Mathematical Sciences: Twistor, Complex Manifolds, and Differential Geometry

数学科学：扭量、复流形和微分几何

• 批准号: 8704401
• 负责人: Claude LeBrun
• 金额: $ 4.7万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-06-01 至 1990-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8704401&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Twistor; Complex; Manifolds

Claude LeBrun will study a number of problems in or related to Riemannian geometry in which complex-analytic techniques may be brought to bear by means of a twistor correspondence - that is, by thinking of the points of a Riemannian manifold as compact complex submanifolds of some associated complex manifold. Several of these problems, notably questions concerning solutions to Einstein's equations and the Yang-Mills equations, are motivated by physics. Others are problems internal to geometry and analysis concerning the existence or non-existence of solutions to particular partial differential equations of a geometric character. There are three principal directions in which the research will proceed. First he will seek large categories of examples of quaternionic manifolds. The second direction concerns further extensions of Witten's interpretation of Yang-Mills fields in terms of third order infinitesimal neighborhoods of embeddings. Such extensions have already been carried out by LeBrun in the case of self dual manifolds. The final part of the program involves defining a quaternionic analog of the notion of CR structure.

克劳德·勒布朗将研究一些问题， 到黎曼几何，其中复分析技术可以 通过扭量对应关系来承担-- 通过把黎曼流形的点看作是紧的 某个相关复流形的复子流形几 这些问题，特别是有关解决办法的问题， 爱因斯坦方程和杨-米尔斯方程 通过物理学。其他的是几何和分析的内部问题 关于存在或不存在解决办法的问题 特殊的几何偏微分方程 性格 研究主要有三个方向 将继续进行。首先，他将寻找大类的例子， 四元数流形第二个方向涉及到更多 推广了维滕对杨-米尔斯场的解释， 嵌入的三阶无穷小邻域项。 这样的扩展已经由LeBrun在 自对偶流形的情形节目的最后一部分 涉及定义CR概念的四元数类似物 结构