论文标题
Maslov $ s^{1} $捆绑包和Maslov数据
Maslov $S^{1}$ Bundles and Maslov Data
论文作者
论文摘要
我们将Maslov $ s^1 $捆绑在符号歧管$（m，ω）$上定义。这些是由与$ω$兼容的统一框架束的决定套件的$γ_j$，而捆绑包$γ_j^2 =γ_j\ big/ \ big/ \ {\ pm1 \} $。我们分析了Maslov $ s^1 $捆绑$γ_j$和$γ_j^2 $的属性，重点是它们的几何形状与紧凑型Lie lie $ g $在$ m $上的符合性动作之间的相互作用，该动力在$ m $上引起了$ g $ g $ g $ g $ g $ g $ g $ g $ g $ g $ g $ conmin $γ_j$和$γ_j_j_j_j^2 $^2 $^2 $。我们表明，当$ m $是均匀的$ g $ -space时，第一个真正的Chern Chern类$C_γ$是不变的，$γ_j$和$γ_j^2 $也是同质$ g $空。此外，我们提供了一个替代证明，证明当$ [ω] = r \，C_γ$对于某些实际数字$ r $时，那么$（m，ω）$上的Symbletic $ g $ Action是Hamiltonian。当Maslov $ s^1 $ bundle $γ_j^2 $很琐碎时，可以定义概括Maslov索引的索引。如果$γ_j^2 $不是微不足道的，这将不再正确。但是，如果$ g = s^1 $在$（m，ω）$上符合起作用，我们定义了一个数量，我们称之为Maslov数据，它是Maslov索引概念的不可汇总版本，如果$γ_j^2 $不是Trivial，并且我们将Maslov数据与$ G = S^1 $ comess的固定点相关联，则可以将其关联到$ g = S^1 $ comenance。最后，我们考虑了对可综合的哈密顿系统的研究所激发的三种应用。首先，我们讨论条件下，可以将$ s^1 $对称性的对称性扩展到$ \ Mathbb t^2 $对称性。其次，我们证明了拉格朗日捏托里的马斯洛夫$ s^1 $捆绑包。第三，我们将$ s^2 \ times s^2 $视为符号歧管，其$ s^1 $操作对应于两个球体的同时旋转，我们计算相应的MASLOV数据。
We define Maslov $S^1$ bundles over a symplectic manifold $(M,ω)$. These are the determinant bundle $Γ_J$ of the unitary frame bundle defined by an almost complex structure compatible with $ω$, and the bundle $Γ_J^2 = Γ_J \big/ \{\pm1\}$. We analyze the properties of the Maslov $S^1$ bundles $Γ_J$ and $Γ_J^2$, focusing on the interplay between their geometry and the dynamics of a symplectic action of a compact Lie group $G$ on $M$ which induces lifted $G$ actions on $Γ_J$ and on $Γ_J^2$. We show that when $M$ is a homogeneous $G$-space and the first real Chern class $c_Γ$ is nonvanishing, $Γ_J$ and $Γ_J^2$ are also homogeneous $G$-spaces. Moreover, we give an alternative proof of the fact that when $[ω]=r\,c_Γ$ for some real number $r$, then the symplectic $G$ action on $(M,ω)$ is Hamiltonian. When the Maslov $S^1$ bundle $Γ_J^2$ is trivial, then an index generalizing the Maslov index can be defined. This is no longer true if $Γ_J^2$ is not trivial. However, if $G=S^1$ acts symplectically on $(M,ω)$ we define a quantity that we call Maslov data which serves as a non-integrable version of the notion of Maslov index in the case where $Γ_J^2$ is not trivial, and we associate the Maslov data at fixed points of the $G=S^1$ action to their resonance type. Finally, we consider three applications motivated by the study of integrable Hamiltonian systems. First, we discuss conditions under which an $S^1$ symmetry of a two degrees of freedom integrable Hamiltonian system can be extended to a $\mathbb T^2$ symmetry. Second, we show that the Maslov $S^1$ bundles over Lagrangian pinched tori are trivial. Third, we consider $S^2 \times S^2$ as a symplectic manifold with an $S^1$ action corresponding to simultaneous rotations of the two spheres, and we compute the corresponding Maslov data.
