The Singularity Of Special Surfaces In Space Forms

Parallel surface and ruled surface are special surfaces in space forms.And they always have singularities like cuspidal edges and swallowtails.In the first part of this paper,we study parallel surfaces.Namely,we introduce parallel surfaces in 3-dimensional non-flat space forms(i.e.~3or~3)when the initial cuspidal edges for fronts.We define the principal curvature and principal direction of cuspidal edge.And give ridge point for cuspidal edge by using those.Then we show relations between geometric properties of initial cuspidal edge and singularities on parallel surface.And initial cuspidal edge is always cuspidal edge in parallel surface if and only if the initial cuspidal edge is not a ridge point;initial cuspidal edge is swallowtail in parallel surface if and only if the initial cuspidal edge is the first ridge point.In the second part of this paper,we study ruled surface when the initial surface is mixed type surface.In Minkowski 3-space,a surface is always a mixed type surface.The first kind of lightlike points in mixed type surface is similar to cuspidal edges in front.So we define ruled surfaces along the first kind of lightlike points in Minkowski 3-space.We study singularities of the ruled surfaces,and classify those singularities.Finally,we give the conditions for the ruled surfaces have cuspidal edges and swallowtails.