On some 3-dimensional CR submanifolds in $S\sp 6$

Abstract

We give two types of 3-dimensional CR-submanifolds of the 6-dimensional sphere. First we study whether there exists a 3-dimensinal CR-submanifold which is obtained as an orbit of a 3-dimensional simple Lie subgroup of $G_2$. There exists a unique (up to $G_2$) 3-dimensional CR-submanifold which is obtained as an orbit of reducible representations of $SU(2)$ on ${\bf R}^7$. As orbits of the subgroup which corresponds to the irreducible representation of $SU(2)$ on ${\bf R}^7$, we obtained 2-parameter family of 3-dimensional CR-submanifolds. Next we give a generalization of the example which was obtained by K. Sekigawa.