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Problems of Lifts in Symplectic Geometry

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Let (M,ω) be a symplectic manifold. In this paper, the authors consider the notions of musical (bemolle and diesis) isomorphisms ωb: TMT*M and ω#: T*MTM 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 \(<mi xmlns:xlink="" mathvariant="fraktur">A</mi>\)-manifold. The transform of complete lifts of vector-valued 2-form is also studied.

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Correspondence to Arif Salimov, Manouchehr Behboudi Asl or Sevil Kazimova.

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Salimov, A., Behboudi Asl, M. & Kazimova, S. Problems of Lifts in Symplectic Geometry. Chin. Ann. Math. Ser. B 40, 321–330 (2019).

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2000 MR Subject Classification