Homogeneity of infinite dimensional isoparametric submanifolds
Homogeneity of infinite dimensional isoparametric submanifolds
复制标题
无限维等参子流形的齐次性
DOI：
10.2307/121022
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发表时间：
1999
影响因子：
4.9
通讯作者：
Xiaobo Liu
中科院分区：
文献类型：
--
作者：
E. Heintze;Xiaobo Liu
黎曼流形N的一个子集S称为超齐次的，如果S是N的等距群的一个子群的轨道。在[Th]中，Thorbergsson证明了秩至少为3的有限维欧氏空间的完备、连通、满、不可约等参子流形是超齐性的。这个结果与[PT1]和[Da]的结果相结合，最终分类了秩至少为3的有限维欧氏空间的不可约等参子流形.虽然Thorbergsson的证明使用了山雀的建筑物，但Olmos给出了一个更简单的证明，而没有使用山雀的建筑物。[O2]）。本文的主要目的是将Thorbergsson的结果推广到有限维情形。Hilbert空间（通常假设是可分的）的有限维等参子流形的研究是由Terng [T2]开始的。有限维等参子流形理论除了其内在的意义外，在研究紧致对称空间的子流形几何方面也是一个非常有用的工具（参见《几何学》）。[TT][1]和[2]。虽然已经取得了很大的进展（特别是，有限维等参子流形的许多基本性质已经成功地推广到有限维的情况下（参见。[T2]，[T5]，和[HL]），但有限维等参子流形的分类问题还远未解决。为此，对有限维等参子流形的齐性的理解肯定会起到非常重要的作用，就像在有限维情况下一样。非齐次Inflnite维等参超曲面的例子已经由Terng和Thorbergsson发现（参见[1]）。[TT]）。在本文中，我们将证明下面的定理，它解决了一个长期存在的开放问题（参见。[T3]、[T4]、[TT]）。定理A.设M是Hilbert空间V中的完备、连通、不可约等参子流形。假设M在某点的所有曲率法线的集合不包含在任何一条直线中。则M在Hilbert空间V中是超齐次的。备注。在这个定理中，M的维数和V的维数可以是有限的或有限的。不失一般性，我们可以假设M是满的，即
A subsetS of a Riemannian manifoldN is called extrinsically homogeneous ifS is an orbit of a subgroup of the isometry group ofN. In [Th], Thorbergsson proved the remarkable result that every complete, connected, full, irreducible isoparametric submanifold of a flnite dimensional Euclidean space of rank at least 3 is extrinsically homogeneous. This result, combined with results of [PT1] and [Da], flnally classifled irreducible isoparametric submanifolds of a flnite dimensional Euclidean space of rank at least 3. While Thorbergsson’s proof used Tits buildings, a simpler proof without using Tits buildings was given by Olmos (cf. [O2]). The main purpose of this paper is to extend Thorbergsson’s result to the inflnite dimensional case. The study of inflnite dimensional isoparametric submanifolds of a Hilbert space (always assumed to be separable) was initiated by Terng [T2]. Besides its intrinsic interest, the theory of inflnite dimensional isoparametric submanifolds is a very useful tool in studying the submanifold geometry of compact symmetric spaces (cf. [TT] as well as [HL] and [Ew]). Although much progress has been made (especially, many basic properties of flnite dimensional isoparametric submanifolds having been successfully extended to the inflnite dimensional case (cf. [T2], [T5], and [HL])), the classiflcation of inflnite dimensional isoparametric submanifolds is far from being solved. To this end, the understanding of the homogeneity of inflnite dimensional isoparametric submanifolds will certainly play a very important role, as it does in the flnite dimensional case. Examples of non-homogeneous inflnite dimensional isoparametric hypersurfaces have been found by Terng and Thorbergsson (cf. [TT]). In this paper, we will prove the following theorem which solves a long standing open problem (cf. [T3], [T4], [TT]). Theorem A. Let M be a complete, connected, irreducible isoparametric submanifold in a Hilbert space V . Assume that the set of all the curvature normals of M at some point is not contained in any a‐ne line. Then M is extrinsically homogeneous in the Hilbert space V . Remark. In this theorem, the dimension of M and thus V could be either flnite or inflnite. Without loss of generality, we may assume that M is full, i.e.