Problems of Lifts in Symplectic Geometry |
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Authors: | Arif SALIMOV Manouchehr BEHBOUDI ASL and Sevil KAZIMOVA |
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Institution: | Department of Algebra and Geometry, Baku State University, AZ1148, Baku, Azerbaijan.,Department of Mathematics, Salmas Branch, Islamic Azad University, Salmas, Iran. and Department of Algebra and Geometry, Baku State University, AZ1148, Baku, Azerbaijan. |
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Abstract: | Let $(M,\omega )$ be a symplectic manifold. In this paper, the
authors consider the notions of musical (bemolle and diesis)
isomorphisms $\omega ^{b}:TM\rightarrow T^{\ast }M$ and $\omega
^{\sharp }:T^{\ast }M\rightarrow TM$ between tangent and cotangent
bundles. The authors prove that the complete lifts of symplectic
vector f\/ield to tangent and cotangent bundles is $\omega
^{b}$-related. As consequence of analyze of connections between the
complete lift $^{c}\omega _{TM}$ of symplectic $2$-form $\omega $ to
tangent bundle and the natural symplectic $2$-form $\rmd p$ on
cotangent bundle, the authors proved that $\rmd p$ is a pullback of
$^{c}\omega _{TM}$ by $\omega ^{\sharp }$. Also, the authors
investigate the complete lift $^{c}\varphi _{T^{\ast }M}$ of almost
complex structure $\varphi $ to cotangent bundle and prove that it
is a transform by $\omega ^{\sharp }$ of complete lift $^{c}\varphi
_{TM}$ to tangent bundle if the triple $(M,\omega ,\varphi )$ is an
almost holomorphic $\mathfrak{A}$-manifold. The transform of
complete lifts of vector-valued $2$-form is also studied. |
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Keywords: | Symplectic manifold Tangent bundle Cotangent bundle Transform oftensor f/ields Pullback Pure tensor Holomorphic manifold |
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