The Singer invariant of homogeneous spaces

我们调查Riemannian Manifold和Curvature Tensor R及其Covariant衍生物之间的关系中的问题，......，是Riemannian Manifold的必要局部变异体，以及以下结果的必要结果。歌手的变化：给一个本地同源空间M，我们可以定义一个非负整数K_M从其曲线张量和可变衍生物的数据中的数据，它被称为M的歌手非变量。我们的第一个问题是计算同源空间的各种歌手变异物。我们已经确定或估计了以下案例的歌手变异体： （a）低尺寸的案例，在特定的4尺寸同源空间中;（b）单位球体中的同源超饱和体;（c）具有左可变性度量的通用海森堡群体。最近，在我们的知识中，只有少数同源空间，其中歌手的变化者是已知的，他们的歌手变化者都是最多的。最近的C。Meusers Proves，通过给出一个带有高歌手变异体的明确表达示例，使一个局部同源的Riemannian多态性的歌手变异体可以变成高的敏感性。这是一个了不起的结果。我们认为这将是一个有趣的问题，以描述他在歌手变化框架中的例子。曲率同源空间谁有较大的曲率张量对称性：给予一个曲率张量R，我们通过G_0拒绝接受谎言组的身份组成部分的线性等测量，使R不变。We study the following problems " Classify本地同源空间或curvature同源空间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？transitively on a unit sphere。

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.