Lightlike Surfaces Of Darboux-like Indicatrixes And Binormal Indicatrixes Of Spacelike Curves In Lightcone 3-space

In this paper,by introducing a new frame on spacelike curves lying in lightcone 3-space,we investigate the geometric properties of the lightlike surface of the Darboux-like indicatrix and the lightlike surface of binormal indicatrix generated by spacelike curves in lightcone 3-space.As an application of singularity theory,the singularities of the lightlike surfaces of Darboux-like indicatrix and the lightlike surface of binormal indicatrix are classified,several new invariants of spacelike curves are discovered to be useful for characterizing these singularities,meanwhile,it is found that the new invariants also measure the order of contact between spacelike curves or principal normal indicatrixes of spacelike curves located in lightcone 3-space and two-dimensional lightcone whose vertices are at the singularities of lightlike surfaces.One concrete example is provided to illustrate our results.