Singularities Of Special Singular Submanifolds And Slant Focal Surfaces Along Lightlike Locus On The Mixed Surfaces

The main work has two parts.One is to investigate the differential geometric properties of singular points of special singular submanifolds by Legendrian dualities in non-flat 3-space.The other is that the differential geometric properties of lightlike locus can be revealed by study of singularities of slant focal surfaces along lightlike locus in Lorentz 3-space.The Legendrian dualities is very useful for the study of the differential geometry on submanifolds in pseudo-spheres of semi-Euclidean space.The objects investigated by Legendrian dualities are framed curves in non-flat Riemannian 3-space and cuspidal edge in 3-sphere.Framed curve is a curve with a moving frame.And It may have singularities.Since the classical "Frenet-Serret type moving frames" cannot be established at singular points,then we can investigate the singular points by moving frame of frame curve.Wave front is a important class of singular surfaces.There exists a smooth unit normal field of wave front even at a singular point.This provides a good idea about the study of singular points on wave front.One of the simplest and generic wave fronts is a cuspidal edge.Set of singular points in the neighbourhood of cuspidal edge may be a regular curve.Then we can investigate properties of cuspidal edge.On the other hand,mixed surface is a generic class of surfaces in Lorentz 3-space.Lightlike point on mixed surfaces can be a singular point of the induced metric.So the lightlike locus is a special curve on the mixed surface.Its differential properties is fascinating for us to study.There are main results in this thesis as follows:1.We give the framed curve in Riemannian 3-space.Then the focal surfaces and evolutes of framed curves in non-flat Riemannian space form can be given from the viewpoint of Legendrian duality.To investigate the local geometric properties of singular points of framed curves in the Riemannian 3-space form,we study singularities of focal surfaces and evolutes and give the relationship between focal surfaces and evolutes from viewpoint of singular theory.Furthermore,the dual surfaces of evolutes of framed curves in non-flat 3space form are given.We study singularities of dual surfaces of evolutes and consider the properties of the dualities of singularities.2.Using the Legendrian dualities,we study three kinds of extrinsic flat surfaces with respect to the singular set of a cuspidal edge in the three-sphere.Moreover,geometrical properties of the cuspidal edge can be revealed by study of singularities of these flat surfaces.Then we also study properties of dualities of singularities.By the dual surfaces,we study the global geometry of singular set of cuspidal edge.3.We define slant focal surfaces and slant evolutes along lightlike locus.Moreover,we obtain that singularities of slant focal surfaces and slant evolutes depend on the differential geometric properties of the lightlike locus.Furthermore,we give the relationship between slat focal surfaces and slat evolutes from viewpoint of singular theory.Finally,we also consider the relationship between slant evolutes and the lightlike locus on the lightcone.