Non-geodesic particle trajectories with vanishing higher accelerations

Geodesics in semi-Riemannian manifolds admit equivalent definitions as curves with a vanishing acceleration vector field and as critical curves for the energy functional. Both definitions can be generalized to what we call mth-parallel curves and m-geodesics, respectively. Hence, the question naturally arises: do unit speed m-parallel curves and m-geodesics (m>1) have to be geodesics necessarily. In Section  3 the question for m-parallel curves is answered in the affirmative for Riemannian manifolds and we give examples showing that, in contrast, the corresponding result is no longer true in the semi-Riemannian setting. Nevertheless, we also prove that it still holds for Lorentzian surfaces. Our results in Section  4 show that, in general, one should expect large families of non-geodesic unit speed m-geodesics. Specifically, we prove that when m=2 (Riemannian cubics) and the ambient space has constant sectional curvature, the family comprises helices in addition to geodesics and that these helices are geodesics in either the Hopf torus of S3 or in a flat B-scroll of the anti-de Sitter space. Finally, in Section  5 we characterize m-parallel vector fields along unit speed curves in semi-Riemannian surfaces and, as an application, we use it to give an alternative proof for the surface version of some results in Section  3.