Lightlike surfaces of Darboux‐like indicatrixes and binormal indicatrixes of spacelike curves in lightcone 3‐space

Abstract

In this paper, by introducing a new frame on spacelike curves lying in lightcone 3-space, we investigate the geometric properties of the lightlike surface of the Darboux-like indicatrix and the lightlike surface of the binormal indicatrix generated by spacelike curves in lightcone 3-space. As an extension of our previous work and an application of the singularity theory, the singularities of the lightlike surfaces of the Darboux-like indicatrix and the lightlike surface of the binormal indicatrix are classified, several new invariants of spacelike curves are discovered to be useful for characterizing these singularities, meanwhile, it is found that the new invariants also measure the order of contact between spacelike curves or principal normal indicatrixes of spacelike curves located in lightcone 3-space and two-dimensional lightcone whose vertices are at the singularities of lightlike surfaces. One concrete example is provided to illustrate our results. KEYWORDS Darboux-like indicatrix, lightcone helix, lightlike surfacesMath Meth Appl Sci. 2020; :5-34.wileyonlinelibrary.com/journal/mmawhere (e 1 , e 2 , e 3 , e 4 ) is the canonical basis of R 4 1 . One can easily show that ⟨a, x ∧ y ∧ z⟩ = det(a, x, y, z). We say that a vector x ∈ R 4 1 ∖{0} is spacelike, lightlike, or timelike if ⟨x, x⟩ is positive, zero, or negative, respectively. The norm of a vector x ∈ R 4 1 is defined by || x|| = √ |⟨x, x⟩|, we call x the unit vector if || x|| = 1. Let ∶ I → R 4 1 be a regular curve in R 4 1 (ie, ′ (t) ≠ 0 for any t ∈ I), where I is an open interval. For any t ∈ I, the curve is called spacelike curve, lightlike curve, or timelike curve if all its velocities are ⟨ ′ (t), ′ (t)⟩ > 0, ⟨ ′ (t), ′ (t)⟩ = 0, or ⟨ ′ (t), ′ (t)⟩ < 0, respectively. We call the nonlightlike curve if is a timelike curve or a spacelike curve. The arc-length of a nonlightlike curve , measured from (t 0 ) (t 0 ∈ I), is s(t) = ∫ t t 0 || ′ (t)||dt. Then, the parameter s is determined such that || ′ (s)|| = 1 for the nonlightlike curve, where ′ (s) = d ds (s). Therefore, we say that a nonlightlike

Slant helix of order n and sequence of Darboux developables of principal‐directional curves

“…Some new results concerning the singularities of submanifolds were established by the second author and his collaborators. [12][13][14][15][16][17][18][19][20][21][22][23][24] As an application of singularity theory, in this paper, we study the singularities of the Darboux developable of nth principal-directional curve of a curve. It is demonstrated that the ratio of torsion to curvature of a curve play a key role in characterizing the singularities of the Darboux developables of the nth principal-directional curve of a curve .…”

“…Better understanding of the local topological structure of singularities of a manifold is very important. Some new results concerning the singularities of submanifolds were established by the second author and his collaborators 12‐24 . As an application of singularity theory, in this paper, we study the singularities of the Darboux developable of n th principal‐directional curve of a curve.…”

Generalized focal surfaces of spacelike curves lying in lightlike surfaces

“…Geometric objects in Minkowski space, regarding singularity, have been studied extensively by, among others, the second author and by previous researchers. [14][15][16][17][18][19][20][21][22][23][24][25][26] However, to the best of the authors' knowledge, the singularities of surfaces and curves as they relate to a curve lying in a general lightlike surface have not been considered in the literature, aside from the case of lightcone. Thus, the current study hopes to serve such a need.…”

Tubular surfaces of center curves on spacelike surfaces in Lorentz‐Minkowski 3‐space

“…It is well known that there exist three kinds of submanifolds, that is, spacelike submanifolds, timelike submanifolds, and lightlike submanifolds in Lorentz‐Minkowski space. Most researchers focus on the submanifolds immersed in Lorentzian space forms with constant curvature from the viewpoint of singularity, the geometric properties of these submanifolds have been explored extensively by the other scholars and the second author . However, the causalities (ie, the structure as a Lorentzian manifold) of submanifolds with nonconstant curvature are quite different from those of the Lorentzian space forms with constant curvature, therefore there will be some different and interesting results when we consider a submanifold immersed in a higher dimensional submanifold from the viewpoint of singularity, for instance, Ito and Izumiya investigate regular curves on a spacelike surfaces in Lorentz‐Minkowski 3‐space, they introduce five special curves that are in Lorentzian subspace forms with constant curvature along the curve associated to the Lorentzian Darboux frame and obtain some good results .…”

“…Most researchers focus on the submanifolds immersed in Lorentzian space forms with constant curvature from the viewpoint of singularity, the geometric properties of these submanifolds have been explored extensively by the other scholars and the second author. [8][9][10][11][12][13][14][15][16][17] However, the causalities (ie, the structure as a Lorentzian manifold) of submanifolds with nonconstant curvature are quite different from those of the Lorentzian space forms with constant curvature, therefore there will be some different and interesting results when we consider a submanifold immersed in a higher dimensional submanifold from the viewpoint of singularity, for instance, Ito and Izumiya investigate regular curves on a spacelike surfaces in Lorentz-Minkowski 3-space, they introduce five special curves that are in Lorentzian subspace forms with constant curvature along the curve associated to the Lorentzian Darboux frame and obtain some good results. 6 However, to the best of the authors' knowledge, we can not find any literature on the study for regarding curves lying in spacelike surfaces as the original objects and considering the singularities of surfaces generated by these curves.…”