Geometric Structures on Manifolds

流形上的几何结构

• 批准号: 1207068
• 负责人: Daryl Cooper
• 金额: $ 19万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-15 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207068&HistoricalAwards=false
• 关键词: Geometric; Structures; Manifolds

This project studies various aspects of geometric structures and in particular real projective structures on manifolds and orbifolds. A uniformly fat degeneration of convex projective structures gives a limit metric locally modeled on the affine building for the projective linear group. For surfaces the general case is called a mixed structure. One project is to understand the picture in higher dimensions, where there is some kind of stratification by dimension depending on rates of growth in different directions. Another project is to study transitions between different homogenous structures within the setting of projective structures. In particular we hope to determine the graph with eight vertices representing the Thurston geometries, and edges representing a transition within projective geometry. Some of this discussion is most naturally done in the language of nonstandard analysis using infinitesimals. The study of geometrical properties of space in various dimensions forms the basis of Einstein's theory of relativity, and the physics of superstring theory. We experience the curvature of space as gravity. This is all based on Riemannian geometry which is a way to study different kinds of distance. On the other hand there is projective geometry. This arose from the study of perspective developed by artists in the Renaissance and is based on the notion of straight line but without a notion of distance. This geometry is useful in computer graphics. Freeing oneself from length and distance introduces added flexibility. Recently there has been interest in developing this in the context of topology and applying it to theoretical physics.

该项目研究了几何结构的各个方面，尤其是在歧管和球形上的实际投影结构。凸射结结构的均匀脂肪变性给出了局部建模的局部模型，该指标是在射影线性群体的仿射建筑物上进行的。对于表面，一般情况称为混合结构。一个项目是在较高的维度上理解图片，其中根据不同方向的增长率，按维度进行某种分层。另一个项目是在投影结构的环境中研究不同同质结构之间的过渡。特别是我们希望用代表瑟斯顿几何形状的八个顶点来确定图形，以及代表投影几何形状过渡的边缘。其中一些讨论最自然地在使用无限量的非标准分析的语言中进行。在各个方面的空间几何特性的研究构成了爱因斯坦相对论的基础，以及超声理论的物理学。我们经历了空间作为重力的曲率。这全都基于Riemannian几何形状，这是研究不同距离的一种方式。另一方面，有投射的几何形状。这是根据艺术家在文艺复兴时期开发的观点的研究，是基于直线的概念，但没有距离的概念。该几何形状在计算机图形方面很有用。释放自己的长度和距离会带来增加的灵活性。最近，人们有兴趣在拓扑的背景下开发这种情况并将其应用于理论物理学。