Triharmonic Curves in 3-Dimensional Homogeneous Spaces

We first prove that, unlike the biharmonic case, there exist tri- harmonic curves with nonconstant curvature in a suitable Riemannian manifold of arbitrary dimension. We then give the complete classifica- tion of triharmonic curves in surfaces with constant Gaussian curvature. Next, restricting to curves in a 3-dimensional Riemannian manifold, we study the family of triharmonic curves with constant curvature, showing that they are Frenet helices. In the last part, we give the full classification of triharmonic Frenet helices in space forms and in Bianchi–Cartan–Vranceanu spaces.