Real hypersurfaces in the complex quadric with Lie invariant normal Jacobi operator

We introduce the notion of Lie invariant normal Jacobi operator for real hypersurfaces in the complex quadric Q m = S O m + 2 / S O m S O 2 . The existence of an invariant normal Jacobi operator implies that the unit normal vector field N becomes A -principal or A -isotropic. Using an analysis of cases, we give a complete classification of real hypersurfaces in Q m = S O m + 2 / S O m S O 2 with Lie invariant normal Jacobi operator.