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.

该项目的目的是研究嵌入光滑的riemannian歧管中的奇异分层空间的微分几何形状。环境度量标准在单一空间的每个阶层上诱导了一个riemannian度量，而感兴趣的对象是地球学，曲率和距离函数在奇异空间上，尤其是它们在每个层边界附近的渐近行为。此外，我们将定义和研究基于单数点（即非最大地层中的一个点）和以单数点为中心的小球的体积渐近性的指数图。圆锥形和尖锐的奇异性案例非常令人理解，我们的目标是将这些结果扩展到半代数表面和具有多面体奇异性的空间。您的方法基于奇异性的分辨率，结合了现代的奇异方法，其奇异分析适用于吉尼尔顿系统，描述了地理位置的奇异系统。如初步工作所示，可以预期该方法会产生有关所需差异几何量的完整渐近信息。