Symplectic, Hermitian and Kahler Structures on Walker 4-Manifolds

We call a pseudo-Riemannian 4-manifold, which admits a field of parallel null 2-planes, a Walker 4-manifold. A pseudo-Riemannian metric of a Walker 4-manifold is necessarily of neutral signature, and it admits an orthogonal almost complex structure. We show that such a Walker 4-manifold can carry various structures with respect to a certain kind of almost complex structure, e.g., symplectic structures, Kahler structures, Hermitian structures, according as the properties of certain functions which define the canonical form of the metric. The combination of these structures are also analyzed.