On Norden-Walker 4-manifolds

Abstract

A Walker 4-manifold is a semi-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 Norden structures on 4-dimensional Walker manifolds with respect to a proper and opposite almost complex structures. We discuss sequently the problem of integrability, Kähler (holomorphic), isotropic quasi-Kähler conditions for these structures. The curvature properties for Norden-Walker metrics is also investigated. Also, we give counterexamples to Goldberg's conjecture in the case of neutral signature.