Complex Finsler Geometry and Related Topics

复杂芬斯勒几何及相关主题

• 批准号: 0713348
• 负责人: Jianguo Cao
• 金额: $ 10.85万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0713348&HistoricalAwards=false
• 关键词: Complex; Finsler; Geometry; Related; Topics

The PI proposes to establish a correspondence between concepts in FINSLER GEOMETRY with concepts in ALGEBRAIC GEOMETRY. To be more specific we want to find the algebraic concepts which correspond to the differential geometric concepts in Complex Finsler Geometry:(a) Holomorphic bisectional curvature,(b) Ricci curvature,(c) Holomorphic sectional curvature,(d) Scalar curvature,(e) Bochner tensor (the complex analogue of the Weyl conformal tensor).The Finsler approach would work for certain coherent sheaves rather than just vector bundles as in the classical theory of Hermitian geometry. It also allows us to extend the theory from complex to varieties defined over other fields such as the p-adic number fields.I am motivated in part by the classical theory of intrinsic metrics (such as the Kobayashi and Caratheodory metrics) which are, in general, Finslerian rather than Hermitian. Another motiviation comes from the recent development in general relativity based on Finsler-Einstein-Lorentze metrics. Thus a rigorous Finsler geometric approach is essential to understand these metrics. In this theory the time cone is not circular (two cones, one on top of the other) but in the shape of pyramids (two pyramids, one on top of the other) and the geometry is much more intricate. The understanding of the Finsler Bochner-Weyl tensor would be very helpful as it is related to Twistor theory and the Yang-Mill self-duality.

PI建议在Finsler几何中的概念与代数几何中的概念之间建立对应关系。更具体地说，我们希望找到与复芬斯勒几何中的微分几何概念相对应的代数概念：(A)全纯对分曲率，(B)Ricci曲率，(C)全纯截面曲率，(D)标量曲率，(E)Bochner张量(Weyl共形张量的复类)。Finsler方法适用于某些相干的层，而不仅仅是经典的厄米几何理论中的矢丛。它还允许我们将理论从复数扩展到定义在其他领域的变种，如p-进数域。我的部分动机是经典的内在度量理论(如Kobayashi和Caratheodory度量)，这些度量通常是芬斯勒的，而不是厄米特的。另一个动因来自基于芬斯勒-爱因斯坦-洛伦泽度量法的广义相对论的最新发展。因此，一个严格的Finsler几何方法对于理解这些指标是必不可少的。在这个理论中，时间锥不是圆形的(两个锥体，一个在另一个的顶部)，而是金字塔的形状(两个金字塔，一个在另一个的顶部)，而且几何形状要复杂得多。理解Finsler Bochner-Weyl张量将是非常有帮助的，因为它与Twistor理论和Yang-Mill自对偶有关。