On Spherical Indicatrices and Their Spherical Image of Null Curves in Minkowski 3-Space

Abstract

In this paper, we investigate the spherical images of null curves and null helixes in Minkowski 3-space. We provide the spherical indicatrices of null curves in Minkowski 3-space with their causal characteristics. We also show the conditions of spherical indicatrices of null curves to be a curve lying on pseudo-sphere in Minkowski 3-space. In addition, we give the properties of spherical indicatrices of null curves satisfying generalized helices and lying on pseudo-sphere in Minkowski 3-space.

Some Characterizations of Translation Surface Generated by Spherical Indicatrices of Timelike Curves in Minkowski 3-space

“…The Minkowski 3-space denoted by E 3 1 is a three dimensional real vector space R 3 endowed with the metric tensor ⟨., .⟩ = −dx 2 + dy 2 + dz 2 . The (Lorentzian) scalar and cross product are defined by…”

“…A vector x ∈ E 3 1 is said to be spacelike when ⟨x, x⟩ > 0 or x = 0, timelike when ⟨x, x⟩ < 0 and lightlike (null) when ⟨x, x⟩ = 0. A curve in E 3 1 is called spacelike, timelike or lightlike when the velocity vector of the curve is spacelike, timelike or lightlike, respectively. Let γ = γ(s) : I → E 3 1 be an arbitrary timelike curve.…”

“…A curve in E 3 1 is called spacelike, timelike or lightlike when the velocity vector of the curve is spacelike, timelike or lightlike, respectively. Let γ = γ(s) : I → E 3 1 be an arbitrary timelike curve. The curve γ is said to be a unit speed (or parameterized by the arc-length parameter s) if ⟨γ ′ (s), γ ′ (s)⟩ = −1 for any s ∈ I.…”

“…It was shown that all non-lightlike ruled surfaces of constant slope are developable but not minimal surfaces. In [3], Arfah discussed the causal characterization of spherical indicatrices of timelike curves in Minkowski 3-space, and provided the concept of spherical indicatrix of tangent, normal and binormal vectors of timelike curves with their casual properties. Acar and Aksoyak studied translation surfaces generated by tangent, normal and binormal indicatrices of space curves in E 3 , respectively, and obtain some characterizations based on the fact that such surfaces are flat or minimal [4].…”

“…Acar and Aksoyak studied translation surfaces generated by tangent, normal and binormal indicatrices of space curves in E 3 , respectively, and obtain some characterizations based on the fact that such surfaces are flat or minimal [4]. Motivated by these studies, in this paper, we study translation surfaces generated by tangent, principal normal and binormal indicatrices of timelike curves α and β in Minkowski 3-space E 3 1 . We obtain some characterizations of such surfaces based when they are flat or minimal.…”