TAILIEUCHUNG - Notes on null curves in Minkowski spaces

We show a correspondence between the evolute of a null curve and the involute of a certain spacelike curve in the 4 -dimensional Minkowski space. Also we characterize pseudo-spherical null curves in the ndimensional Minkowski space in terms of the curvature functions. | Turk J Math 34 (2010) , 417 – 424. ¨ ITAK ˙ c TUB doi: Notes on null curves in Minkowski spaces Makoto Sakaki Abstract We show a correspondence between the evolute of a null curve and the involute of a certain spacelike curve in the 4 -dimensional Minkowski space. Also we characterize pseudo-spherical null curves in the n dimensional Minkowski space in terms of the curvature functions. Key Words: Null curve, Minkowski space, Frenet frame, Cartan curvature. 1. Introduction In a semi-Riemannian manifold, there exist three families of curves, that is, spacelike, timelike, and null or lightlike curves, according to their causal characters. In the case of null curves, many different situations appear compared with the cases of spacelike and timelike curves. The theory of Frenet frames for a null curve has been studied and developed by several researchers in this field (cf. [2], [4], [1] and [3]). In [4] Ferrandez, Gimenez and Lucas introduced a Frenet frame with curvature functions for a null curve in a Lorentzian manifold, and studied null helices in Lorentzian space forms. In [1] C¨ o ken and Ciftci studied null curves in the 4 -dimensional Minkowski space R41 , and characterized pseudo-spherical null curves and Bertrand null curves. In this paper we discuss null curves in the n-dimensional Minkowski space Rn1 . We define the evolute of a null curve in R41 and the involute of a spacelike curve in R41 , and show a correspondence between them which is similar to that between the plane evolute and involute. Also, we characterize pseudo-spherical null curves in Rn1 in terms of the curvature functions, which is a generalization of [1, Theorem ] for R41 . 2. Preliminaries In this section, following [4] and [1], we recall the Frenet frame and curvature functions for a null curve in Rn1 . Let , denote the metric on Rn1 . A curve γ(t) in Rn1 is called a null curve if γ (t), γ (t) = 0 and γ (t) = 0 for all t. We note that a null curve