φ

g)

= (Φ

g)

= cc

+ b

+ 1

2 (a + b)c

, (Φ

g)

= cb

+ a

− 2c

+ 1

2 (a + b)b

, (Φ

g)

= ca

− b

− 1

2 (a + b)a

, (Φ

g)

= (Φ

g)

= cc

− a

+ 2c

− 1

2 (a + b)c

, (Φ

g)

= cb

+ b

− 1

2 (a + b)b

. From the above equations, we have

Theorem 5. A triple (M

, ϕ, g) is a K¨ ahler-Norden-Walker manifold if and only if the following PDEs hold:

a

= a

= b

= b

= b

= c

= c

= 0, a

− 2c

= 0. (12) A Norden-Walker manifold (M

, ϕ, g) satisfying the condition Φ

g

+ 2∇

G

to be zero is called a quasi-K¨ ahler manifold, where G is defined by G

= ϕ

g

.

Remark 2. From (2) and (3) we easily see that, the twin Norden metric G is non-Walker.

For the covariant derivative ∇G of the associated metric G put (∇G)

= ∇

G

. The non-vanishing components of ∇

G

are

∇

G

= ∇

G

= c

, ∇

G

= ∇

G

= c

, (13)

∇

G

= ∇

G

= −∇

G

= −∇

G

= 1

2 (a

+ c

),

∇

G

= ∇

G

= ∇

G

= ∇

G

= 1

2 (c

− a

),

∇

G

= 2c

− a

+ 1

2 a

(a + b) + ca

,

∇

G

= ∇

G

= 1

2 (ca

+ cc

) − 1

4 ((a + b)(a

−c

)),

∇

G

= 2c

− a

− 1

2 c

(a + b) + cc

,

∇

G

= ∇

G

= −∇

G

= −∇

G

= 1

2 (b

+ c

),

∇

G

= ∇

G

= ∇

G

= ∇

G

= 1

2 (b

− c

),

∇

G

= b

+ cc

+ 1

2 c

(a + b),

∇

G

= ∇

G

= 1

2 c(b

+ c

) − 1

4 ((c

− b

)(a + b)),

∇

G

= b

+ cb

− 1

2 b

(a + b).

From (11) and (13) we have

Theorem 6. A triple (M

, ϕ, g) is a quasi-K¨ ahler Norden-Walker manifold if and only if the following PDEs hold:

b

= b

= b

= 0, a

− 2c

= 0, a

− 2c

= 0, ca

− a

+ 2c

− (a + b)c

= 0.

4 Curvature properties of Norden-Walker manifolds

If R and r are respectively the curvature and the scalar curvature of the Walker metric, then the components of R and r have, respectively, expressions (see [15], Appendix A and C) R

= −

a

, R

= −

c

, R

= −

a

, R

= −

c

, (14)

R

=

a

−

c

−

a

b

+

c

c

, R

= −

b

, R

= −

c

, R

= −

b

, R

=

c

−

b

−

(c

)

+

a

b

−

b

c

+

b

c

, R

= −

a

, R

= −

c

,

R

=

a

−

c

−

a

c

+

a

c

−

a

b

+

(c

)

, R

= −

b

, R

=

c

−

b

−

c

c

+

a

b

,

R

= c

−

a

−

b

−

a(c

)

+

aa

b

+

ca

b

−

cc

c

−

a

c

+

a

c

−

a

b

+

ca

b

+

ba

b

−

b(c

)

−

b

c

+

a

b

+

a

b

+

b

c

−

a

b

. and

r = a

+ 2c

+ b

. (15)

Suppose that the triple (M

, ϕ, g) is K¨ ahler-Norden-Walker. Then from the last equation in (12) and (14), we see that

R

= c

− 1

2 a

= − 1

2 (a

− 2c

)

= 0.

From (12) we easily we see that the another components of R in (14) directly all vanish. Thus we have

Theorem 7. If a Norden-Walker manifold (M

, ϕ, g) is K¨ ahler-Norden-Walker, then M

is flat.

Remark 3. We note that a K¨ ahler-Norden manifold is non-flat, in such manifold curva- ture tensor pure and holomorphic [8].

