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 A new classification of real hypersurfaces with Reeb parallel structure Jacobi operator in the complex quadric J. Korean Math. Soc.Published online December 23, 2020 Hyunjin Lee and Young Jin Suh Kyungpook National University Abstract : In this paper, first we introduce the full expression of the Riemannian curvature tensor of a real hypersurface $M$ in complex quadric $Q^{m}$ from the equation of Gauss and some important formulas for the structure Jacobi operator $R_{\xi}$ and its derivatives $\nabla R_{\xi}$ under the Levi-Civita connection $\nabla$ of $M$. Next we give a complete classification of Hopf real hypersurfaces with Reeb parallel structure Jacobi operator, $\nabla_{\xi}R_{\xi} =0$ in the complex quadric $Q^{m}$ for $m \geq 3$. In addition, we also consider a new notion of $\mathcal C$-parallel structure Jacobi operator of $M$ and give a nonexistence theorem for all Hopf real hypersurfaces with $\mathcal C$-parallel structure Jacobi operator in $Q^{m}$ for $m \geq 3$. Keywords : Reeb parallel structure Jacobi operator, $\mathcal C$-parallel structure Jacobi operator, singular normal vector field, K\"{a}hler structure, complex conjugation, complex quadric MSC numbers : 53C40, 53C55 Full-Text :