Journal of the
Korean Mathematical Society JKMS

In this paper we consider the notion of $\xi$-invariant or ${\frak D}^{\bot}$-invariant real hypersurfaces in a complex two-plane Grassmannian $G_2({\Bbb C}^{m+2})$ and prove that there do not exist such kinds of real hypersurfaces in $G_2({\Bbb C}^{m+2})$ with parallel second fundamental tensor on a distribution $\frak F$ defined by ${\frak F}={\xi}\cup{\frak D}^{\bot}$, where ${\frak D}^{\bot}=\text{Span}\ \{{\xi}_1,{\xi}_2,{\xi}_3\}$.

Keywords: complex Grassmannians, real hypersurfaces, tubes, shape operator, Kahler structure, quaternionic Kahler structure

MSC numbers: Primary 53C40; Secondary 53C15