On J-parallel totally real three-dimensional submanifolds of S6(1)S6(1)

In this paper, we study totally real submanifolds of the nearly Kähler six-dimensional unit sphere. Since in this case also, parallel submanifolds are totally geodesic, we introduce a weaker condition, namely that for any tangent vector vv〈(∇h)(v,v,v),Jv〉=0.〈(∇h)(v,v,v),Jv〉=0. We obtain a complete classification of totally real three-dimensional submanifolds of S6(1)S6(1) satisfying the above condition.