Let (M

, ϕ, g) be a Norden-Walker manifold with the integrable proper structure ϕ, i.e.,

N

= 0. If a = b, then from proof of the Theorem 4 we see that the equation (8) hold. If

c = c(y, z, t) and c = c(x, z, t), then c

= (c

)

= (c

)

= 0. In these cases, by virtue of (8)

we find a = a(x, z, t) and a = (y, z, t) respectively. Using of c

= 0 and a

+ b

= 0 (see

(9)), we from (15) obtain r = 0. Thus we have

Theorem 8. If (M

, ϕ, g) is a Norden-Walker non-K¨ ahler manifold with metrics

g =



 



0 0 1 0

0 0 0 1

1 0 a(x, z, t) c(y, z, t) 0 1 c(y, z, t) a(x, z, t)



 

 , ˜ g =



 



0 0 1 0

0 0 0 1

1 0 a(y, z, t) c(x, z, t) 0 1 c(x, z, t) a(y, z, t)



 

 ,

then M

is scalar flat.

5 On the Goldberg conjecture

Let (M

, J, g) be an almost Hermitian manifold. Then, Goldberg’s conjecture states that an almost Hermitian manifold must be K¨ ahler if the following three conditions are imposed:

(G

) the manifold M

is compact; (G

) the Riemannian metric g is Einstein; (G

) the fun- damental 2-form Ω defined by Ω(X, Y ) = g(JX, Y ) is closed (dΩ = 0).

It should be noted that no progress has been made on the Goldberg conjecture, and the orginal conjecture is stil an open problem.

Let (M

, ϕ, g) be an almost Norden manifold. Given an almost complex structure ϕ on M

, take any Riemannian metric ˜ g, which exists provided M

is compact (paracompact) [9, p. 60]. We obtain a Hermitian metric h by setting

h(X, Y ) = ˜ g(X, Y ) + ˜ g(ϕX, ϕY )

for any X, Y ∈ ℑ

(M

). The pair (ϕ, ˜ g) defines a fundamental 2-form Ω

by Ω

(X, Y ) = h(ϕX, Y ).

We call it a ϕ-compatible 2-form.

Let (M

, ϕ, g) be an almost Norden manifold, and choose a ϕ-compatible 2-form Ω

on M

. Then we can propose an almost Norden version of Goldberg conjecture as follows [16]:

if (G

) M

is compact, (G

) g is Einstein, and if (G

) a ϕ-compatible 2-form Ω

is closed, then ϕ must be integrable.

Let now (M

, ϕ, g) be an almost Norden-Walker 4-manifold. The pair (ϕ, g) defines as usual, a rank two tensor G(X, Y ) = g(ϕX, Y ), but G is symmetric (in fact another neutral metric) and pure, rather than a 2-form. We call it a twin Norden metric, which plays a role similar to the fundamental 2-form Ω in Hermitian geometry. If we define an operator Φ

applied to a pure twin metric G, then we have

(Φ

G)(X, Y, Z) = (Φ

g)(ϕX, Y, Z) + g(N

(X, Y ), Z).

If G ∈ KerΦ

, then by virtue of Theorem 1, we have ∇

ϕ = 0, where ∇

is the Levi-Civita connection of the twin Norden metric G, which coincides with the Levi-Civita connection of the orginal Norden metric g in K¨ ahler-Norden-Walker manifolds. Since ∇

is a torsion-free connection, then ϕ must be integrable. Thus, we can propose a result concerning the Norden version of Goldberg conjecture as follows: (N G) if G ∈ KerΦ

, then ϕ must be integrable.

6 Opposite almost complex structure ϕ ^′

It is known that an oriented 4-manifold with a field of 2-planes, or equivalently endowed

with a neutral indefinite metric, admits a pair of almost comlex structure ϕ and an opposite

almost complex structure ϕ

, which satisfy the following properties ([11]-[13], [15]):

i) ϕ

= ϕ

= −1,

ii) g(ϕX, ϕY ) = g(ϕ

X, ϕ

Y ) = g(X, Y ), iii) ϕϕ

= ϕ

ϕ,

iv) the preferred orientation of ϕ coincides with that of M

, v) the preferred orientation of ϕ

is opposite to that of M

.

