Differential geometry of singular spaces

奇异空间的微分几何

• 批准号: 287897279
• 负责人: Professor Dr. Daniel Grieser
• 金额: --
• 依托单位: Institut für Mathematik (IFM)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2015
• 资助国家: 德国
• 起止时间: 2014-12-31 至 2018-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/287897279?language=en
• 关键词: Differential; geometry; singular; spaces

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.

这个项目的目的是研究嵌入到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诱导出黎曼度量，其中感兴趣的对象是奇异空间上的测地线、曲率和距离函数，特别是它们在每一层边界附近的渐近行为。此外，我们还将定义和研究基于奇点(即非极大层中的点)的指数映射和以奇点为中心的小球的体积渐近。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，结合应用于描述测地线流动的哈密顿系统的现代奇性分析方法。如初步工作所示，该方法有望产生关于所需微分几何量的全部渐近信息。