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Regarding the generalized Tanaka-Webster connection, we considered a new notion of $${𝔇}^{\perp }$$ -parallel structure Jacobi operator for a real hypersurface in a complex two-plane Grassmannian G 2(ℂm+2) and proved that a real hypersurface in G 2(ℂm+2) with generalized Tanaka-Webster $${𝔇}^{\perp }$$ -parallel structure Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ℍP n in G 2(ℂm+2), where m = 2n.

We study classifying problems of real hypersurfaces in a complex two-plane Grassmannian $G_2\left({ℂ}^{m+2}\right)$. In relation to the generalized Tanaka-Webster connection, we consider that the generalized Tanaka-Webster derivative of the normal Jacobi operator coincides with the covariant derivative. In this case, we prove complete classifications for real hypersurfaces in $G_2\left({ℂ}^{m+2}\right)$ satisfying such conditions.

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...