Mathematical Sciences: Non-Positive Curvature, Triangulations and Topology

数学科学：非正曲率、三角剖分和拓扑

• 批准号: 9505136
• 负责人: Pedro Ontaneda
• 金额: $ 3.36万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-08-01 至 1997-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9505136&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Non; Positive; Curvature

9505136 Ontaneda A basic problem in Geometry and Topology is to study the relation between curvature (local information about a space) and topology (global information about a space). The investigator studies a particular case of curvature: spaces of non-positive curvature, in the riemannian case and in the PL (piecewise linear) case. He also studies some problems concerning riemannian and PL rigidity (in the non-positively curved case); the relation between riemannian and PL non-positively curved geometry; and how the topology could determine the curvature: which spaces admit a non-positively curved geometry. A space with non-positively curved geometry is a space on which, essentially, an object, as it moves away from an observer, seems to decrease in size at a rate equal to or faster than the rate in the "real world" (euclidean geometry). In general, geometry determines some properties about the (global) shape of the space. For instance, if a space has a non-positively curved geometry, after "unwrapping" the space we obtain another space that, unlike a doughnut, can be shrunk within itself to a single point (Cartan-Hadamard theorem). This research project is oriented towards studying such relations on spaces with non-positive geometry: how the geometry (local data) determines the global shape (topology) and also how topology could determine the geometry. Although the relevance of this to the "real world" is not obvious if one considers only the apparently Euclidean environs of the earth, questions of cosmology -- the nature of the universe as a whole and not merely our small corner of it -- are intimately related to these mathematical questions. ***

几何与拓扑学中的一个基本问题是研究曲率（关于空间的局部信息）与拓扑（关于空间的全局信息）之间的关系。研究者研究了曲率的一种特殊情况：非正曲率空间，在黎曼情况下和在PL（分段线性）情况下。他还研究了黎曼刚性和PL刚性（在非正弯曲情况下）的一些问题；黎曼几何与PL非正弯曲几何的关系；以及拓扑如何决定曲率：哪些空间允许非正弯曲几何。具有非正弯曲几何的空间本质上是这样一个空间，在这个空间中，当一个物体远离观察者时，它的体积缩小的速度似乎等于或快于“真实世界”（欧几里得几何）中的速度。一般来说，几何形状决定了空间（整体）形状的一些属性。例如，如果一个空间有一个非正弯曲的几何，在“展开”空间之后，我们得到另一个空间，不像甜甜圈，它可以在自身内部收缩到一个点（Cartan-Hadamard定理）。本研究项目旨在研究非正几何空间上的这种关系：几何（局部数据）如何决定全局形状（拓扑），以及拓扑如何决定几何。虽然如果只考虑地球表面的欧几里得环境，它与“真实世界”的相关性并不明显，但宇宙学问题——宇宙作为一个整体的本质，而不仅仅是我们的一个小角落——与这些数学问题密切相关。＊＊＊