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Suh ( Math. Nachr . 290 (2017) 442–451) proved that there are no Hopf real hypersurfaces in the complex quadric that have parallel normal Jacobi operators. Motivated by this result, in this paper, we introduce the notions of   C   ‐parallel  and  Reeb parallel  for normal Jacobi operators, which generalize the notion ‘parallel’. First we obtain a non‐existence theorem of Hopf real hypersurfaces with   C  ‐parallel normal Jacobi operator in the complex quadric    Q  m    for    m  ⩾  3   . We then prove that a Hopf real hypersurface has a Reeb parallel normal Jacobi operator if and only if it has an   A  ‐isotropic singular normal vector field.

Derivatives of normal Jacobi operator on real hypersurfaces in the complex quadric