标题： 叶层的喷射和$b^k$-代数胚 显示英文标题

标题： Jets of foliations and $b^k$-algebroids

摘要： 在本文中，我们介绍并研究了$b^k$型的奇异叶状结构。 这些奇异叶状结构形式化了在子流形$W \subset M$上与阶数$k$相切的向量场的性质。 我们的第一个结果是对这些叶状结构的分类，将其与在子流形的形式邻域中定义的几何结构相关联，例如到阶数$k-1$的可积分布的喷射。 当$W$是一个超曲面时，$b^k$型的奇异叶状结构是李代数胚。 在这一特殊情况下，它们是Scott引入的$b^k$切丛的推广。 确实，它们总是局部同构于$b^k$-切丛，但全局上这种同构被一个Holonomy不变量所阻碍。 我们的第二个主要结果是用Holonomy表示对$b^k$型奇异叶状结构进行类似Riemann-Hilbert的分类。 在本文中，我们从几个不同的角度研究了$b^k$型奇异叶状结构。 特别是：(1) 我们研究将$k$阶叶状结构扩展为$(k+1)$阶叶状结构的问题，并证明这是被一个特征类所阻碍的。 (2) 当$W$是一个超曲面时，我们对代数微分形式进行了详细研究，并扩展了Scott对上同调的计算。 (3) 我们根据$W$上诱导的几何结构来研究代数辛形式。 特别是，我们发现上述扩张的障碍类与在$W$上诱导的辛叶层的辛变分之间存在密切关系。 显示英文摘要

摘要： In this article, we introduce and study singular foliations of $b^k$-type. These singular foliations formalize the properties of vector fields that are tangent to order $k$ along a submanifold $W \subset M$. Our first result is a classification of these foliations, relating them to geometric structures defined in a formal neighborhood of the submanifold, such as jets of distributions that are involutive up to order $k-1$. When $W$ is a hypersurface, singular foliations of $b^k$-type are Lie algebroids. In this particular case, they are generalizations of the $b^k$-tangent bundles introduced by Scott. Indeed, they are always locally isomorphic to $b^k$-tangent bundles, but globally such an isomorphism is obstructed by a holonomy invariant. Our second main result is a Riemann-Hilbert-style classification of singular foliations of $b^k$-type in terms of holonomy representations. In this paper, we study singular foliations of $b^k$-type from several different perspectives. In particular: (1) We study the problem of extending a $k$-th-order foliation to a $(k+1)$-th order foliation and prove that this is obstructed by a characteristic class. (2) When $W$ is a hypersurface, we give a detailed study of algebroid differential forms and extend Scott's calculation of the cohomology. (3) We study algebroid symplectic forms in terms of the geometric structures induced on $W$. In particular, we find that there is a close relationship between the above obstruction class for extensions and the symplectic variation of the symplectic foliation induced on $W$.