Lightlike hypersurfaces and lightlike focal sets with respect to Bishop frame in 4-dimensional Minkowski Space E1-4.

Zehra Ozdemir

doi:10.31801/cfsuasmas.456764

Abstract

In this article, light-like hypersurfaces that is derived by null Cartan curves will be examined and discussed. The singularities of lightlike hypersurfaces and light-like focal sets are investigated by using the Bishop frame on the Null Cartan curves. We obtain that the types of these singularities and the order of contact between the null Cartan curves are closely related to the Bishop curvatures of the null Cartan curves. Moreover, two examples of light-like hypersurfaces and light-like focal sets are given to illustrate our theoretical results.

Highlights

In 4-dimensional Minkowski space, due to the causal character there are three categories of vectors, namely, space-like, time-like and light-like ones
The induced metric g is degenerate on M and it has a constant rank n. These hypersurfaces are usually used in modeling objects that are di¢ cult to understand
Light-like hypersurfaces are of interest to physicists because Kerr black holes, and various horizons can be modeled with these hypersurfaces [2, 4, 10, 11, 13, 17, 18, 20, 24]

Summary

Introduction

In 4-dimensional Minkowski space, due to the causal character there are three categories of vectors, namely, space-like, time-like and light-like (null) ones. The geometry of light-like hypersurfaces becomes more di¢ cult and is completely di¤erent from that of the space-like and time-like hypersurfaces. The induced metric g is degenerate on M and it has a constant rank n These hypersurfaces are usually used in modeling objects that are di¢ cult to understand. Nersessian and Ramos have shown that there is a geometric particle-model based on the geometry of null curves in Minkowski 4-space [14]. The singularities of the hypersurfaces are de...ned by using the Bishop frame of the null Cartan curve in Minkowski 4 space. We visualized light-like hypersurfaces and light-like focal set to demonstrate our theoretical results

Preliminaries

Lightlike Hypersurfaces and Singularities

Applications be a following parameterized curve in E41 p p s

), References