Three-dimensional CR submanifolds of the nearly Kahler sphere S⁶(1) that admit foliation by S²(1)

Abstract

A submanifold M of an almost Hermitian manifold (N, J) is a CR submanifold if admits a C∞-differentiable almost complex distribution D (JD⊆D), such that its orthogonal complement D⊥⊂TM is totally real, i.e. JD⊥ ⊆ TM⊥ and they represent the most natural generalization of the notions of almost complex and totally real submanifolds. Here, we are interested in three-dimensional CR submanifolds of the nearly Kähler, six-dimensional sphere S6(1). In particular, we note that S6(1) is one of the four six-dimensional, homogeneous nearly Kähler manifolds. One of the first known families of the three dimensional minimal CR submanifolds in S6(1) was introduced in [2] and [1]. We recall that a submanifold M of a Riemannian manifold (N, g) is said to be ruled, if it admits a foliation with leaves that are totally geodesically immersed into N. We investigate three dimensional CR submanifold of S6(1) ruled by S2(1) and give their explicit classification. In particular, we show that the examples given in [1] are of this type.