On general helices and pseudo-Riemannian manifolds

Abstract

In a Riemannian manifold, a regular curve is called a general helix if $frac{alpha}{beta}$ is constant and its first and second curvatures are not constant [4]. If its first and second curvatures are constant the third curvature is zero then the regular curve is called helix. For helices in a Lorentzdan manifold, there is a research of T. Ikawa, who investigated and obtained the differential equation; $D_x D_x D_ x X = KD_x X$ , $(K = alpha^2 - beta^2)$for the circular helix which corresponds to the case that the curvatures $alpha$ and $beta$ of the timelike curve c(t) on the Lorentzian manifold M are constant [3]. Later, N. Ekmekçi and H.H. Hacısalihoğlu obtained the differential equation$D_x D_x D_ x X = KD_x X + 3alpha D_x Y$, $(K =frac{alpha^{''}{beta} + alpha^2 - beta^2)$ for the case of general helix [2]. Recently, T. Nakanishi [5] prove the following lemma about a helix in Pseudo-Riemannian manifold which is stated as, "A unit speed curve c in $M_{alpha}$ is a helix if and only if there exist a constant $lambda$ such that $D_x D_x D_ x X = lambda D_x X^{''}$This paper generalizes the lemma stated above to the case of a general helix.