Real Hypersurfaces in Qm with Commuting Structure Jacobi Operator

In this paper, we study real hypersurfaces in the complex quadric space Qm whose structure Jacobi operator commutes with their structure tensor field. When the normal vector field is A-principal we show that the Reeb curvature α is non-vanishing and determine principal curvatures of the hypersurface. In the case of A-isotropic normal vector field, we prove that the hypersurface is Hopf if it has vanishing Reeb curvature or commuting shape operator. We also consider Reeb flat hypersurfaces, namely when the Reeb curvature is zero. We see that this family of hypersurfaces is non-empty and among other results we prove that if the Ricci tensor of a Reeb flat Hopf hypersurfaces is Killing, then the Ricci tensor is parallel.