Three-dimensional CR submanifolds of the nearly Kähler S ³ × S ³

Abstract

It is known that there exist only four six-dimensional homogeneous non-Kähler, nearly Kähler manifolds: the sphere S 6 , the complex projective space CP 3 , the flag manifold F 3 and S 3 × S 3 . So far, most of the results about submanifolds have been obtained when the ambient space is the nearly Kähler S 6 . Recently, the investigation of almost complex and Lagrangian submanifolds of the nearly Kähler S 3 × S 3 has been initiated. Here we start the investigation of three-dimensional CR submanifolds of S 3 × S 3 . The tangent space of three-dimensional CR submanifold can be naturally split into two distributions D 1 and D1⊥. In this paper, we found conditions that three-dimensional CR submanifolds with integrable almost complex distribution D 1 should satisfy, and we give some constructions which allow us to define a wide-range family of examples of this type of submanifolds. Our main result is classification of the three-dimensional CR submanifolds with totally geodesics both, almost complex distribution D 1 and totally real distribution D1⊥.