Polyharmonic curves in semi-Riemannian manifolds

Abstract

Let $(M_t^m,g)$ be a semi-Riemannian manifold of dimension $m$ with a non-degenerate metric of \textit{index} $t$, $m\ge 2$, $1\le t\le m-1$. The main aim of this paper is to investigate the existence of Frenet curves in $(M_t^m,g)$ which are polyharmonic of order $r$, shortly, $r$-harmonic. We shall focus primarily on the cases that the ambient space is a semi-Riemannian space form $N_t^m(c)$ of sectional curvature $c$, a ruled Lorentzian surface or a suitable, possibly warped, product space. We shall obtain existence, non-existence and classification results.