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

Authors : Miroslava ANTİC, Djordje KOCİC

Pages : 125-131

Doi:10.36890/iejg.790910

View : 38 | Download : 14

Publication Date : 2021-04-15

Article Type : Research Paper

Abstract :It is well known that the sphere $S^6insert ignore into journalissuearticles values(1);$ 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 $S^6insert ignore into journalissuearticles values(1);$ admitting an umbilic direction orthogonal to $\mathcal{D}_3$ and show that such submanifolds do not exist.  
Keywords : CR submanifolds, umbilical direction, nearly Kähler 6 sphere

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