On Walker 4-manifolds with pseudo bi-Hermitian structures

(M2n, g, D) is a 4-dimensional Walker manifold and this triple is also a pseudo-Riemannian manifold (M2n, g ) of signature (+ + −−) (or neutral), which is admitted a field of null 2-plane. In this paper, we consider bi-Hermitian structures (φ1, φ2) on 4-dimensional Walker manifolds. We discuss when these structures are integrable and when the bi-Kähler forms are symplectic.


Introduction
Let M 2n be a manifold with a neutral metric which is a pseudo-Rieamnnian metric g of signature (n, n) . Let ℑ p q (M 2n ) be the set of all tensor fields of type (p, q) on M 2n . Manifolds, tensor fields, and connections are assumed to be differentiable and of class C ∞ .
The pair (M 2n , φ) is called an almost complex manifold if the condition φ 2 = −I is hold, where I is a field of identity endomorphisms and φ is an affinor field φ ∈ ℑ 1 1 (M 2n ). The affinor field φ is integrable if and only if there exists a torsion-free affine connection ∇ with respect to which the structure tensor φ is covariantly constant, i.e., ∇φ = 0 . Moreover, if the Nijenhuis tensor of such an affinor field φ defined by is equivalent to the vanish, then the almost complex structure φ is called integrable. In this case, the almost complex manifold (M 2n , φ) is called a complex manifold.
Let M 2n be a 4-dimensional complex manifold and φ i , for i = 1, 2, be two independent compatible integrable almost complex structures. Here φ 1 (x) ̸ = φ 2 (x) for a point x in M 2n . Also, g metric is a Hermitian metric with respect to both complex structures φ 1 and φ 2 , i.e., In this case, the quartet (M 2n , g, φ 1 , φ 2 ) is called bi-Hermitian manifold. If φ 1 (x) ̸ = φ 2 (x) everywhere on M 2n , a bi-Hermitian structure (g, φ 1 , φ 2 ) is called strongly bi-Hermitian. The real function p is defined by or equivalently where p is the angle function of a bi-Hermitian structure and where I is the field of identity endomorphisms [1,13].
An almost Hermitian structure on a manifold M 2n consists of a nondegenerate 2-form w , an almost complex structure φ and a metric g satisfying the compatibility condition w (X, Y) =g(φX, Y ) . If the 2-form w is closed, i.e., dw = 0 , a triple (g, φ, w) is called an almost Kähler structure. Also, the triple (g, φ, w) is called Kähler structure if the almost complex structure φ is integrable [4].

Walker metrics
Let M 2n be a 4-dimensional manifold and g w be a neutral metric (or g w is of signature (+ + −−). g w is called Walker metric if there exists a 2-dimensional null distribution D on M 2n , which is parallel with respect to g w . Such metrics are studied by Walker [15] and canonical form of the metric g w is given by where a, b , and c are some functions depending on the coordinates ( . Note that the parallel . Such Walker manifolds are intensively investigated (see, e.g. [4][5][6][7][8][9][10][11][12]14,15]).

Almost bi-Hermitian structures on a neutral 4-manifold
In this section, we consider 4-dimensional pseudo-Riemannian manifolds of neutral signature. For the next step, it is appropriate to state a neutral metric g and the almost complex structure φ in terms of an orthonormal frame {e i } , (i = 1, 2, 3, 4 ) of vectors and its dual frame { e j } , (j = 1, 2, 3, 4 ) of 1-forms. The metric g is given by , two almost complex structures φ 1 and φ 2 can be written as: According to g, φ 1 , and φ 2 , we have two kinds of Kähler forms on 4-manifolds which are given by Equation (3.4) is equivalent to in matrix notations in the following equation 3) and (3.5), we can write These Kähler forms in terms of the local orthonormal basis { e j } (j = 1, 2, 3, 4 ) of 1-forms are written as: (3.9)

Almost bi-Hermitian structures and bi-Kähler forms on Walker 4-manifolds
Let (M 2n , g w ) be a Walker-4 manifold which is given in (2.1), where g w is Walker metric and let {e i } and of the change of coordinates satisfies: where A T is the transpose matrix of A .
Substituting (2.1) and (3.1) in (4.1), one of the matrices which we apply in the present analysis, we obtain as: of the change of coordinates satisfies: where A −1 is the inverse matrix of A and it is given by: of the change of coordinates for the tensor fields of type (0, 2) satisfies: where A T is the transpose matrix of A .
Substituting (3.6) and (4.2) in (4.7), the bi-Kähler form in (3.6) is obtained as: The bi-Kähler form in (4.8) is written in terms of the coordinate basis as follows: Similarly, substituting (3.7) and (4.2) in (4.7), we obtain the bi-Kähler form in (3.7) as: Also, in terms of the coordinate basis, the bi-Kähler form in (4.10) is written as follows: (4.11)

Integrability of φ 1 ′ and φ 2 ′ (bi-Hermitian structures)
The almost complex structure φ ′ is integrable if and only if From (4.5) and (5.1), the Nijenhuis tensor of φ 1 ′ in (4.5) has nonzero components as follows: From these equations, we have: From (4.6) and (5.1), the Nijenhuis tensor of φ 2 ′ in (4.6) has nonzero components as follows: From these equations, we have: