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On lifts of paracomplex structures

In this paper, we obtain vertical, complete and horizontal lifts of paracomplex geometric structures on paracomplex manifolds to its tangent bundle. Also, we obtain integrability on paracomplex tangent bundle. | Turk J Math 30 (2006) , 197 – 210. ¨ ITAK ˙ c TUB On Lifts of Paracomplex Structures Mehmet Tekkoyun Abstract In this paper, we obtain vertical, complete and horizontal lifts of paracomplex geometric structures on paracomplex manifolds to its tangent bundle. Also, we obtain integrability on paracomplex tangent bundle. Key Words: Paracomplex structure, paracomplex manifold, vertical lift, complete lift, horizontal lift, integrability. 1. Introduction The method of lift has an important role in modern differentiable geometry. With the lift function it is possible to generalize to differentiable structures on any manifold to its extensions. Vertical, complete and horizontal lifts of functions, vector fields, 1-forms and other tensor fields defined on any manifold M to tangent manifold T M has been obtained by Yano and Ishihara [9, 11], Yano and Patterson [10]. Vertical, complete and horizontal lifts of geometric structures defined on any complex manifold M to its tangent bundle T M had been obtained by Tekkoyun [5, 6, 7] and Civelek [5, 6]. In this study, we obtain vertical, complete and horizontal lifts of differential geometric structures on paracomplex manifolds to its tangent bundles. Also, we conclude integrability conditions on paracomplex tangent bundle. Along this paper, all mappings and manifolds will be understood to be of class differentiable and the sum is taken over repeated indices. AMS Mathematics Subject Classification: Primary 55S10, 55S05 197 TEKKOYUN 1.1. Paracomplex manifolds An almost product structure J on manifold M of dimension 2m is a (1, 1) tensor field J on M such that J 2 = I. The pair (M, J) is called an almost product manifold. An almost paracomplex manifold is an almost product manifold (M, J) such that the two eigenbundles T + M and T − M associated to the eigenvalues +1 and -1 of J, respectively, have the same rank. The dimension of an almost paracomplex manifold is necessarily even. Equivalently, a splitting of the tangent