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 \(<mi xmlns:xlink="http://www.w3.org/1999/xlink" mathvariant="fraktur">A</mi>\)-manifold. The transform of complete lifts of vector-valued 2-form is also studied.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
Bejan, C.-L. and Druta-Romaniuc, S.-L., Connections which are harmonic with respect to general natural metrics, Differential Geom. Appl, 30(4), 2012, 306–317.
Bejan, C.-L. and Kowalski, O., On some differential operators on natural Riemann extensions, Ann. Global Anal. Geom., 48(2), 2015, 171–180.
Cakan, R., Akbulut, K. and Salimov, A., Musical isomorphisms and problems of lifts, Chin. Ann. Math. Ser. B, 37(3), 2016, 323–330.
Druta-Romaniuc, S.-L., Natural diagonal Riemannian almost product and para-Hermitian cotangent bundles, Czechoslovak Math. J., 62(4), 2012, 937–949.
Etayo, F. and Santamaría, R., (J2 = ±1) -metric manifolds, Publ. Math. Debrecen, 57(3–4), 2000, 435–444.
Norden, A. P., On a class of four-dimensional A-spaces, Izv. Vyssh. Uchebn. Zaved. Matematika, 17(4), 1960, 145–157.
Oproiu, V. and Papaghiuc, N., Some classes of almost anti-Hermitian structures on the tangent bundle, Mediterr. J. Math., 1(3), 2004, 269–282.
Oproiu, V. and Papaghiuc, N., An anti-Kählerian Einstein structure on the tangent bundle of a space form, Colloq. Math., 103(1), 2005, 41–46.
Salimov, A., Almost ψ-holomorphic tensors and their properties, Doklady Akademii Nauk, 324(3), 1992, 533–536.
Salimov, A., On operators associated with tensor fields, J. Geom., 99(1–2), 2010, 107–145.
Salimov, A., Tensor Operators and Their Applications, Mathematics Research Developments Series, Nova Science Publishers, New York, 2013.
Salimov, A., On anti-Hermitian metric connections, C. R. Math. Acad. Sci. Paris, 352(9), 2014, 731–735.
Salimov, A. and Cakan, R., Problems of g-lifts, Proc. Inst. Math. Mech., 43(1), 2017, 161–170.
Salimov, A. and Iscan, M., Some properties of Norden-Walker metrics, Kodai Math. J., 33(2), 2010, 283–293.
Tachibana, S., Analytic tensor and its generalization, Tohoku Math. J., 12, 1960, 208–221.
Vishnevskii, V. V., Shirokov, A. P. and Shurygin, V. V., Spaces Over Algebras, Kazanskii Gosudarstvennii Universitet, Kazan, 1985.
Yano, K. and Ako, M., On certain operators associated with tensor fields, Kodai Math. Sem. Rep., 20, 1968, 414–436.
Yano, K. and Ishihara, S., Tangent and Cotangent Bundles: Differential Geometry, Marcel Dekker, New York, 1973.
Rights and permissions
About this article
Cite this article
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
- Symplectic manifold
- Tangent bundle
- Cotangent bundle
- Transform of tensor fields
- Pure tensor
- Holomorphic manifold
2000 MR Subject Classification