Three-Dimensional CR Submanifolds in $S^6(1)$ with Umbilical Direction Normal to $\mathcal{D}_3$

Abstract

It is well known that the sphere $S^6(1)$ admits an almost complex structure $J$ which is nearly K\"{a}hler. A submanifold $M$ of an almost Hermitian manifold is called a CR submanifold if it admits a differentiable almost complex distribution $\mathcal{D}_1$ such that its orthogonal complement is a totally real distribution. In this case the normal bundle of the submanifold also splits into two distributions $\mathcal{D}_3$, which is almost complex, and a totally real complement. In the case of the proper three-dimensional CR submanifold of a six-dimensional manifold the distribution $\mathcal{D}_3$ is non-trivial. Here, we investigate three-dimensional CR submanifolds of the sphere $S^6(1)$ admitting an umbilic direction orthogonal to $\mathcal{D}_3$ and show that such submanifolds do not exist.