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 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 ω _TM 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 ω _TM by ω ^#. Also, the authors investigate the complete lift ^cϕ _T*M of almost complex structure ϕ to cotangent bundle and prove that it is a transform by of complete lift ^c ϕ _TM to tangent bundle if the triple (M, ω,ϕ) is an almost holomorphic $<mi xmlns:xlink="http://www.w3.org/1999/xlink" mathvariant="fraktur">A</mi>$-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

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