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

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, $T{M}^{\perp }$ the normal bundle over $M$ and $\mathfrak D^\perp=\mathfrak JTM^\perp$\mathrm{.}Therealhypersurface$M$issaidtobe$\mathfrak D^\perp$-invariantif$\mathfrak D^\perp$ 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.} $\mathbf{\text{23}}$(2012) 1250103] and [{Monatsh. Math.} $\mathbf{\text{127}}$(1999), 1–14]. We also classified ${\mathfrak{D}}^{\perp }$ real hypersurface in complex Grassmannians of rank two with constant principal curvatures.