Real Hypersurfaces in the Complex Quadric with Normal Jacobi Operator of Codazzi Type

Abstract

We introduce the notion of normal Jacobi operator of Codazzi type for real hypersurfaces in the complex quadric \(Q^m = \text {SO}_{m+2}/\text {SO}_m\text {SO}_2\) . The normal Jacobi operator of Codazzi type implies that the unit normal vector field N becomes \({\mathfrak {A}}\)-principal or \({\mathfrak {A}}\)-isotropic. Then, according to each case, we give a complete classification of Hopf real hypersurfaces in \(Q^m = \text {SO}_{m+2}/\text {SO}_m\text {SO}_2\) with normal Jacobi operator of Codazzi type. The result of the classification is that no such hypersurfaces exist.