Differential Geometric Aspects of Rigidity

刚度的微分几何方面

• 批准号: 1007136
• 负责人: Karin Melnick
• 金额: $ 12.63万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-15 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007136&HistoricalAwards=false
• 关键词: Differential; Geometric; Aspects; Rigidity

The PI proposes to study differential-geometric aspects of rigidity: which groups can act on a compact manifold preserving a rigid geometric structure of given type? When a given group acts, what can be the geometric and topological properties of the underlying manifold? The PI's work focuses on these questions in the setting of pseudo-Riemannian metrics and conformal structures.The proposed research draws from three active branches of mathematics: differential geometry, dynamics, and Lie groups. The objects of study, pseudo-Riemannian and, in particular, Lorentzian metrics, arise in general relativity. The spacetimes of interest to relativists are noncompact, but they often consider conformal compactifications. Isometric or conformal automorphisms are important in the search for models of spacetime, because they simplify the background partial differential equations, and they correspond to conservation laws.

PI建议研究刚性的微分几何方面：哪些群体可以作用于保持给定类型的刚性几何结构的紧致流形？ 当一个给定的群起作用时，其基础流形的几何和拓扑性质是什么？ PI的工作集中在伪黎曼度量和共形结构的设置中的这些问题。拟议的研究借鉴了数学的三个活跃分支：微分几何，动力学和李群。 研究的对象，伪黎曼度规，特别是洛伦兹度规，产生于广义相对论。 相对论者感兴趣的时空是非紧的，但他们经常考虑共形紧化。 等距或共形自同构在寻找时空模型中很重要，因为它们简化了背景偏微分方程，并且它们对应于守恒定律。