Hyperbolic Geometry

双曲几何

• 批准号: 9803445
• 负责人: Francis Bonahon
• 金额: $ 8.94万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-15 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803445&HistoricalAwards=false
• 关键词: Hyperbolic; Geometry

9803445 Bonahon The leading theme of this project is the study of rigidity problems for hyperbolic structures on 3-manifolds. In particular, it aims at proving that the geometry of a 3-dimensional hyperbolic convex hull is completely determined by partial data on the geometry of its boundary. The project also involves several connected problems on hyperbolic and euclidean cone manifolds, and on a generalization of curves on a surface called geodesic laminations. These problems are of interest by themselves, but they are also a laboratory for developing and testing new technical tools for application to a wider range of problems in low-dimensional geometry. It is anticipated that a solution to these problems will deepen our understanding of hyperbolic geometry in low dimensions and have applications to other branches of mathematics as well. The project will study several geometric problems on 3-manifolds, namely on spaces of dimension 3. Twenty years ago, revolutionary developments in the field of low-dimensional topology showed that hyperbolic (non-euclidean) geometry was a very powerful tool to analyze the topology of 3-manifolds. This breakthrough also led to the establishment of unexpected connections between low-dimensional topology and other branches of mathematics, such as differential geometry, complex analysis and dynamical systems. ***

小行星9803445 本计画的主要主题是研究三维流形上双曲型结构的刚性问题。 特别是，它的目的是证明的几何形状的3维双曲凸船体是完全由部分数据的几何形状的边界。 该项目还涉及几个连接的问题，双曲和欧几里得锥流形，并在一个推广的曲线上的表面称为测地线叠层。 这些问题本身是有趣的，但它们也是一个实验室，用于开发和测试新的技术工具，以应用于更广泛的低维几何问题。 预计这些问题的解决将加深我们对低维双曲几何的理解，并对其他数学分支也有应用。 该项目将研究三维流形上的几个几何问题，即三维空间。 二十年前，低维拓扑领域的革命性发展表明，双曲（非欧几里德）几何是分析三维流形拓扑的一个非常强大的工具。 这一突破也导致了低维拓扑学和其他数学分支之间意想不到的联系，如微分几何，复分析和动力系统。 ***