\(s\)-degenerate curves in Lorentzian space forms (Q1864947)

A curve \(\gamma: I \rightarrow M_1^n\) in a Lorentzian manifold \(M_1^n\) is called \(s\)-degenerate for \(s \in \mathbb{N}\), \(s > 1\) if for any \(t\) all derivatives of order \( < s\) are space-like and the derivative of order \(s\) is a null-vector. In the paper Frenet frames with the minimal number of curvatures \(k_i\) are constructed such that they are invariant under Lorentzian transformations. \(s\)-degenerate curves for which such a frame exists are called \(s\)-degenerate Cartan curves. The main theorem proved in this article is: - Given differentiable curvature functions \(k_i\), then up to a Lorentzian transformation there exists a unique \(s\)-degenerate Cartan curve \(\gamma(t)\) with these curvatures. Additionally, for the 4-dimensional case, the authors give a complete classification of \(s\)-degenerate helices (i.e. \(s\)-degenerate Cartan curves with constant curvature functions).