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| Problems of Lifts in Symplectic Geometry |
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Citation： |
Arif SALIMOV,Manouchehr BEHBOUDI ASL,Sevil KAZIMOVA.Problems of Lifts in Symplectic Geometry[J].Chinese Annals of Mathematics B,2019,40(3):321~330 |
| Page view： 1021
Net amount： 635 |
Authors： |
Arif SALIMOV; Manouchehr BEHBOUDI ASL;Sevil KAZIMOVA |
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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. |
Keywords： |
Symplectic manifold, Tangent bundle, Cotangent bundle, Transform oftensor f/ields, Pullback, Pure tensor, Holomorphic manifold |
Classification： |
53D05, 53C12, 55R10 |
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