Classification results for polyharmonic helices in space forms (Q6639479)

A biharmonic map is a \(C^\infty\) map \(\phi : M \to N\) between Riemannian manifolds \((M, \, g)\) and \((N, \, h)\), which is a critical point to the energy functional\N\[\NE_\Omega (\phi ) = \frac{1}{2} \int_\Omega \big\| \tau (\phi ) \big\|^2 \; d \, \mathrm{vol} (g)\N\]\Nwhere \(\Omega \subset \subset M\) is a relatively compact domain and\N\[\N\tau (\phi ) \in C^\infty \big( \phi^{-1} T (N) \big)\N\]\Nis the tension field of \(\phi\), i.e., locally\N\[\N\tau (\phi )^\alpha = - \Delta_g \phi^\alpha + \Big( \Big\{ \begin{array}{c} \alpha \\\N\mu \nu \end{array} \Big\} \circ \phi \Big) \, \frac{\partial \phi^\mu}{\partial x^i} \,\frac{\partial \phi^\nu}{\partial x^j} \, g^{ij}\, ,\N\]\Nwith \(\Delta_g\) the Laplacian of the Riemannian metric \(g\) and the curly brackets standing for the Christoffel symbols.\N\NA biharmonic curve is a \(C^\infty\) map \(\gamma : {\mathbb R} \to N\), from \({\mathbb R}\) endowed with the Euclidean metric, which is a biharmonic map. In the paper under review, the author introduces a similar notion of polyharmonic curve of order \(r\), as a critical point of the functional\N\[\NE(\gamma ) = \frac{1}{2} \int_{\mathscr I} \big\| \nabla^r_{\dot{\gamma}} \dot{\gamma} \big\|^2_{\gamma (s)} \; d \, s \, , \;\;\; {\mathscr I} \subset\subset {\mathbb R}.\N\]\NThe (higher order) Euler-Lagrange equations associated with the variational principle \(\delta \, E(\gamma )= 0\) involve explicitly the curvature of the target Riemannian manifold \((N, \, h)\), and assume a simpler form when \((N, \, h)\) is a space form (a Riemannian manifold of constant sectional curvature), allowing the author to classify all polyharmonic helices with \(r = 3\) in a sphere \(N = S^m\).