On tangent bundles of Walker four-manifolds

1.1 Norden (anti-Hermitian) and holomorphic Norden (Kähler-Norden) Metrics

A metric g is considered a Norden metric [2] if it satisfies the following condition:

or equivalently

for any U , V ∈ ℑ 0 1 ( M 2 n ) , where ℑ 1 0 ( M 2 n ) denotes the set of 1-form on the manifold ( M 2 n ) .

A Norden metric g is termed holomorphic if it satisfies the following condition:

for any U , V , W ∈ ℑ 0 1 ( M 2 n ) . Here ∅ J g represents the Tachibana operator applied to g metric.

By putting U = ∂ k , V = ∂ i , W = ∂ j in equation (4), the components ( ∅ J g ) k i j of ∅ J g according to the local coordinate system x 1 , … , x n can be expressed by

If ( M 2 n , J , g ) is a Norden manifold with a holomorphic Norden metric g , then it is referred to as a holomorphic Norden manifold.

1.2 Holomorphic tensor fields

We introduce an operator called the Tachibana operator:

that acts on the pure tensor field ω defined in [2]

where L V denotes Lie differentiation according to V .

A holomorphic tensor field ω ⁎ on X n ( C ) can be expressed on M 2 n in the form of a pure tensor field ω , meeting the condition

for any U , V 1 , … , V q ∈ ℑ 0 1 ( M 2 n ) .

2.1 Complete lifts of the Norden-Walker metrics

A Walker metric defined by [3]

where α , β , θ are smooth functions of the coordinates ( x , y , z , t ) . The parallel null 2-plane D is spanned locally by { ∂ x , ∂ y } , where ∂ x = ∂ ∂ x , ∂ y = ∂ ∂ y .

The complete lift g C of a neutral metric g on a four-manifolds M 4 with local components g i j is defined by

In addition, we obtain the inverse of the metric tensor g C defined by (6) as follows:

where α , β , θ are smooth functions and the complete lifts α C , β C , θ C of the functions α , β , θ have the local expressions θ C = y i ∂ i θ = ∂ θ , β C = y i ∂ i β = ∂ β , α C = y i ∂ i α = ∂ α of the coordinates ( x , y , z , t , k , l , m , n ) .

2.2 Complete lifts of the proper almost complex structure J

Consider Φ represents an almost complex structure on a Walker manifold M 4 , meeting the condition

(i) Φ 2 = − I ,

(ii) g ( Φ U , V ) = g ( U , Φ V ) (Nordenian property),

(iii) Φ ∂ x = ∂ y , Φ ∂ y = − ∂ x ( Φ induces a positive π 2 -rotation on D ).

It is evident that these three properties do not uniquely define Φ , i.e.,

and Φ has the local components

in the context of the natural frame { ∂ x , ∂ y , ∂ z , ∂ t } , where ω = ω ( x , y , z , t ) is an arbitrary function, we proceed to set ω = θ . Subsequently, the metric g defines a unique almost complex structure

( M 4 , J , g ) is denoted as an almost Norden-Walker manifold. In accordance with the terminology found in [4–8], we refer to J as the proper almost complex structure.

From equation (1), the complete lifts J C of J on four-manifolds M 4 with local components J j i are determined by:

where α , β , θ are smooth functions, the complete lifts α C , β C , θ C of these functions exhibit local expressions as θ C = y i ∂ i θ = ∂ θ , β C = y i ∂ i β = ∂ β , α C = y i ∂ i α = ∂ α in terms of the coordinates ( x , y , z , t , k , l , m , n ) .

2.3 Integrability conditions of J C

Thus, we see that if α = − β and θ = 0 , then J C is integrable.

Let ( T ( M 4 ) , J C , g C ) be a Norden-Walker manifolds ( N J C = 0 ) and α = β . Then, equation (10) reduces to

from which follows

e.g., the functions α and θ are harmonic according to the arguments x , y and k , l .

2.4 Example of Norden-Walker metric

Suppose α equals β and let h ( x , y , k , l ) represent a function of variables x , y , k and l that is harmonic with respect to α , for instance.

We define

where ω represents any smooth functions z , t , m and n .

In that case, α is also harmonic according to x , y , k and l . We obtain

From equations (11), we derive the partial differential equation (PDE) for θ to fulfill as

For these PDE S , we obtain a value of θ 1 by solutions of the equation θ y = − α x = − e x cos y and a value of θ 2 by solutions of the equation θ l = − α k = − e k cos l . Thus, we obtain the results

where β 1 and β 2 represent arbitrary smooth functions of z , t , k , l , m , n as well as x , y , z , t , m , n , respectively. Consequently, the complete lift of the Norden-Walker metric exhibits components in relation to θ 1 and θ 2 , respectively.

2.5 Holomorphic conditions of g C

If ( M 4 , J , g ) represents an almost Norden-Walker manifold and

then, as demonstrated in Section 1, J is integrable and ( M 4 , J , g ) is denoted as a holomorphic Norden-Walker or a Kähler-Norden-Walker manifold.

By substituting equations (6) and (9) in (12), we observe the non-vanishing components of ( ∅ J C g C ) k i j .

2.6 Quasi-Kähler-Norden-Walker manifold

A Norden-Walker manifold ( M , J , g ) meeting the condition ∅ k g i j + 2 ∇ k G i j being zero is termed a quasi-Kähler manifold, where G is defined by G i j = J i m g m j [9]. Furthermore, for the covariant derivative ∇ G of the associated metric G set ( ∇ G ) i j k = ∇ i G j k on the Norden-Walker four-manifolds.

2.7 Isotropic Kähler-Norden structures J C on T ( M 4 )

A proper almost complex structure on a Norden-Walker manifold ( M 4 , J , g ) is termed isotropic Kähler if ∥ ∇ J 2 ∥ = 0 but ∇ J ≠ 0 . In addition, the complete lift of connection coefficient is written as

Due to Γ i , j k C = Γ j , i k C , we only need to consider Γ i , j k C ( i ⟨ j ) . Let ( g C ) − 1 be the inverse of the metric tensor g C defined by (6) and the covariant derivative ∇ C J C of the almost complex structure is expressed as ( ∇ C J C ) i j k = ∇ i C ( J C ) j k . If the following conditions hold:

then, the non-vanishing components of ( ∇ C ( J C ) ) i j k = ∇ i C ( J C ) j k are as follows: