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

Abstract

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