In this paper we give a characterization of a real hypersurface of Type~$(A)$ in complex two-plane Grassmannians ${ { {G_2({\mathbb C}^{m+2})} } }$, which means a tube over a totally geodesic $G_{2}(\mathbb C^{m+1})$ in ${G_2({\mathbb C}^{m+2})}$, by the Reeb parallel structure Jacobi operator ${\nabla}_{\xi}R_{\xi}=0$.

We prove the non-existence of real hypersurfaces in complex projective space whose structure Jacobi operator is Lie $\mathbb{D}$-parallel and satisfies a further condition.

An improved estimate is provided for the number of $\mathbb{F}_q$-rational points on a geometrically irreducible, projective, cubic hypersurface that is not equal to a cone.

In this paper we study real hypersurfaces in a non-flat complex space form with $\eta$-parallel shape operator. Several partial characterizations of these real hypersurfaces are obtained.

We classify real hypersurfaces in complex projective space whose structure Jacobi operator satisfies two conditions at the same time.

Let P be a maximal proper parabolic subgroup of a connected simple linear algebraic group G, defined over $\mathbb C$, such that $n := \dim_{\mathbb C} G/P \geq 4$. Let $\iota \colon Z \hookrightarrow G/P$ be a reduced smooth hypersurface of degree at least $(n-1)\cdot \operatorname{degree}(T(G/P))/n$. We prove that the restriction of the tangent bundle $\iota^*TG/P$ is semistable.

Real hypersurfaces in a complex space form whose structure Jacobi operator is symmetric along the Reeb flow are studied. Among them, homogeneous real hypersurfaces of type $(A)$ in a complex projective or hyperbolic space are characterized as those whose structure Jacobi operator commutes with the shape operator.