Real hypersurfaces with cyclic parallel normal Jacobi operator in non-flat quaternionic space forms

In this paper, we investigate the properties of the normal Jacobi operator of a real hypersurface in non-flat quaternionic space forms, focused on cyclic parallelism and Codazzi type properties. We classify real hypersurfaces that satisfy these properties. As a solution of the classification problem for such real hypersurfaces, we present the following results: Firstly, we prove that a real hypersurface satisfying a certain condition for the normal Jacobi operator is curvature-adapted. Based on this fact, we further show that in the quaternionic hyperbolic space, the shape operator A of M commutes with the three structure tensors  \(\phi _{\nu }\) ${\varphi }_{\nu }$ , \(\nu =1,2,3\) $\nu =1,2,3$ , of M on the maximal quaternionic subbundle  \(\mathfrak D\) $\mathfrak{D}$ .