Real hypersurfaces with recurrent normal Jacobi operator in the complex quadric

In this paper, first we give a non-existence theorem for a real hypersurface M with anti-commuting shape operator S in the complex quadric Qm, that is, Sϕ+ϕS=0, where ϕ denotes the structure tensor of M in Qm. By using such a non-existence theorem, we give a classification of Hopf hypersurfaces with recurrent normal Jacobi operator in the complex quadric Qm.