Examples of infinite dimensional isoparametric submanifolds
Examples of infinite dimensional isoparametric submanifolds
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无限维等参子流形的示例
DOI：
10.1007/bf02571240
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发表时间：
1990
影响因子：
0.8
通讯作者：
G. Thorbergsson
中科院分区：
文献类型：
--
作者：
U. Pinkall;G. Thorbergsson
(PF)如果端点映射Y是Fredholm，且Y对每个有限半径r的正规盘丛的限制是适当的. Hilbert空间的PF子流形M称为等参子流形，如果（i）法丛是平坦的且具有平凡完整性，（ii）如果v是M上的平行法域，则形状算子Av（x）和Av（y）对所有x，yeM正交等价.设G是紧李群，r是S1上的平凡主G-丛.则Sobolev类HI闭回路的回路群G= HI（S1，G）是~的规范群，它通过规范变换作用于H~的Hilbert空间H~，其中g是G的李代数. Terng在[Tel]中显示主轨道是等参的。更确切地说，在[PT]中证明了G的作用是极坐标的，即作用是等距的，适当的，Fredholm，并且在H~中存在一个仿射子空间2;，它满足每个轨道，并且对于每个xe 2;与通过x的轨道正交。证明了极作用量的主轨道是等参的。有限维向量空间上紧李群的极作用由Dadok [IDa]进行了分类。在无限维中，没有分类结果是已知的。在本文中，我们以下面的方式推广Terng的例子。设M= G/K是对称空间。设/s是端点在K中的H1路空间H：[0，g]~ G.则f ~的规范变换在Hilbert空间H~（[0，hi，g）上的作用是极坐标的，主轨道是等参的.在第一节中，我们回顾了一些关于联系和规范变换的基本材料。在第二节中，我们证明了f（）作用的主轨道是等参的，并确定了它们的主曲率和重数。
(PF) if the endpoint map Y is Fredholm and the restriction of Y to each normal disk bundle of finite radius r is proper. A PF submanifold M of a Hilbert space is called isoparametric if (i) the normal bundle is flat and with trivial holonomy, and (ii) if v is a parallel normal field on M, then the shape operators Av (x) and Av (y) are orthogonally equivalent for all x, yeM. Let G be a compact Lie group and let r be the trivial principal G-bundle over S 1. Then the loop group G= HI (S 1, G) of closed loops of Sobolev class HI is the gauge group of~ and it acts on the Hilbert space H~ of H~ by gauge transformations, where g is the Lie algebra of G. Terng shows in [Tel that the principal orbits are isoparametric. More precisely, it is shown in [PT] that the action of G is polar, ie, the action is isometric, proper, Fredholm and there is an affine subspace 2; in H~ that meets every orbit and is orthogonal to the orbit through x for every xe2;. It turns out that the principal orbits of a polar action are isoparametric. Polar actions of compact Lie groups on finite dimensional vector spaces were classified by Dadok IDa]. In infinite dimensions, no classification results are known. In this paper we generalize the examples of Terng in the following manner. Let M= G/K be a symmetric space. Let/s be the space ofH 1 paths H:[0, g]~ G with endpoints in K. Then the action of the gauge transformations of/~ on the Hilbert space H~([0, hi, g) is polar and the principal orbits are isoparametric. In Sect. 1 we review some basic material on connections and gauge transformations. In Sect. 2 we prove that the principal orbits of the action of/(are isoparametric and determine their principal curvatures and multiplicities.