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This paper is concerned with the problem of the geometry of Norden manifolds. Some properties of Riemannian curvature tensors and curvature scalars of Kähler-Norden manifolds using the theory of Tachibana operators is presented.

A Walker 4-manifold is a pseudo-Riemannian manifold, (M4, 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.

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The authors consider a differentiable manifold with Π-structure which is an isomorphic representation of an associative, commutative and unitial algebra. For Riemannian metric tensor fields, the Φ-operators associated with r-regular Π-structure are introduced. With the help of Φ-operators, the hyperholomorphity condition of B-manifolds is established.