Problems of Lifts in Symplectic Geometry

Abstract

Let (M,ω) be a symplectic manifold. In this paper, the authors consider the notions of musical (bemolle and diesis) isomorphisms ωb: 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 ωb-related. As consequence of analyze of connections between the complete lift cωTM 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 cωTM by ω#. Also, the authors investigate the complete lift cϕT*M of almost complex structure ϕ to cotangent bundle and prove that it is a transform by of complete lift cϕTM 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.