Three-Dimensional CR Submanifolds in S6(1) with Umbilical Direction Normal to D3

Abstract

It is well known that the sphere S6(1) admits an almost complex structure J which is nearly Kähler.A submanifold M of an almost Hermitian manifold is called a CR submanifold if it admits adifferentiable almost complex distribution D1 such that its orthogonal complement is a totally realdistribution. In this case the normal bundle of the submanifold also splits into two distributionsD3, which is almost complex, and a totally real complement. In the case of the proper threedimensionalCR submanifold of a six-dimensional manifold the distribution D3 is non-trivial.Here, we investigate three-dimensional CR submanifolds of the sphere S6(1) admitting an umbilicdirection orthogonal to D3 and show that such submanifolds do not exist.