Focal set of curves in the Minkowski space near lightlike points

TL;DR

This paper investigates the geometric properties of curves in Minkowski and de Sitter spaces at lightlike points, focusing on the structure of their focal sets using singularity theory, especially near points where the tangent is lightlike.

Contribution

It introduces a detailed analysis of focal sets of curves at lightlike points in Minkowski space using singularity theory, addressing the undefined nature of focal sets at these points.

Findings

Focal sets are not defined at lightlike points.

Singularity theory techniques reveal the local structure of focal sets near lightlike points.

The study enhances understanding of curve geometry in Lorentzian manifolds.

Abstract

We study the geometry of curves in the Minkowski space and in the de Sitter space, specially at points where the tangent direction is lightlike (i.e. has length zero) called lightlike points of the curve. We define the focal sets of these curves and study the metric structure of them. At the lightlike points, the focal set is not defined. We use singularity theory techniques to carry out our study and investigate the focal set near lightlike points.