W-Curves in Lorentz-Minkowski Space

Abstract

In this paper, we investigate the chord properties of the non-null W-curves in Lorentz-Minkowski space. We give the general equation form for W-curves in (2n+1)-dimension. We define some special curves and give the relations between these curves and isoparametric surfaces. Finally we obtain the geodesics of the pseudospherical cylinder and pseudohyperbolic cylinder in 4-dimensional space.

Gauss Map and Local Approach of Isoparametric Surfaces in Lorentz and Euclidean Space

“…It follows from the definitions above and (2.1) that γ is non-null hyperbolic curve in Lorentz plane. Now we extend the chord idea to the surfaces in the high dimensions and we give some characterizations about these surfaces by the help of Gauss map itself, in terms of [1] and [7], in Lorentz space. Throughout this chapter, the metric tensor will be considered as a Lorentzian unless otherwise mentioned.…”

Application of Matrix Methods On W-Curves

“…One can consider these curves as an integral curve of the helicoidal vector field in Lie algebra or as a solution of the system of linear homogeneous ordinary differential equations of first order with constant (Frenet curvatures) coefficients. These curves widely investigated in Euclidean space ( [1], [2]) and in Lorentz-Minkowski space ( [5], [6]). The generic parametric equations of unit speed W-curves in R 2n+1 Euclidean space and R 2n+1 .., a n sin λ n s + b n cos λ n s, a n cos λ n s − b n sin λ n s, cs)…”

“…Walrave [6] classified all of W −curves by using some analysis technics in three dimensional Minkowski space. Besides in [5], curves which are satisfy the condition (C) are investigated and given some characterizations about the Frenet vectors of the curve. The metric tensor (inner product) of Lorentz-Minkowski space is given by…”

“…where T = γ (s) is unit tangent vector field of the curve. Öztürk and Yaylı ( [5], Definition 3.1) called these curves as C−curve and investigated the relations between W −curves and this curve in Lorentz-Minkowski space. In [2], these curves characterized by algebraic methods with respect to study of [1].…”

Elementary Approachs on De Sitter Space

“…Also the derivatives of X (k) (s) are constant for all k, 1 ≤ k ≤ n, in n−dimensional Euclidean space. In [9], the unit speed nonnull curves with constant curvature are called as "C−curve" in Minkowski space. The authors proved that the norm of the high order derivatives of the C−curves are constant and the unit tangent vector field of the curve is T (s) = AX(s) + c, for suitable constant semi skew symmetric matrix A, in Minkowski space.…”