Exceptional Lie Groups and Grassmann Geomtry

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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.