Geometry of Integrable Systems and Submanifold Geometry

可积系统的几何和子流形几何

• 批准号: 0529756
• 负责人: Chuu-lian Terng
• 金额: $ 7.03万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-01-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0529756&HistoricalAwards=false
• 关键词: Geometry; Integrable; Systems; Submanifold

The proposal of Chuu-Lian Terng contains threeprojects. The first project is joint with KarenUhlenbeck (University of Texas at Austin). Terng andUhlenbeck have collaborated for the past several yearsto give geometric interpretations and applications ofanalytic and algebraic constructions in integrablesystems. They propose to continue their investigationof geometric aspects of integrable systems, Virasoroactions and topological conformal field theory. Thesecond project is on submanifolds in symmetric spaceswhose Gauss-Codazzi equations are integrable systems.Terng has made several contributions to this area whensubmanifolds lie in a space form, and she plans tocontinue the research for the symmetric space case. The third project is joint with Gudlaugur Thorbergsson(University of Koln). Terng and Thorbergsson havemade contributions to the theory of isoparametric submanifolds in space forms and equifocal submanifoldsin symmetric spaces. In this project, they proposeto study the geometry of submanifolds in complex andquaternionic n-space whose U(n) and Sp(n)-invariantsare constant in various senses. The success of thisproject should give better understanding ofsubmanifold geometry in Hermitian and quaternionicKahler symmetric spaces. The theory of integrable systems has deep relationswith mechanics and dynamics, applied mathematics,algebra, theoretical physics, partial differentialequations, algebraic geometry, and differentialgeometry. It has also been used in other sciences. For example, (1) the sine-Gordon equation, which isthe Gauss-Codazzi equation for constant negativeGaussian curvature surfaces in 3-space, also arises inplasma physics; (2) the non-linear Schrodingerequation models the propergation of wave envelope inoptic fiber. Success of the proposed projects willgive better understanding of the structure ofintegrable systems and their geometrizations.

Chuu-Lian Terng的提案包含三个项目。 第一个项目是与KarenUhlenbeck（德克萨斯大学奥斯汀分校）联合开展的。Terng和Uhlenbeck在过去的几年里一直在合作，在积分系统中给出了几何解释和解析及代数构造的应用。他们建议继续他们的investigationof几何方面的可积系统，Virasoractions和拓扑共形场论。第二个项目是对称空间中的子流形，其Gauss-Codazzi方程是可积系统。Terng在这一领域做出了一些贡献，当子流形位于空间形式中时，她计划继续研究对称空间的情况。第三个项目是与Gudlaugur Thorbergsson（喀布尔大学）联合开展的。 Terng和Thorbergsson对空间形式中的等参子流形和对称空间中的等焦子流形的理论作出了贡献。 在这个项目中，他们提出研究复空间和四元数n-空间中U（n）和Sp（n）-不变量在各种意义下都是常数的子流形的几何。 该项目的成功将使我们更好地理解厄米特和四元Kahler对称空间中的子流形几何。 可积系统理论与力学、动力学、应用数学、代数、理论物理、偏微分方程、代数几何、微分几何等学科有着密切的联系。 它也被用于其他科学。例如：（1）sine-Gordon方程，即三维空间中常负高斯曲率曲面的Gauss-Codazzi方程，也出现在等离子体物理中;（2）非线性Schroding方程模拟了光纤中波包的传播。 所提出的项目的成功将使我们更好地理解可积系统的结构及其几何化。