Singular Riemannian foliation

庞加莱姆-霍普定理指出，通过适当地计算向量场的奇点，或者更准确地说，通过添加它们的指标，可以得到欧拉特征。用现代术语来说，这是欧拉类的局部化，指标是残差数据。对于一个杀戮场，一个无限小的等距运动，Bott能够将流形的庞特里亚金类中的上次多项式局部化到它的奇异点，奇异黎曼叶理是杀戮场的高维类比。在这个项目中，我们想要为奇异黎曼叶导出上述类型的残差公式，并特别关注黎曼叶的叶闭包。作为一个应用，我们想要在给定流形上导出黎曼叶化存在的拓扑障碍。

The Poincaré-Hopf Theorem states that by properly counting the singularities of a vector field, or more precisely by adding their indices, one obtains the Euler characteristic. In modern terms this is the localization of the Euler class and the indices are residual data. For a Killing field, an infinitesimal isometric motion, Bott was able to localize polynomials of top degree in the Pontryagin classes of the manifold to its singularities, A singular Riemannian foliation is the higher dimensional analogue of a Killing field. In this project we want to derive a residue formula of the above kind for singular Riemannian foliations with special attention to those that arise as leaf closures of a Riemannian foliation. As an application we want to derive topological obstructions to the existence of a Riemannian foliation on a given manifold.