Let (M

, ϕ, g) be an almost Norden-Walker manifolds. For a Walker manifold M

, with the proper almost complex structure ϕ, the g-orthogonal opposite almost complex structure ϕ

takes the form

ϕ

∂

= −(θ

c +

a)∂

−

b∂

+ θ

∂

+ θ

∂

, ϕ

∂

= (−

a + θ

c)∂

+

b∂

+ θ

∂

− θ

∂

, ϕ

∂

= −(

ac +

a

+

1+θ²₂

)∂

− (

ab +

1+θ²₂

)∂

+

a∂

+

a∂

, ϕ

∂

= −(θ

c

+

ab +

1+θ₂²

+

(ac − bc))∂

+ (−

bc +

b

+

)∂

+ (

b + θ

c)∂

+ (θ

c −

b)∂

, where θ

and θ

are two parameters.

In the present paper, we shall focus our attention to one of explicit forms of ϕ

, obtained by fixing two parameters as θ

= 1 and θ

= 0 (only for simplicity), as follows:

ϕ

∂

= −c∂

−

b∂

+ ∂

, ϕ

∂

= −

a∂

+ ∂

, ϕ

∂

= −

ac∂

− (

ab + 1)∂

+

a∂

,

ϕ

∂

= −(c

+

ab + 1)∂

−

bc∂

+

b∂

+ c∂

,

(16)

and ϕ

has the local components

ϕ

= (ϕ

) =



 



−c −

a −

ac −(c

+

ab + 1)

−

b 0 −(

ab + 1) −

bc

0 1 0

b

1 0

a c



 

 . (17)

For the covariant derivative ∇ϕ

of the opposite almost complex structure ϕ

, the non- vanishing components of which are

∇

ϕ

= −∇

ϕ

= ∇

ϕ

= −∇

ϕ

= 1

2 ∇

ϕ

= − 1

2 c

, (18)

