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The Singer invariant of homogeneous spaces

齐次空间的辛格不变量

基本信息

批准号：
13640066
负责人：
TSUKADA Kazumi
金额：
$ 1.47万
依托单位：
OCHANOMIZU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640066/
关键词：
locally homogeneous spaces curvature homogeneous spaces The Singer invariant curvature tensors generalized Heisenberg groups 曲所等質空間局率等質空間 Singer不変量

项目摘要

We investigate the problems on the relation between the homogeneity or the local homogeneity of a Riemannian manifold and the curvature tensor R and its covariant derivatives ∇R, ∇^2R,・・・, which are essential local invariants of a Riemannian manifold and obtain the following results.1. The Singer invariant : Given a locally homogeneous space M, we can define a non-negative integer κ_M from the data of its curvature tensor and covariant derivatives, which is called the Singer invariant of M. Our first problem is to compute the Singer invariant of various kinds of homogeneous spaces. We determined or estimated the Singer invariant for the following cases : (a) low-dimensional cases, in particular 4-dimensional homogeneous spaces, (b) homogeneous hypersurfaces in a unit sphere, (c) generalized Heisenberg groups with left invariant metric. Up to recently, at our knowledge, there were only a few homogeneous spaces whose Singer invariants are known and their Singer invariants are all at most 1. Recently C. Meusers proves, by giving explicit examples of solvmanifolds with high Singer invariant, that the Singer invariant of a locally homogeneous Riemannian manifold can become arbitrarily high. It is a remarkable result. We think that it will be an interesting problem to characterize his examples in the frame work of the Singer invariant.2. Curvature homogeneous spaces whose curvature tensors have large symmetries : Given a curvature tensor R, we denote by G_0 the identity component of the Lie group consisting of linear isometrics which preserve R invariantly. We study the following problems " Classify locally homogeneous spaces or curvature homogeneous spaces whose G_0 are large ". We obtained the results for the following cases : (a) G_0 = SO R x SO (n-r) (b) G_0 = S0 (n - 2) (c) G_0 acts transitively on a unit sphere.

我们研究了黎曼流形的齐性或局部齐性与曲率张量R及其协变导数∇R，∇^2R，···之间的关系问题，得到了如下结果。辛格不变量：给定一个局部齐次空间M，我们可以由它的曲率张量和协变导数定义一个非负整数κ_M，称为M的辛格不变量。我们的第一个问题是计算各种齐次空间的辛格不变量。我们确定或估计了以下情形的Singer不变量：(A)低维情形，特别是4维齐次空间；(B)单位球面上的齐次超曲面；(C)具有左不变度量的广义Heisenberg群。直到最近，据我们所知，只有少数齐次空间的Singer不变量是已知的，并且它们的Singer不变量都至多为1。最近，C.Meuser通过给出具有高Singer不变量的解流形的显式例子证明了局部齐次黎曼流形的Singer不变量可以变得任意高。这是一个了不起的结果。我们认为，在《歌手不变量》的框架中描述他的例子将是一个有趣的问题。曲率张量具有大对称性的曲率齐次空间：给定一个曲率张量R，我们用G_0表示由保持R不变的线性等距构成的李群的单位分量。我们研究了“局部齐次空间或G_0大的曲率齐次空间的分类”问题。我们得到了如下结果：(A)G_0=so R x so(n-r)(B)G_0=S_0(n-2)(C)G_0在单位球面上传递作用。