Some new properties of null curves on 3-null cone and unit semi-Euclidean 3-spheres

Abstract

The null curves on 3-null cone have the applications in the studying of horizon types. Via the pseudo-scalar product and Frenet equations, the differential geometry of null curves on 3-null cone is obtained. In the local sense, the curvature describes the contact of submanifolds with pseudo-spheres. We introduce the geometric properties of the null curves on 3-null cone and unit semi-Euclidean 3-spheres, respectively. On the other hand, we give the existence conditions of null Bertrand curves on 3-null cone and unit semi-Euclidean 3-spheres. c ⃝2015 All rights reserved.

On lightlike hypersurfaces and lightlike focal sets of null Cartan curves in Lorentz-Minkowski spacetime

“…Because many difficulties arise in generalizing the use of a singularity theory approach from nonlightlike submanifolds to lightlike submanifolds, the study of the singularity of submanifolds in Minkowski space remained at the nonlightlike submanifolds over a long period of time until we extended it to the lightlike submanifolds in 2010 and provided meaningful results [23][24][25]. Pei et al also described the properties of the local differential geometry of the null curve and investigated the singularity of the null surface of the null curve on the 3-null cone [19,20]. However, to the best of the authors knowledge, we are not aware of any literature report addressing the singularities of lightlike hypersurfaces involving null curves in R 4 1 .…”

Investigation of a curve using Frenet frame in the lightlike cone

“…For example, in [6], Liu considered curves in the null cone and in [7], Liu and Mong gave some formulas of curves in Q 2 and Q 3 . Furthermore, Sun and Pei studied the null curves on Q 3 and unit semi-Euclidean 3-spheres in [8].…”

Singularity analysis of pseudo null hypersurfaces and pseudo hyperbolic hypersurfaces

“…On the other hand, singularity theory, which is a direct descendant of differential calculus, is certain to have a great deal of interest to say about geometry, equation, physic, astronomy and other disciplines [3,5,8,9,10]. In general, the current theory always does not allow for singularities.…”