Classification of four-dimensional CR submanifolds of the homogenous nearly Kähler S^3×S^3 whose almost complex distribution is almost product orthogonal to itself

Abstract

Butruille has shown that S^3×S^3 is one of the four homogeneous six-dimensional nearly Kähler manifolds. Besides its almost complex structure J, it also admits a canonical almost product structure P. The investigation of CR submanifolds of S^3×S^3 started recently. In this article an investigation of four-dimensional CR submanifolds is launched, because an investigation of some specific types of CR submanifolds leads to different types of submanifolds classified with respect to different positions of base vector fields under an action of the almost product structure P. The main result is the classification of four-dimensional CR submanifolds of S^3×S^3 whose almost complex distribution D_1 is almost product orthogonal to itself. First, it is obtained that such submanifold M has no integrable almost complex distribution D_1 and and further it is proved that such submanifolds belong to the same type of CR submanifolds, whose almost complex distribution has an arbitrary vector field E_1 such as PE_1∈TM^⊥. These submanifolds are locally product manifolds of curves and the three-dimensional CR submanifolds with a non-integrable almost complex distribution D_1 for which PD_1⊥D_1 holds, as well.