A real hypersurface is an immersed submanifold of real co-dimension one in complex space form. The Kähler structure of the ambient space induces on the real hypersurface an almost contact metric structure ($\varphi, \xi, \eta, g$). Cho extended the notion of Tanaka-Webster connection of contact metric manifolds in the case of real hypersurfaces in complex space forms. In this case the connection is called \emph{generalized Tanaka-Webster connection}. The aim of this talk is to present new results concerning three dimensional real hypersurfaces in non-flat complex space forms, i.e. complex projective or hyperbolic space, when 1) the structure Jacobi operator, 2) the shape operator of them satisfies geometric conditions with respect to the generalized Tanaka-Webster connection of them.

We present results on the classification and non-existence of real hypersurfaces in $\mathbb{C}P^{2}$ and $\mathbb{C}H^{2}$, when the structure Jacobi operator of them satisfies conditions such as pseudo-parallellness, or $\mathcal{L}_{X}l=\nabla_{X}l$, where $X$ is orthogonal to $\xi$ or Lie recurrence. Furthermore, we present results, which complete the work that so far has been done in dimensions greater than three, when the structure Jacobi operator is $\xi$-parallel, or Lie $\mathbb{D}$- parallel or it satisfies the relation $\mathcal{L}_{\xi}l=\nabla_{\xi}l$. These results are based on a joint work with Philippos J. Xenos and Georgios Kaimakamis.