将双拉格朗日结构上的辛同胚群作用提升到惠特尼和
Bi-Lagrangian and transverse Dirac structures induced on the Whitney sum
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中文总结 AI 辅助
研究可平行化流形\(M\)上双拉格朗日结构相关问题，证明其可在切丛、余切丛及惠特尼和上诱导相应结构，还表明若\(M\)的双拉格朗日结构能提升到切丛或余切丛，辛同胚群作用也能自然提升到这些空间。
中文摘要 AI 辅助
设\(M\)是赋予双拉格朗日结构\((\omega,\mathcal{F}_{1},\mathcal{F}_{2})\)的流形。其中，\(\omega\)是辛形式，\((\mathcal{F}_{1},\mathcal{F}_{2})\)是辛流形\((M,\omega)\)上的一对横截拉格朗日叶状结构。我们证明，若\(M\)是可平行化的，则\(M\)上的每个双拉格朗日结构自然地在切丛\(TM\)、余切丛\(T^*M\)以及惠特尼和\(W = TM \oplus T^*M\)上诱导出双拉格朗日结构。我们还表明，若\(M\)的双拉格朗日结构可提升到\(TM\)或\(T^*M\)，则文献\(\cite{TNB}\)中定义的辛同胚群在双拉格朗日结构集上的作用可自然地提升到\(TM\)、\(T^*M\)，进而提升到\(W = TM \oplus T^*M\)。
英文摘要
Let $(M,ω,\mathcal{F}_1,\mathcal{F}_2)$ be a bi-Lagrangian manifold. For a distribution $\mathcal{D}\subseteq TM$, we characterize when the subbundle $N^{*t}\mathcal{D}:=\mathcal{D}\oplus N^*\mathcal{D}$ of the Whitney sum $W=TM\oplus T^*M$ is a Dirac structure for both the untwisted and $(H,θ)$-twisted Courant brackets. Consequently, every bi-Lagrangian structure canonically determines a pair of transverse Dirac structures. the associated bi-Lagrangian connection, also called the Hess connection $\nabla$ induces a linear connection on $W$ with respect to which both Dirac subbundles are parallel, thereby establishing a natural link between bi-Lagrangian geometry and Courant Dirac geometry. We also construct induced bi-Lagrangian structures on $TM$, $T^*M$, and $W$, and study the relation between their affineness and that of the original structure. We also lift to these bundles the canonical action of the symplectomorphism group on bi-Lagrangian structures.
发表机构
- University of Ngaoundere（昂旺德雷大学）
- University of Douala（杜阿拉大学）
- University of Maroua（马鲁阿大学）
- University of Kinshasa（金沙萨大学）
- Center for Research on Mathematics Education in the DRC（刚果民主共和国数学教育研究中心）
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