${\mathfrak{D}}^{\perp }$-invariant real hypersurfaces in complex Grassmannians of rank two

Abstract

Let $M$ be a real hypersurface in complex Grassmannians of rank two. Denote by $\mathfrak{J}$ the quaternionic Kähler structure of the ambient space, $T{M}^{\perp }$ the normal bundle over $M$ and ${\mathfrak{D}}^{\perp }=\mathfrak{J}T{M}^{\perp }$. The real hypersurface $M$ is said to be ${\mathfrak{D}}^{\perp }$-invariant if ${\mathfrak{D}}^{\perp }$ is invariant under the shape operator of $M$. We showed that if $M$ is ${\mathfrak{D}}^{\perp }$-invariant, then $M$ is Hopf. This improves the results of Berndt and Suh in [{Int. J. Math.} \textbf{23}(2012) 1250103] and [{Monatsh. Math.} \textbf{127}(1999), 1--14]. We also classified ${\mathfrak{D}}^{\perp }$ real hypersurface in complex Grassmannians of rank two with constant principal curvatures.