$k$-harmonic maps into a Riemannian manifold with constant sectional curvature

J. Eells and L. Lemaire introduced $k$-harmonic maps, and Shaobo Wang showed the first variational formula. When $k=2$, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, we consider the relationship between biharmonic maps and $k$-harmonic maps, and we show the non-existence theorem of 3-harmonic maps. We also give the definition of $k$-harmonic submanifolds of Euclidean spaces and study $k$-harmonic curves in Euclidean spaces. Furthermore, we give a conjecture for $k$-harmonic submanifolds of Euclidean spaces.