Focal surfaces of lightcone framed surfaces in the Lorentz-Minkowski 3-space

TL;DR

This paper explores the differential geometry of focal surfaces derived from lightcone framed surfaces in Lorentz-Minkowski 3-space, introducing new concepts and analyzing their properties, especially for mixed type surfaces with singularities.

Contribution

It introduces the concept of lightcone framed surfaces and studies their focal surfaces' differential geometry, extending understanding of mixed type surfaces with singular points in Lorentz-Minkowski space.

Findings

Defined the lightcone frame for such surfaces

Analyzed the differential geometry of focal surfaces

Provided properties of focal surfaces in Lorentz-Minkowski space

Abstract

In this paper, we consider the differential geometry properties of focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space. In general, a mixed type surface is a connected regular surface with non-empty spacelike and timelike point sets. In order to investigate the mixed type surface with singular points, we introduce the lightcone framed surface. First, we give the lightcone frame, by which we define the lightcone framed surface. Next, we consider the differential geometry properties of lightcone framed surfaces by using the lightcone frame. At last, we give the definition of focal surfaces of lightcone framed surfaces and investigate the differential geometry properties of focal surfaces.