Lightcone framed surfaces in the Lorentz-Minkowski 3-space

Abstract

For the study of mixed type surfaces with singular points, we define a lightcone frame of a surface in the Lorentz-Minkowski 3-space. We give the existence and uniqueness for the basic invariants of the lightcone framed surface. By using the lightcone frame, we investigate differential geometric properties of the mixed type surface. Then we show a method to recognize the type of points on the surface. Moreover, we give the necessary and sufficient conditions for a surface become a lightcone framed surface. We also discuss a relation between lightcone framed surfaces and generalised framed surfaces. As a special case of a lightcone frame, we introduce the lightcone circle frame and call surfaces with such frames lightcone circle framed surfaces. In addition, the conditions for a lightcone framed surface to become a lightcone circle framed surface are also given.