Null generalized helices in Lorentz Minkowski spaces

Abstract

We obtain a Lancret-type theorem for null generalized helices in Lorentz-Minkowski spaces. In the 3-dimensional space we get that the only null generalized helices are the ordinary null helices. In the 5-dimensional space, we distinguish between null generalized helices with non-null or null axis and in both cases we solve the natural equations problem.

On k‐type null Cartan slant helices in Minkowski 3‐space

“…19 In semi-Euclidean spaces E 4 1 and E 5 2 , null Cartan slant helices are characterized in Nešović et al 20 and Uçum et al 21 In particular, null Cartan slant helices in E 3 1 are studied in Ali and López 22 as null Cartan curves that satisfy the condition that the scalar product of their principal normal vector field N and a constant vector is constant. Null Cartan general helices in Lorentz-Minkowski spaces L n , n ≥ 3, are introduced in Ferrández et al 23 It is proved in Ferrández et al 23 that null Cartan curve in E 3 1 is the general helix if and only if it is null Cartan helix. Now, we can ask the following question: "Can we determine the conditions under which a null Cartan curve lying on a timelike surface in E 3 1 is null Cartan general helix or null Cartan slant helix, by using its Darboux frame?"…”

Characterizations of Spacelike Slant Helices In Minkowski 3-Space

“…For general helices in semi-Riemannian settings, including Lorentzian ones, we refer the reader to [5,6,7,8,9].…”

Eikonal slant helices and Eikonal Darboux helices in 3-dimensional Riemannian manifold

“…The same definition is also valid in Lorentzian space and spacelike, timelike and null helices have been studied by some mathematicians [7][8][9].…”