D⊥-INVARIANT REAL HYPERSURFACES IN COMPLEX GRASSMANNIANS OF RANK TWO

Let M be a real hypersurface in a complex Grassmannian of rank two. Denote by J the quaternionic Kahler structure of the ambient space, TM perpendicular to the normal bundle over M, and D-perpendicular to = JTM(perpendicular to). The real hypersurface M is said to be D-perpendicular to-invariant if D-perpendicular to is invariant under the shape operator of M. We show that if M is D-perpendicular to-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 D-perpendicular to real hypersurfaces in complex Grassmannians of rank two of noncompact type with constant principal curvatures.