Some new results on real hypersurfaces with generalized Tanaka-Webster connection

Let M be a real hypersurface in nonflat complex space forms of complex dimension two. In this paper, we prove that the shape operator of M is transversally Killing with respect to the generalized Tanaka-Webster connection if and only if M is locally congruent to a type (A) or (B) real hypersurface. We also prove that shape operator of M commutes with Cho operator on holomorphic distribution if and only if M is locally congruent to a ruled real hypersurface.

Generating Curves of Minimal Ruled Real Hypersurfaces in a Nonflat Complex Space Form

We first provide a necessary and sufficient condition for a ruled real hypersurface in a nonflat complex space form to have constant mean curvature in terms of integral curves of the characteristic vector field on it. This yields a characterization of minimal ruled real hypersurfaces by circles. We next characterize the homogeneous minimal ruled real hypersurface in a complex hyperbolic space by using the notion of strong congruency of curves.

Real hypersurfaces with η-parallel shape operator in complex two-plane Grassmannians

In this paper we give a characterisation of -invariant real hypersurfaces of type A; that is, a tube over a totally geodesic G2(ℂm+1) in complex two-plane Grassmannians G2(ℂm+2) or a ruled real hypersurface foliated by complex hypersurfaces which includes a maximal totally geodesic submanifold G2(ℂm+1) in G2(ℂm+2) in terms of η-parallel shape operator.