In the present paper we study an almost paracontact Riemannian manifold which is closely related to the pure Riemannian metric. We construct a new tensor field corresponding to pure metric of almost paracontact structure. An integrability condition and curvature properties of structure by using this tensor field are presented. Finally, we study almost paracontact structures with structural exact 1-form.

Abstract:The differential geometry of tangent bundles was studied by several authors, for example: Yano and Ishihara [8]and among others. It is well known that different structures defined on a manifold M can be lifted to the same type of structures on its tangent bundle. In addition, several authors was studied on operators too, for example: A.A. Salimov [5]. Our goal is to study Tachibana and Vishnevskii Operators Applied to X V and X H in almost paracontact structure on tangent bundle T (M). In addition, this results which obtained shall be studied for some special values in almost paracontact structure.

Abstract:The differential geometry of tangent bundles was studied by several authors, for example: Yano and Ishihara [8] and among others. It is well known that differant structures deffined on a manifold M can be lifted to the same type of structures on its tangent bundle. Our goal is to study Lie derivatives of almost contact structure and almost paracontact structure with respect to X C and X V on tangent bundle T (M). In addition, this Lie derivatives which obtained shall be studied for some special values.

There are many structures in the cotangent bundle. These include the complete and horizontal lifts of the F λ (7, 1) -structure. The F λ (7, 1) -structure was first extended in M n to T * (M n ) by Das, Nivas, and Pathak. Later, the horizontal and complete lift of the Fa(K, 1) -structure in the tangent bundle was given by Prasad and Chauhan. This paper consists of two main sections. In the first part, we find the integrability conditions by calculating Nijenhuis tensors of the complete lifts of the F λ (7, 1) -structure. Later, we get the results of the Tachibana operators applied to vector and covector fields according to the complete lifts of the F λ (7, 1) -structure in the cotangent bundle T * (Mn) . Finally, we study the purity conditions of the Sasakian metric with respect to the complete lifts of the F λ (7, 1) -structure. In the second part, all results obtained in the first section are obtained according to the horizontal lifts of the F λ (7, 1) -structure in cotangent bundle T * (Mn) .

The F v+1 2 F v 1 = 0 structure ( 3) have been studied by Kim J. B. [14]. Later, Srivastava S.K studied on the complete lifts of (1; 1) tensor …eld F satisfying structure F v+1 2 F v 1 = 0 and extended in M n to cotangent bundle. This paper consists of two main sections. In the …rst part, we …nd the integrability conditions by calculating Nijenhuis tensors of the complete and horizontal lifts of F v+1 2 F v 1 = 0: Later, we get the results of Tachibana operators applied to vector and covector …elds according to the complete and horizontal lifts of F ( + 1) ; 2 ( 1) -structure and the conditions of almost holomor…c vector …elds in cotangent bundle T (M n ). Finally, we have studied the purity conditions of Sasakian metric with respect to the lifts of F v+1 2 F v 1 = 0 structure. In the second part, all results obtained in the …rst section were investigated according to the complete and horizontal lifts of the F v+1 2 F v 1 = 0 structure in tangent bundle T (M n ).