Biharmonic submanifolds in \(3\)-dimensional \((\kappa, \mu)\)-manifolds (Q2368481)

A submanifold of a space form is called pseudo-parallel, if its curvature operator acts on its second fundamental form as the curvature operator of a constant curvature manifold. The first named author, in collaboration with others, introduced this notion and investigated properties of pseudo-parallel hypersurfaces and pseudoparallel two-dimensional surfaces in five-dimensional manifolds. In the present paper the authors study the relation between the property of pseudo parallelity and the flatness of the normal bundle of a submanifold. In particular they prove that an immersion with flat normal bundle and two independent principal normal vector fields is pseudo-parallel and is a composition of Dupin cyclide and umbilical inclusion provided that the vector fields are parallel along the corresponding subbundles. The main result uses this to obtain a local classification of pseudo-parallel isometric immersions with flat normal bundles.