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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.

On the basis of the phase completion the notion of vertical and horizontal lifts of vector fields is defined in the tensor bundles over a Riemannian manifold. Such a tensor bundle is made into a manifold with a Riemannian structure of special type by endowing it with Sasakian metric. The components of the Levi-Civita and other metric connections with respect to Sasakian metrics on tensor bundles with respect to the adapted frame are presented. This having been done, it is shown that it is possible to study geodesics of Sasakian metrics dealing with geodesics of the base manifolds. Mathematics Subject Classification (2000). Primary 53C22; Secondary 53C25.

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.