论文标题
奇异的riemannian叶子和$ \ Mathcal {i} $ - 泊松歧管
Singular Riemannian foliations and $\mathcal{I}$-Poisson manifolds
论文作者
论文摘要
我们回想起在歧管$ m $上的单数叶酸（SF）的概念，被视为$ \ m athfrak {x}（m）$的适当子模块，并将其改编成Riemannian Metric $ g $的存在，产生了单一的Riemannian Foliation（Srf）的模块版本。在SFS的Hausdorff Morita等效性的Garmendia-Zambon之后，我们定义了SRF的Morita等效性（无论是在模块的意义上以及在Molino的传统几何上），并表明Morita等效的Srf的叶子是异性含量等于Pseudo-pseudo-Meto-Mets-Metric-Metric-Metric-Metric-Metric-Metric-Metric-Metric-Metric-Metric-Metric-Metric-Metric-Metric-Metric-Metric-Metric-Metric-Metric-Metric-Metric。 在第二部分中，我们介绍了$ \ mathcal {i} $ - 泊松歧管的类别。它的对象和形态学在存在适当理想的情况下概括了泊松的流形和形态，$ \ nathcal {i} $在流形上的平稳功能的函数可以满足两个条件：$（i）$ poisson多种多样的类别在选择$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ii} = 0 $（ii} $（ii} $（ii）类别时Poisson代数，该代数将串联减少为单一环境。 $ m $上的每个sf都会产生$ \ mathcal {i} $ - $ t^*m $上的泊松歧管，而$ g $则在且仅当诱导的汉密尔顿人在$ \ nathcal {i} $的标准时才将其增强到srf。一方面，这种观点提供了一个简单的证据，证明了每个模块SRF都是几何SRF，另一方面，构造了单数叶子的代数不变性的结构：Hausdorff Morita Morita等效的SF具有同构的poisson algebras。
We recall the notion of a singular foliation (SF) on a manifold $M$, viewed as an appropriate submodule of $\mathfrak{X}(M)$, and adapt it to the presence of a Riemannian metric $g$, yielding a module version of a singular Riemannian foliation (SRF). Following Garmendia-Zambon on Hausdorff Morita equivalence of SFs, we define the Morita equivalence of SRFs (both in the module sense as well as in the more traditional geometric one of Molino) and show that the leaf spaces of Morita equivalent SRFs are isomrophic as pseudo-metric spaces. In a second part, we introduce the category of $\mathcal{I}$-Poisson manifolds. Its objects and morphisms generalize Poisson manifolds and morphisms in the presence of appropriate ideals $\mathcal{I}$ of the smooth functions on the manifold such that two conditions are satisfied: $(i)$ The category of Poisson manifolds becomes a full subcategory when choosing $\mathcal{I}=0$ and $(ii)$ there is a reduction functor from this new category to the category of Poisson algebras, which generalizes coistropic reduction to the singular setting. Every SF on $M$ gives rise to an $\mathcal{I}$-Poisson manifold on $T^*M$ and $g$ enhances this to an SRF if and only if the induced Hamiltonian lies in the normalizer of $\mathcal{I}$. This perspective provides, on the one hand, a simple proof of the fact that every module SRF is a geometric SRF and, on the other hand, a construction of an algebraic invariant of singular foliations: Hausdorff Morita equivalent SFs have isomorphic reduced Poisson algebras.
