On helices of a Lorentzian manifold

Abstract

T. Ikawa obtained in [1] the following differential equation $D_xD_xD_x X - KD_x X = 0 , K = k_1^2- k_2^2$ for the circular helix which corresponds to the case that the curvatures $k_1$and $k_2$ of a time-like curve $alpha$ on the Lorentzian manifold $M_1$ are constants. In this paper, T. Ikawa's result is generalized to the case of general helix, i.e. $k_1$and $k_2$, are non-constant functions of t, but $frac{k_1}{k_2}$ is constant.