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.

In the present paper a generalized Kählerian space $𝔾{\underset{1}{𝕂}}_N$ of the first kind is considered as a generalized Riemannian space ${\mathrm{𝔾ℝ}}_N$ with almost complex structure $F_i^h$ that is covariantly constant with respect to the first kind of covariant derivative. Using a non-symmetric metric tensor we find necessary and sufficient conditions for geodesic mappings $f:{\mathrm{𝔾ℝ}}_N\to 𝔾{\underset{1}{\overline{𝕂}}}_N$ with respect to the four kinds of covariant derivatives. These conditions have the form of a closed system of partial differential equations in covariant derivatives...