辛同胚群作用在双拉格朗日结构集合上的无穷提升
Infinite Lifting of an Action of Symplectomorphism Group on the set of Bi-Lagrangian Structures
AI总结：
本研究证明流形上的双拉格朗日结构可提升到平凡丛上，定义了辛同胚群作用下的动力系统，该系统与Hess联络相容、保持仿射结构且可迭代提升，可平行化流形的切/余切丛情形下结论仍成立。
摘要：
双拉格朗日结构记为$(ω,\n\n我们证明，流形$M$上的任意双拉格朗日结构都可提升到平凡丛$M\times\mathbb{R}^{2n}$上的双拉格朗日结构；仿射双拉格朗日结构的提升结果仍为仿射结构。随后我们通过辛同胚群在双拉格朗日结构空间上的作用定义了一个动力系统，该动力学与Hess联络相容，保持仿射双拉格朗日结构，可提升到$M\times\mathbb{R}^{n}$上，还可迭代提升到$\left(M\times\mathbb{R}^{2n}\right)\times\mathbb{R}^{4n}$上。对于合适的结构，所得提升动力学与$M\times\mathbb{R}^{n}$上的初始动力学一致。最后，若$M$是可平行化的，将$M\times\mathbb{R}^{2n}$替换为切丛$TM$或余切丛$T^{*}M$时，上述结论同样成立。
英文摘要：
$(ω,\mathcal{F}_{1},\mathcal{F}_{2})$, where $ω$ is symplectic and $(\mathcal{F}_{1},\mathcal{F}_{2})$ is a pair of transversal Lagrangian foliations. Such structures admit a natural geometric object, the Hess connection, which is central to the classification of affine bi-Lagrangian structures. We show that any bi-Lagrangian structure on $M$ lifts to a bi-Lagrangian structure on the trivial bundle $M\times\mathbb{R}^{2n}$. Moreover, the lift of an affine bi-Lagrangian structure is again affine. We then define a dynamical system in terms of an action of the symplectomorphism group on the space of bi-Lagrangian structures. This dynamics is compatible with Hess connections, preserves affine bi-Lagrangian structures, and can be lifted to $M\times\mathbb{R}^{n}$, and iterated to $\left(M\times\mathbb{R}^{2n}\right)\times\mathbb{R}^{4n}$. The resulting lifted dynamics agrees with the initial one on $M\times\mathbb{R}^{n}$ for suitable structures. Finally, the same results hold when $M\times\mathbb{R}^{2n}$ is replaced by $TM$ or $T^{*}M$, provided that $M$ is parallelizable.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
