Umbilical CR submanifolds of the nearly Kahler S^3xS^3

Abstract

Recently, the investigation of a CR submanifolds of the nearly Kähler manifold S^3×S^3 was started. After obtained classifications with respect to positions which the distributions D_1, D_2 and D_3 have under an action of the almost product structure P, obtained are some classes of the CR submanifolds of S^3×S^3 which have an umbilical section. It is proved that CR submanifolds of the nearly Kähler manifold S^3×S^3 with umbilical sections must have dimension three and then we obtain some examples of them with distinguished vector fields. Also, we classify minimal submanifolds that have a vector field E4 as an umbilical section. The main result is classification of the three-dimensional umbilical CR submanifolds with totally geodesic an almost complex distribution.