Gaussian surfaces and nullcone dual surfaces of null curves in a three-dimensional nullcone with index 2

Singularities of Indefinite Hypersurfaces and Indefinite Surfaces along Timelike Curves in Nullcone with Index Two

“…There exist spacelike curves, timelike curves, and null curves in * 0 ; among them, the null curves had been investigated by the second author [15]; in this paper, we will restrict our attention to the case of timelike curve in * 0 ; the case of space curve would be discussed in future work. Let …”

“…L. Kong and D. Pei [8] investigated the singularities of lightlike surfaces and focal surfaces of spacelike curves in hyperbolic spacetime sphere, and some geometry properties of the spacelike curves are obtained. The second author and his collaborators presented abundant research results concerning the singularities of submanifolds in semiEuclidean space [12][13][14][15]. In [13], they investigated the null developables of timelike curves that lie on nullcone in 3-dimensional semi-Euclidean space with index 2 and classified the singularities of the null developables of timelike curves.…”

Lightcone dual surfaces and hyperbolic dual surfaces of spacelike curves in de Sitter 3-space

“…We introduce the Legendrian dualities between pseudo-spheres in Minkowski space-time which has been proved to be a basic tool for the study of surfaces in pseudo-spheres in Minkowski space [3,4,5,6,12]. We define one-forms dv,…”

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 .…”