Pseudo‐spherical normal Darboux images of curves on a lightlike surface

Abstract

We introduce the notions of the pseudospherical normal Darboux images for the curve on a lightlike surface in Minkowski 3-space and study these Darboux images by using technics of the singularity theory. Furthermore, we give a relation between these Darboux images and Darboux frame from the viewpoint of Legendrian dualities.

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

“…When we investigate a space curve lying in a surface, it is more convenient for us to use the Lorentzian Darboux frame along the curve as the basic tool than the Frenet‐Serret type frame in Lorentzian space. There are several papers about Lorentzian Darboux frame …”

“…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 .…”

“…There are several papers about Lorentzian Darboux frame. [6][7][8] 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.…”

“…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.…”

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

“…Lorentz-Minkowski space, as the mathematical setting of Einstein's theory of special relativity, is closely related to physics and provide the supports of theory and methodology for the study of astrophysics and cosmology by considering various of geometric invariants under Lorentzian transformations. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] It is well-known that there exist spacelike curves, timelike curves, and null curves in Lorentz-Minkowski space, which makes the study for them have more abundant contents and more diverse than curves in Euclidean space. The study of problems on curves lying in Lorentz-Minkowski space has received much attention from scholars in the fields of geometry and mathematical physics.…”