Three-Dimensional CR Submanifolds in S⁶(1) with Umbilical Direction Normal to D<inf>3</inf>

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 a differentiable almost complex distribution D1 such that its orthogonal complement is a totally real distribution. In this case the normal bundle of the submanifold also splits into two distributions D3, which is almost complex, and a totally real complement. In the case of the proper threedimensional CR 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 umbilic direction orthogonal to D3 and show that such submanifolds do not exist.