We study the classifying problem of immersed submanifolds in Hermitian symmetric spaces. Typically in this paper, we deal with real hypersurfaces in a complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$ which has a remarkable geometric structure as a Hermitian symmetric space of rank 2. In relation to the generalized Tanaka-Webster connection, we consider a new concept of the parallel normal Jacobi operator for real hypersurfaces in $G_2({\mathbb C}^{m+2})$ and prove non-existence of real hypersurfaces in $G_2({\mathbb C}^{m+2})$ with generalized Tanaka-Webster parallel normal Jacobi operator.
We give a classification of Hopf real hypersurfaces in complex hyperbolic two-plane Grassmannians
${\rm SU}_{2,m}/S(U_2{\cdot}U_m)$ with commuting conditions between the restricted normal Jacobi operator $\overline{R}_N\phi$ and the shape operator $A$ (or the Ricci tensor $S$)., Doo Hyun Hwang, Eunmi Pak, Changhwa Woo., and Obsahuje bibliografii