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-Kähler (paraholomorphic), quasi-para-Kähler and isotropic para-Kähler 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.