Null scrolls asB-scrolls in Lorentz–Minkowski 3-space

Abstract

Null scrolls, i.e. ruled surfaces whose base curve and rulings are both lightlike (null), are Lorentzian surfaces having no Euclidean counterparts. In this work we present reparametrization of nondegenerate null scroll as a Bscroll, i.e. as a ruled surface whose rulings correspond to the binormal vectors of a base curve. We prove that the curvature of a base curve, which determines the Gaussian and mean curvature of a null scroll, is invariant under such a reparametrization. We also determine a one-parameter family of null curves on null scroll which serve as base curves for this kind of reparametrization.

Null scrolls with prescribed curvatures in Lorentz-Minkowski 3-space

“…In [11] it was shown that if the base curve of a null scroll is framed by a null frame L = (A, B, C) whose vector field B is collinear to the rulings of a null scroll, then the mean curvature H of a null scroll is given by H = k 3 , where k 3 is the curvature function of c with respect to L. Following this approach, in this paper we establish relations which will be used in defining null scrolls with a given base curve and prescribed curvatures. Results are presented in the next section.…”

“…If we introduce another null frame for the base curve, related to the rulings of the null scroll, curvatures of the base curve are given by the following lemma. It is also known ( [11]) that the mean curvature H of S satisfies H(u) = k 3 (u), whereas for the Gaussian curvature K(u) = k 2 3 (u) holds. In order to obtain relation between the curvature of a base curve and the mean curvature of a null scroll, first we establish relations between vector fields of the frames L andL = (Ā(u),B(u),C(u)) = (c , e, c × e).…”

“…If the mean curvature of a null scroll is prescribed, then the Gaussian curvature is also given, and vice versa. In [11], we showed that if we introduce a null frame for a base curve of a null scroll which is related to its rulings, then its mean curvature is determined by the curvature of a base curve. If an arbitrary null curve is given (assumed to be parametrized by the pseudo-arc parameter), it is of interest to find all null scrolls (assumed to be parametrized by arbitrary parameters) with the given null curve as a base curve and prescribed curvatures.…”