On a 2-form Derived by Riemannian Metric in the Tangent Bundle

Year 2022, , 225 - 228, 31.10.2022

Narmina Gurbanova

https://doi.org/10.36890/iejg.1137820

Abstract

In a recent paper [Salimov, A., Asl, M.B., Kazimova, S.: Problems of lifts in symplectic geometry. Chin Ann. Math. Ser. B. 40(3), (2019), 321-330] the authors have investigated the curious fact that the canonıcal symplectic structure dp = dpi ∧ dxi on cotangent bundle may be given by the introduction of symplectic isomorphism between tangent and cotangent bundles. Our analysis began with the observation that the complete lift of the symplectic structure from the base manifold to its tangent bundle is being a closed 2-form and consequently we proved that its image by the simplectic isomorphism is the natural 2-form dp. We apply this construction in the case where the basic manifold of bundles is a Riemannian manifold with metric g and consider a new 1-form ω = gijyjdxi and its exterior differential on the tangent bundle, from which the symplectic structure is derived.

Keywords

Tangent bundle, 2-form, symplectic manifold, complete lift, musical isomorphism

References

• [1] Salimov, A., Asl, M.B., Kazimova, S.: Problems of lifts in symplectic geometry. Chin Ann. Math. Ser. B. 40(3), (2019), 321-330.
• [2] Salimov, A.: Tensor operators and their applications. Nova Science Publishers Inc., New York, (2013).
• [3] Yano, K., Ishihara, S.:Tangent and cotangent bundles. Marcel Dekker Inc., New York, (1973).