In this paper, by applying for a new approach of the so-called Tsinghua principle, we prove the nonexistence of locally conformally flat real hypersurfaces in both the m-dimensional complex quadric $Q^m$ and the complex hyperbolic quadric $Q^{m\ast }$ for $m\ge 3$ .

We introduce the notion of Lie invariant structure Jacobi operators for real hypersurfaces in the complex quadric $Q^{m}=SO_{m+2}/SO_{m}SO_{2}$. The existence of invariant structure Jacobi operators implies that the unit normal vector field $N$ becomes $\mathfrak{A}$-principal or $\mathfrak{A}$-isotropic. Then, according to each case, we give a complete classification of real hypersurfaces in $Q^{m}=SO_{m+2}/SO_{m}SO_{2}$ with Lie invariant structure Jacobi operators.

We introduce the notion of Killing normal Jacobi operator for real hypersurfaces in the complex quadric Qm = SOm+2/SOmSO2. The Killing normal Jacobi operator implies that the unit normal vector field N becomes 𝔄-principal or 𝔄-isotropic. Then according to each case, we give a complete classification of real hypersurfaces in Qm = SOm+2/SOmSO2 with Killing normal Jacobi operator.