Non-geodesic particle trajectories with vanishing higher accelerations (Q391050)

This is a report of potential interest to researchers in the field of classical differential geometry, and in particular to those working on Riemannian and Lorentzian geometries. The authors consider a generalization of the notion of geodesics to \(m\)-parallel curves (which generalizes the concept of curves with vanishing accelerations) as well as \(m\)-geodesics (which generalize geodesics as critical curves of the energy functionals). The authors ask whether unit speed \(m\)-parallel curves and \(m\)-geodesics have to be geodesics (necessarily). It is shown that the answer to such a question is affirmative for Riemannian manifolds but, on the other hand, the corresponding answer is not affirmative in the semi-Riemannian context. Several additional results are provided, in a very well organized and natural manner. The paper is appropriate for a large audience of researchers and graduate students in differential geometry and related fields.