Some differential-geometric properties of $R$-spaces
Some differential-geometric properties of $R$-spaces
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$R$空间的一些微分几何性质
DOI：
10.21099/tkbjm/1496164288
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发表时间：
2001
影响因子：
0.7
通讯作者：
Hyunjung Song
中科院分区：
文献类型：
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作者：
Hyunjung Song
设G/K是不可约的黎曼对称空间，G是连通的紧半单李群，K是其闭子群.伴随表示群Ad（K）作为等距群作用在G/K在原点o处的切空间T0（G/K）上。设S表示T0（G/K）中以原点o为中心的单位超球面。对于S中的每一点a，a在Ad（K）下的轨道Ad（K）a称为R-空间。i？-空间形成了一类丰富的齐次黎曼流形，并且具有作为S的子流形的几个显著性质，因此许多作者从微分几何的角度对它们进行了研究，（例如，[5]，[10]，[12]，[13]，[16]，[17]，[21]，[22]，[24]，[31]，[32]，[33]）在本文中，对于这些i？（1）在G/K是Hermitian的情况下，研究了G/K的复结构与限制根系之间的关系。(II)我们表示每个i的第二基本形式的协变导数，空间关于G的李代数中的李括号。作为（Ⅰ）的应用，我们得到了复射影空间中齐次CR子流形的许多新的例子，如定理3.2所述。作为（II）的应用，我们可以给出S.前田的问题，这是陈述作为推论4.5。在此，作者对R教授表示感谢。感谢高木先生的宝贵建议和不断的感谢。
Let G/K be an irreducible Riemannian symmetric space, where G is a connected compact semisimple Lie group and K its closed subgroup. The adjoint representation group Ad(K) acts on the tangent space T0(G/K) of G/K at the origin o as an isometry group. Let S denote a unit hypersphere in the T0(G/K) centered at the origin o. For each point a of S, the orbit Ad(K)a of a under Ad(K) is called an R-space. The i?-spaces form an abundant class of homogeneous Riemannian manifolds and have several distinguished properties as submanifolds of S, and so they have been investigated by many authors from the point of view of differential geometry, (e.g., [5], [10], [12], [13], [16], [17], [21], [22], [24], [31], [32], [33]) In this paper, for these i?-spaces we shall study the following: (I) In the case where G/K is Hermitian, we investigate some relations between the complex structure and the restricted root system with respect to G/K. (II) We express the covariant derivative of the second fundamental form of every i?-space in S with respect to the Lie brackets in the Lie algebra of G. As an application of (I), we obtain many new examples of homogeneous CRsubmanifold in a complex projective space, which is stated as Theorem 3.2. As an application of (II), we can give a partial solution to the S. Maeda's Problem, which is stated as Corollary 4.5. The author would like to express her thanks to Professor R. Takagi for his valuable suggestions and constant encouragements.