Differential geometry of singular spaces

本课题的目的是研究嵌入在光滑黎曼流形中的奇异分层空间的微分几何。环境度量在奇异空间的每一层上推导出一个黎曼度量，奇异空间上的测地线、曲率和距离函数，尤其是它们在每一层边界附近的渐近行为。此外，我们将定义和研究基于奇点（即非极大层中的点）的指数映射以及以奇点为中心的小球的体积渐近性。圆锥和尖角奇点的情况已经很好地理解了，我们的目标是将这些结果推广到半代数曲面和具有多面体奇点的空间。我们的方法是基于奇点的分辨率，结合奇异分析的现代方法应用于描述测地线流的哈密顿系统。正如在初步工作中所显示的，这种方法可以期望在期望的微分几何量上产生完全的渐近信息。

The goal of this project is to study the differential geometry of singular stratified spaces embedded in a smooth Riemannian manifold. The ambient metric induces a Riemannian metric on each stratum of the singular space, and among the objects of interest are geodesics, curvature and distance function on the singular space, especially their asymptotic behavior near the boundary of each stratum. Also, we will define and study the exponential map based at a singular point (i.e. a point in a non-maximal stratum) and the volume asymptotics of small balls centered at singular points. The cases of conical and cuspidal singularities are quite well-understood, and our goal is to extend these results to semi-algebraic surfaces and to spaces with polyhedral singularities.Our approach is based on resolutions of the singularities, combined with modern methods of singular analysis applied to the Hamiltonian system describing the geodesic flow. As shown in preliminary work, this method can be expected to yield full asymptotic information on the desired differential geometric quantities.