Three-dimensional CR submanifolds of the nearly Kähler $$\mathbb {S}^3\times \mathbb {S}^3$$ S 3 × S 3
Résumé
It is known that there exist only four six-dimensional homogeneous non-Kähler, nearly Kähler manifolds: the sphere S6, the complex projective space CP3, the flag manifold F3 and S3×S3. So far, most of the results about submanifolds have been obtained when the ambient space is the nearly Kähler S6. Recently, the investigation of almost complex and Lagrangian submanifolds of the nearly Kähler S3×S3 has been initiated. Here we start the investigation of three-dimensional CR submanifolds of S3×S3. The tangent space of three-dimensional CR submanifold can be naturally split into two distributions D1 and D⊥1. In this paper, we found conditions that three-dimensional CR submanifolds with integrable almost complex distribution D1 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 D1 and totally real distribution D⊥1.