Totally biharmonic hypersurfaces in space forms and 3-dimensional BCV spaces

A hypersurface is said to be totally biharmonic if all its geodesics are curves in the ambient space. We prove that a into space form an isoparametric hypersurface, which allows us give full classification of hypersurfaces these spaces. Moreover, restricting ourselves 3-dimensional case, we show surfaces Bianchi–Cartan–Vranceanu spaces and their classification. In particular, that, leaving aside sphere, only nontrivial example surface appears product [Formula: see text].