Singularities Of Timelike And Lightlike Surface Along Framed Base Curves In De Sitter Space And Duality

In this paper,we mainly consider the problem of type of singularity of the lightlike curve,timelike surface and lightlike surface generated from Sabban frame curve in de Sitter 3-space.According to the characteristic that the frame curve may have singularities,we establish two kinds of Sabban frames of regular curves and singular curves respectively.And we define three new geometric invariants.Using the approach of the unfolding theory in singularity theory,the singularities of curves and surfaces mentioned above are classified in detail.We establish the relationships between the curve and surfaces and the corresponding three geometric invariants to characterize the type of singularities of the lightlike curve,timelike surface and lightlike surface via these three invariants,respectively.The results show that under appropriate conditions,lightlike curve,timelike surface and lightlike surface are locally diffeomorphic to cuspidal edge or swallowtail.In addition,according to the Legendrian dualities between pseudospheres,we also describe the tangent curve t of framed base curve γ and lightlike curve Lγ are Δ5-dual each other.And the framed base curve γ and lightlike surface LSγ are Δ3-dual each other.Finally,three concrete examples are presented to interpret our theoretical results.