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In this paper we prove a non-existence of real hypersurfaces in complex hyperbolic two-plane Grassmannians SU2.m/S(U2·Um), m≥3, whose structure tensors {ɸi}i=1,2,3 commute with the shape operator.

We give a classification of Hopf real hypersurfaces in complex hyperbolic two-plane Grassmannians ${\mathrm{SU}}_{2,m}/S(U_2·U_m)$ with commuting conditions between the restricted normal Jacobi operator ${\overline{R}}_N\phi$ and the shape operator $A$ (or the Ricci tensor $S$).

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 $G_2\left({ℂ}^{m+2}\right)$ 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 $G_2\left({ℂ}^{m+2}\right)$ and prove non-existence of real hypersurfaces in $G_2\left({ℂ}^{m+2}\right)$ with generalized Tanaka-Webster...