Problems of Lifts in Symplectic Geometry

Let $(M,\omega )$ be a symplectic manifold. In this paper, the authors consider the notions of musical (bemolle and diesis) isomorphisms $\omega ^{b}:TM\rightarrow T^{\ast }M$ and $\omega ^{\sharp }:T^{\ast }M\rightarrow TM$ between tangent and cotangent bundles. The authors prove that the complete lifts of symplectic vector f\/ield to tangent and cotangent bundles is $\omega ^{b}$-related. As consequence of analyze of connections between the complete lift $^{c}\omega _{TM}$ of symplectic $2$-form $\omega $ to tangent bundle and the natural symplectic $2$-form $\rmd p$ on cotangent bundle, the authors proved that $\rmd p$ is a pullback of $^{c}\omega _{TM}$ by $\omega ^{\sharp }$. Also, the authors investigate the complete lift $^{c}\varphi _{T^{\ast }M}$ of almost complex structure $\varphi $ to cotangent bundle and prove that it is a transform by $\omega ^{\sharp }$ of complete lift $^{c}\varphi _{TM}$ to tangent bundle if the triple $(M,\omega ,\varphi )$ is an almost holomorphic $\mathfrak{A}$-manifold. The transform of complete lifts of vector-valued $2$-form is also studied.