Smooth Only Torsion-free G_3 Holonomy, Normal Holonomy and Isoparametric Hypersurfaces in Spheres

美国国家科学基金会建议#0103838的摘要迟浩田博士建议，通过探索由这种联系引出的某个偏微分方程组，找出非解析的光滑无扭G_3联系。这项工作的肯定解决方案将开辟一条研究这些联系的新途径，到目前为止，所有已知的例子都是分析的。他还建议通过研究球面上具有四个主曲率的等参超曲面的焦点子流形的正规完整李代数来研究球面上具有四个主曲率的等参超曲面的分类问题，其中包含了焦点子流形的第二基本形式的信息，根据他的研究，这对分类是至关重要的。如果一个人从北极出发，带着指南针沿着环路旅行，你会发现在返回北极的旅程结束时，指南针已经转了一度。这意味着从数学上讲，沿循环的矢量的平行四边形平移(由线性连接引起)可以检测空间是否平坦，这是爱因斯坦相对论的基础。用赫尔曼·韦尔的话说，宇宙中惯性系统的存在保证了它的线性联系是无挠度的。关于无扭转G_3连接的项目是对所有奇异完整系统中这些连接的相当反常性质的研究，这些完整系统的发展已经被PI参与。等参超曲面的概念最早是与欧氏空间中的波前有关的，最近更广泛的Dupin超曲面被与流体力学类型的可积哈密顿系统联系在一起。在引入等参超曲面后不久，它们都被归类在所有维度的欧几里得空间和双曲空间中。尽管在过去的三十年里取得了许多杰出的成果，但对任意维度球面上的这种超曲面的完整分类已经挑战了人们近六十年的努力。PI的第二个项目是利用正规完整理论和他、T.Cecil和G.Jensen的新方法来分类四个主曲率的情况。

Abstract of NSF Proposal #0103838 Dr. Chi proposes to find smooth torsion-free G_3 connections that are notanalytic by exploring a certain system of partial differential equations induced from such connections. The affirmative solution to this work willopen a new avenue of research about these connections, for which all the known examples by far have been analytic. He also proposes to investigatethe classification problem of isoparametric hypersurfaces with fourprincipal curvatures in spheres by investigating the normal holonomy Lie algebra of a focal submanifold of the hypersurface, which contains theinformation about the 2nd fundamental form of the focal submanifold, whichis, by his study, crucial for the classification. If one sets off from the North Pole and travels with a compass along a loop, one will discover that at the end of the trip back at the North Polethe compass needle has turned a degree. This says mathematically that paralleltranslation (induced by a linear connection) of vectors along loops candetect whether the space is flat, which is fundamental to Einstein's Theory of Relativity. In Hermann Weyl's terms, the existence of inertia systems in the universe warrants that its linear connection istorsion-free. The project on torsion-free G_3 connections is a study of the rather anomalous nature of these connections among all exotic holonomies whose development has been participated by the PI. The notion ofisoparametric hypersurfaces was first proposed in connection with wave fronts in Euclidean space and recently the more general Dupin hypersurfaces were tied with integrable hamiltonian systems of hydrodynamic type. Shortly after their introduction the isoparametric hypersurfaces were all classified in Euclidean and hyperbolic spaces of all dimensions. The complete classification of such hypersurfaces in spheres of arbitrary dimensions has defied people's efforts for nearly six decades, despite many outstandingresults achieved in the past three decades. The PI's second project is to utilize the normal holonomy and the new approach of him, T. Cecil and G. Jensen toward the classification in the case of four principal curvatures.