Exceptional Lie Groups and Grassmann Geomtry

设S^6为7维欧克里希空间中以原点为中心的6维单位球。我们用纯虚的八元离子IrnO（或Cayley代数）确定了7维欧几里德空间。考虑到八元数的代数性质，我们可以定义S^6上的齐次几乎Herinitian结构，用G_2表示八元数0的自同构李群。然后是S^6 - G_2/SU(3)这个几乎复杂的结构满足近Kah1er条件（（*xJ)X=0），其中*是S^6的Levi-Civita连接，X是S^6的任意向量场。首先，我们给出了S^6=（S^6， J,<, >）的不变子流形（J-全纯曲线，全实曲线和CR-submani[olds））对应的齐次grassmann子束的表示。其次，我们得到了S^6=（S^6， J, <, >）的不变子流形的一些构造。我们给出了许多三维齐次cr -子流形的例子，即S^6和扩展cC，这是由K.Sekigawa得到的。得到了三维cr -子流形在G2作用下的一些刚性定理，并确定了G2不变量。我们构造了三维全实子流形和三维cr -子流形作为j -全纯曲线的第一或第二正规束的一个管。我们给出了一些四维cr子流形的例子，并研究了这些子流形存在的障碍。

Let S^6 be the 6-dimensional unit sphere centered at the origin in a 7-dimensional Eu-cliclean spacc. We identified 7-dimensional Euclidean space with purely imaginary octo-nions IrnO (or Cayley algebra). Taking account of algebraic properties of octonions we can define the homogeneous almost Herinitian structure on S^6 We denote by G_2 the Lie group of automorphisms of the octonions 0. Then we have S^6 - G_2/SU(3).This almost complex structure satisfy the nearly Kah1er condition ((*xJ)X=0) where * is the Levi-Civita connection of S^6, and X is any vector field of S^6First, we gave the representaion of homogeneous grassmann subbundles corresponding to the invariant submanifolds (J-holomorphic curves, totally real and CR-submani[olds) of S^6=( S^6, J,<, >).Next, we obtained some constructions of invariant submanifolds of S^6=(S^6, J, < , >).(1). We gave many examples of 3-dimensional homogeneous CR-submanifolds or S^6 and extension cC the 3-dimensional CR-submanifold which is obtained by K.Sekigawa.(2). We obtained some rigidity theorem of 3-dimensional CR-submanifolds up to the action or G2 and determine G2 invariants.(3). We construct 3-dimensional totally real submanifolds and 3-dimensional CR-submanifolcls as a tube of the first or second normal bundle of J-holomorphic curves.(4). We give some examples of 4-dimensional CRsubmanifolds and studied the obstructions of the existence of such submanifolds.