∇

ϕ

= −∇

ϕ

= ∇

ϕ

= −∇

ϕ

= 1

2 ∇

ϕ

= − 1 2 c

,

∇

ϕ

= −cc

, ∇

ϕ

= −cc

, ∇

ϕ

= −c

− 1 4 ba

+ 1

2 a

+ 1 2 cc

+ 3

4 ac

,

∇

ϕ

= 1 4 bc

+ 1

4 ba

, ∇

ϕ

= ∇

ϕ

= − 1 2 c

− 1

2 a

,

∇

ϕ

= 1

4 aa

+ ca

+ 1

4 ac

, ∇

ϕ

= c

− 1 4 ac

− 1

2 a

+ 1 2 ca

+ 3

4 ba

,

∇

ϕ

= −a

, ∇

ϕ

= 1

4 aca

− a

+ 1

4 acc

+ 1 4 a

c

,

∇

ϕ

= 1

8 abc

+ 1

8 aba

− 1 2 c

− 1

2 a

,

∇

ϕ

= −c

− 1 4 ac

+ 1

2 a

− 1 2 ca

− 1

4 ba

, ∇

ϕ

= − 1 4 ac

− 1

4 aa

,

∇

ϕ

= −2cc

+ ( 1 8 ab − 1

2 c

+ ac − 1

2 )a

+ ca

+ ( 1 8 ab + 1

2 c

− 1

2 )c

,

∇

ϕ

= −c

+ 1

4 b

a

+ 1

4 bcc

+ 1

4 bca

, ∇

ϕ

= −cc

− 1 4 ba

− 1

4 bc

,

∇

ϕ

= c

− 1 4 ba

− 1

2 a

− 1 2 cc

− 1

4 ac

, ∇

ϕ

= − 1 2 b

− 1

4 bc

+ 1 2 cb

+ 3

4 ab

,

∇

ϕ

= 1 4 bb

+ 1

4 bc

, ∇

ϕ

= −b

, ∇

ϕ

= ∇

ϕ

= − 1 2 b

− 1

2 c

,

∇

ϕ

= 1

4 ac

+ cc

+ 1

4 ab

, ∇

ϕ

= − 1 4 ab

+ 1

2 b

+ 1 2 cc

+ 3

4 bc

,

∇

ϕ

= 1

4 acb

− c

+ 1

4 acc

+ 1

4 a

b

, ∇

ϕ

= 1

8 abb

+ 1

8 abc

− 1 2 b

− 1

2 c

,

∇

ϕ

= − 1 4 ab

− 1

2 b

− 1 4 bc

− 1

2 cc

, ∇

ϕ

= − 1 4 ab

− 1

4 ac

,

∇

ϕ

= −cb

+ ( 1 8 ab − 1

2 c

− 1

2 )c

+ acb

+ ( 1 8 ab + 1

2 c

− 1 2 )b

,

∇

ϕ

= −b

+ 1

4 b

c

+ 1

4 bcc

+ 1

4 bcb

, ∇

ϕ

= −cb

− 1 4 bc

− 1

4 bb

,

∇

ϕ

= − 1 4 bc

− 1

2 cb

+ 1 2 b

− 1

4 ab

. From (2), (4) and (18) we have

Theorem 9. The opposite almost complex structure of an almost Norden-Walker manifold (M

, ϕ

, g) is isotropic K¨ ahler if and only if the following PDEs hold:

c

(2ba

− 2ac

+ 4c

− 2a

+ 2ca

) + c

(2b

− 2ab

) = 0. (19) From (19) we have

Corollary 1. The triple (M

, ϕ

, g) with metric

g = (g

) =



 



0 0 1 0

0 0 0 1

1 0 a (x, y, z, t) c (z, t) 0 1 c (z, t) b (x, y, z, t)



 



is always isotropic K¨ ahler.

6.1 Integrability of ϕ

The opposite almost complex structure ϕ

is integrable if the analogue of the PDE’s (6) for ϕ

in (17) vanish. From some calculation, we have explicitly the following theorem.

Theorem 10. The opposite almost complex structure ϕ

of an almost Norden-Walker manifold is integrable if and only if the following PDEs hold:

b

= 0, a

− 2c

= 0, ab

− 2b

= 0,

ba

− 2a

− 2ac

+ 4cc

+ 4c

= 0. (20) Let (M

, ϕ

, g) be a Norden-Walker manifold with the integrable almost complex structure ϕ

, i.e. N

= 0. If a = 0, then from (20) b

= b

= c

= c

= 0.

Thus we have

Theorem 11. Let a = 0. The triple (M

, ϕ

, g) with metric

g = (g

) =



 



0 0 1 0

0 0 0 1

1 0 0 c(x, t) 0 1 c(x, t) b(x, t)



 



is always Norden-Walker.

7 Norden-Walker-Einstein metrics

We now turn our attention to the Einstein conditions for the Norden-Walker metric g in (2).

Let R

and S denote the Ricci curvature and the scalar curvature of the metric g in (2).

The Einstein tensor is defined by G

= R

−

Sg

and has non zero components as follows (see [15], Appendix D):

G

=

a

−

b

, G

=

c

+

b

, G

=

a

+

c

, G

=

b

−

a

,

G

=

aa

+ ca

+

ba

− a

+ c

−

a

c

+

a

c

+

a

b

−

(c

)

−

ac

−

ab

,

G

=

ac

+

cc

+

a

−

c

−

a

b

+

c

c

+

bc

−

c

+

b

−

ca

−

cb

,

G

=

ab

+ cb

+ c

− b

−

(c

)

+

a

b

−

b

c

+

b

c

+

bb

−

ba

−

bc

.

(21)

The metric g in (2) is almost Norden-Walker-Einstein if all the above components G

vanish (G

= 0).

Theorem 12. Let (M

, ϕ

, g) be a Norden-Walker manifold. If

a

= b

= c

= c

= 0 (or a

= a

= c

= c

= 0), (22) then g is a Norden-Walker-Einstein.

Proof. Suppose that the triple (M

, ϕ

, g) be a Norden-Walker manifold. Then from (20) and (22), we see that the assertion is clear, i.e., G

= 0.

Corollary 2. The triple (M

, ϕ

, g) with metric

g = (g

) =



 



0 0 1 0

0 0 0 1

1 0 a(y, z, t) c(t) 0 1 c(t) b(t)



 



is always Norden-Walker-Einstein.

8 Counterexamples to Goldberg’s conjecture

1. Let (M

, ϕ, g) be an almost Norden-Walker manifold.

Consider the metric

g = (g

) =



 



0 0 1 0

0 0 0 1

1 0 a(x, y, z, t) 0 0 1 0 a(x, y, z, t)



 

 .