NOTES ON PARA-NORDEN-WALKER 4-MANIFOLDS

A Walker 4-manifold is a pseudo-Riemannian manifold, (M(4),g) of neutral signature, which admits a field of parallel null 2-plane. The main purpose of the present paper is to study almost paracomplex structures on 4-dimensional Walker manifolds. We discuss sequently the problem of integrability, para-Kahler (paraholomorphic), quasi-para-Kahler and isotropic para-Kahler conditions for these structures. The curvature properties for para-Norden-Walker metrics with respect to the almost paracomplex structure and some properties of para-Norden-Walker metrics in context of almost product Riemannian manifolds are also investigated. Also, we discuss the Einstein conditions for these structures.