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: 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 \(<mi xmlns:xlink="http://www.w3.org/1999/xlink" mathvariant="fraktur">A</mi>\)-manifold. The transform of complete lifts of vector-valued 2-form is also studied.

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Correspondence to Arif Salimov or 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). https://doi.org/10.1007/s11401-019-0135-7

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Keywords

  • Symplectic manifold
  • Tangent bundle
  • Cotangent bundle
  • Transform of tensor fields
  • Pullback
  • Pure tensor
  • Holomorphic manifold

2000 MR Subject Classification

  • 53D05
  • 53C12
  • 55R10