论文标题
由Sigma-actions和仿射Kac-Moody对称空间引起的路径组动作
Path group actions induced by sigma-actions and affine Kac-Moody symmetric spaces of group type
论文作者
论文摘要
1995年，C.-L。与每种超极作用在紧凑的对称空间上相关的TERNG，这是希尔伯特空间上的超极的弗雷德霍尔姆（PF）作用。这是一个无限维路径组的小组作用，它通过仪表转换在希尔伯特空间上。这两个超极作用是通过称为平行传输图的那一层次的riemannian浸没相关的，它们与称为Aggine kac-moody对称空间的无限尺寸对称空间有着密切的关系。在本文中，我们定义了希尔伯特空间之间的线性同构，并表明它相对于仪表变换是等等的，并且与平行传输图兼容。使用这种同构，我们扩展并统一了希尔伯特空间中PF子延伸的主要曲率的所有已知计算结果。尤其是我们研究了与Sigma-Actions相关的高极PF作用的轨道的子序列几何，并给出了希尔伯特空间中Austere PF Submanifolds的新例子。此外，我们表明，此处给出的希尔伯特空间之间的同构对应于群体类型的仿射kac-moody对称空间之间的天然同构。
In 1995, C.-L. Terng associated to each hyperpolar action on a compact symmetric space, a hyperpolar proper Fredholm (PF) action on a Hilbert space. This is a group action by an infinite dimensional path group and it acts on a Hilbert space via the gauge transformations. Those two hyperpolar actions are related through an equivariant Riemannian submersion called the parallel transport map and they have close relations to the infinite dimensional symmetric spaces called affine Kac-Moody symmetric spaces. In this paper we define a linear isomorphism between Hilbert spaces and show that it is equivariant with respect to the gauge transformations and is compatible with the parallel transport map. Using this isomorphism we extend and unify all known computational results of principal curvatures of PF submanifolds in Hilbert spaces. Especially we study the submanifold geometry of orbits of hyperpolar PF actions associated to sigma-actions and give new examples of austere PF submanifolds in Hilbert spaces. Moreover we show that the isomorphism between Hilbert spaces given here corresponds to a natural isomorphism between affine Kac-Moody symmetric spaces of group type.
