Chin. Ann. Math. Ser. B 40(3), 2019, 321–330 DOI: 10.1007/s11401-019-0135-7 Chinese Annals of Mathematics, Series B c  The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2019 Problems of Lifts in Symplectic Geometry Arif SALIMOV 1 Manouchehr BEHBOUDI ASL 2 Sevil KAZIMOVA 1 Abstract Let (M,ω) be a symplectic manifold. In this paper, the authors consider the notions of musical (bemolle and diesis) isomorphisms ω b : TM → T * M and ω ♯ : T * M → TM between tangent and cotangent bundles. The authors prove that the complete lifts of symplectic vector field to tangent and cotangent bundles is ω b -related. As consequence of analyze of connections between the complete lift c ωTM of symplectic 2-form ω to tangent bundle and the natural symplectic 2-form dp on cotangent bundle, the authors proved that dp is a pullback of c ωTM by ω ♯ . Also, the authors investigate the complete lift c ϕT * M of almost complex structure ϕ to cotangent bundle and prove that it is a transform by ω ♯ of complete lift c ϕTM to tangent bundle if the triple (M,ω,ϕ) is an almost holomorphic A-manifold. The transform of complete lifts of vector-valued 2-form is also studied. Keywords Symplectic manifold, Tangent bundle, Cotangent bundle, Transform of tensor fields, Pullback, Pure tensor, Holomorphic manifold 2000 MR Subject Classification 53D05, 53C12, 55R10 1 Introduction Let M be an n-dimensional C ∞ -manifold and T P (M )(T * P (M )) be the tangent (cotangent) vector space at a point P ∈ M . Then the set T (M )= ∪ P ∈M T P (M )(T * (M )= ∪ P ∈M T * P (M )) is, by definition, the tangent (cotangent) bundle over the manifold M . For any point  P of T P (M )(T * P (M )) such that  P ∈ T P (M )(  P ∈ T * P (M )), the correspondence  P → P determines the natural tensor bundle projection π : T (M ) → M (π : T * (M ) → M ), that is, π(  P )= P . Suppose that the base space M is covered by a system of coordinate neighborhoods (U, x i ), where x i ,i =1, ··· ,n are local coordinates in the neighborhood U . The open set π -1 (U ) ⊂ T (M )(π -1 (U ) ⊂ T * (M )) is naturally diffeomorphic to the direct product U × R n in such a way that a point  P ∈ T P (M )(  P ∈ T * P (M )) is represented by an ordered pair (P, ν ) ((P, p)) of the point P ∈ M and a vector (covector) ν ∈ R n (p ∈ R n ) whose components are given by ν i (p i ) of  P in T P (M )(T * P (M )) with respect to the frame (coframe) ∂ i (dx i ). Denoting (x i ) by the coordinates of P = π(  P ) in U and establishing the correspondence (x i ,ν i ) →  P ∈ Manuscript received August 23, 2017. Revised May 15, 2018. 1 Department of Algebra and Geometry, Baku State University, AZ1148, Baku, Azerbaijan. E-mail: asalimov@hotmail.com sevilkazimova@hotmail.com 2 Department of Mathematics, Salmas Branch, Islamic Azad University, Salmas, Iran. E-mail: behboudi@iausalmas.ac.ir