几乎Kahler流形的抽象机械联络与Abel重构
Abstract mechanical connection and Abelian reconstruction for almost Kahler manifolds
摘要：
当Hamilton G-系（P，ω，G，J，H）的相空间P具有几乎Kahler结构时，可以定义一个优选联络，称为抽象机械联络，其方法是将每一点处的水平空间定义为度量正交于群轨道的切线。根据动量映射、辛结构和复结构，导出了相应的联络一形式A的显式公式。这种连接可以起到重建连接的作用（见A. Blaom，通过泊松还原的重建阶段。Diff. Geom. Appl.，12：231- 252，2000.），从而大大简化了阿贝尔群G的相应动力学和几何相位的计算。这些想法是用共振三波相互作用的例子来说明。给出了联系一形式和相位的显式公式，并给出了Poisson结构对称约化的一些新结果。
英文摘要：
When the phase space P of a Hamiltonian G-system (P, ω, G, J, H) has an almost Kahler structure a preferred connection, called abstract mechanical connection, can be defined by declaring horizontal spaces at each point to be metric orthogonal to the tangent to the group orbit. Explicit formulas for the corresponding connection one-form A are derived in terms of the momentum map, symplectic and complex structures. Such connection can play the role of the reconstruction connection (see A. Blaom, Reconstruction phases via Poisson reduction. Diff. Geom. Appl., 12:231--252, 2000.), thus significantly simplifying computations of the corresponding dynamic and geometric phases for an Abelian group G. These ideas are illustrated using the example of the resonant three-wave interaction. Explicit formulas for the connection one-form and the phases are given together with some new results on the symmetry reduction of the Poisson structure.
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