Polyharmonic curves in semi-Riemannian manifolds

Let ( M t m , g ) be a semi-Riemannian manifold of dimension m with a non-degenerate metric of index t , m ≥ 2 , 1 ≤ t ≤ 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.