$\mathfrak D^\perp$-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 K\"{a}hler structure of the ambient space, $TM^\perp$ the normal bundle over $M$ and $\mathfrak D^\perp=\mathfrak JTM^\perp$. 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.