Homogeneous totally real submanifolds of \(S^ 6\)
Let \(S^ 6\) be the 6-dimensional unit sphere in the 7-dimensional Euclidean space \(R^ 7\). The exceptional simple Lie group \(G_ 2\) operates transitively on \(S^ 6\). There exists essentially a unique \(G_ 2\) invariant almost complex structure J on \(S^ 6\). Therefore it has a meaning to study totally real 3-dimensional submanifolds of \((S^ 6,J)\). In this paper the author defines the four totally real 3- dimensional compact submanifolds \(M_ j\) of \((S^ 6,J)\) as an orbit of a certain subgroup \(L_ j\) of \(G_ 2\) \((j=1,2,3,4)\). Then he proves the following Theorem. Let M be a 3-dimensional totally real submanifold of \(S^ 6\) which is obtained as an orbit of a closed subgroup of \(G_ 2\). Then M is congruent (by an element of \(G_ 2)\) to one of the \(M_ 1\), \(M_ 2\), \(M_ 3\) or \(M_ 4\), unless it is a great sphere.
scientific article; zbMATH DE number 3923616