The paper is devoted to the study of metallic Riemannian structures. An integrability condition and curvature properties for these structures by means of a Φ -operator applied to pure tensor fields are presented. Examples of these structures are also given.

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.

Curvature properties are studied for the Sasaki metric on the (1, 1) tensor bundle of a Riemannian manifold. As an application, examples of almost para-Nordenian and para-Kähler-Nordenian B-metrics are constructed on the (1, 1) tensor bundle by looking at the Sasaki metric. Also, with respect to the para-Nordenian B-structure, paraholomorphic conditions for the complete lifts of vector fields are analyzed.

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The main purpose of this article is to investigate the paraholomorphy property of the Sasaki and Cheeger-Gromoll metrics by using compatible paracomplex stuctures on the tangent bundle. Introduction.Let M be an n-dimensional Riemannian manifold with metric g. We denote by p q (M ) the set of all tensor fields of type (p, q) on M . Manifolds, tensor fields and connections are always assumed to be differentiable and of class C ∞ .An almost paracomplex manifold is an almost product manifold (M, ϕ), ϕ 2 = id, ϕ = ±id, such that the two eigenbundles T + M and T − M associated to the two eigenvalues +1 and −1 of ϕ, respectively, have the same rank. Note that the dimension of an almost paracomplex manifold is necessarily even. Considering the paracomplex structure ϕ, we obtain the following set of affinors on M 2k : {id, ϕ}, ϕ 2 = id, which is an isomorphic representation of the algebra of order 2 over the field R of real numbers, which is called the algebra of paracomplex (or double) numbers and is denoted by R(j) = {a 0 + a 1 j | j 2 = 1, j = ±1; a 0 , a 1 ∈ R}. Obviously, it is associative, commutative and unital, i.e., it admits principal unit 1. The canonical base of this algebra {1, j}. The structure constants of this algebra are= 1, all the others being zero, with respect to the canonical base {e 1 , e 2 } = {1, j} of R(j), i.e. e i e j = C k ij e k . Consider R(j) endowed with the usual topology of R 2 and a domain U of R(j). Let X = x 1 + jx 2 be a variable in R(j), where x i are real coordinates of a point of U for i = 1, 2. Using two real-valued functions f i (x 1 , x 2 ), i = 1, 2, we introduce a