Pseudo‐spherical Darboux images and lightcone images of principal‐directional curves of nonlightlike curves in Minkowski 3‐space

Abstract

Choosing an alternative frame, which is the Frenet frame of the principal-directional curve along a nonlightlike Frenet curve , we define de Sitter Darboux images, hyperbolic Darboux images, and lightcone images generated by the principal directional curves of nonlightlike Frenet curves and investigate geometric properties of these associated curves under considerations of singularity theory, contact, and Legendrian duality. It is shown that pseudo-spherical Darboux images and lightcone images can occur singularities (ordinary cusp) characterized by some important invariants. More interestingly, the cusp is closely related to the contact between nonlightlike Frenet curve and a slant helix, the principal-directional curve of and a helix or the principal-directional curve and a slant helix. In addition, some relations of Legendrian dualities between C-curves and pseudo-spherical Darboux images or lightcone images are shown. Some concrete examples are provided to illustrate our results. KEYWORDS contact, legendrian duality, singularity, slant helix Math Meth Appl Sci. 2020;43:35-77.wileyonlinelibrary.com/journal/mma

On Generalized Darboux Frame of a Pseudo Null Curve Lying on a Lightlike Surface in Minkowski 3-space

“…Pseudo null curves lying on non-degenerate or degenerate surfaces in Minkowski 3space, defined in terms of the Darboux frame's vector fields, are introduced as k-type pseudo null slant helices and isophote curves in [10,8]. Darboux frame's vector fields are also used in characterizations of different types of curves, such as pseudo-spherical Darboux images and lightcone images of principal-directional curves ( [16]) and in investigations of generalized focal surfaces ( [6]) and lightlike surfaces along pseudo-spherical normal Darboux images of spacelike curves ( [14]).…”

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.…”